Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux
SStandard monomial theory and toric degenerations ofSchubert varieties from matching field tableaux
Oliver Clarke and Fatemeh Mohammadi
Abstract
We study Gr¨obner degenerations of Schubert varieties inside flag varieties. We consider toricdegenerations of flag varieties induced by matching fields and semi-standard Young tableaux.We describe an analogue of matching field ideals for Schubert varieties inside the flag varietyand give a complete characterization of toric ideals among them. We use a combinatorialapproach to standard monomial theory to show that block diagonal matching fields giverise to toric degenerations. Our methods and results use the combinatorics of permutationsassociated to Schubert varieties, matching fields and their corresponding tableaux.
Keywords:
Toric degenerations, flag varieties, Schubert varieties, semi-standard tableaux
Contents1 Introduction 22 Preliminaries 4 n . . . . . . . . . . . . . . . . . . 8 a r X i v : . [ m a t h . A C ] S e p . Introduction In this note we provide a new family of toric degenerations of Schubert varieties insidethe full flag variety. Computing toric degenerations of a variety is a valuable tool thatallows us to study general spaces using results from toric geometry and combinatorics, forinstance, see [And13, CLS11]. A toric degeneration of a given variety X is a 1-parameterfamily over the affine line F → A such that the fiber over 0, often called the special fiber,is a toric variety and all other fibers are isomorphic to X . Most algebraic invariants oftoric varieties have combinatorial counterparts such as polyhedral fans and polytopes. Thismakes the study of toric varieties particularly fruitful and motivates the search for toricdegenerations of varieties. More precisely, a toric degeneration is a flat family and so wecan calculate invariants of the original variety by calculating them for the toric fiber. Thisconverts various abstract problems in algebraic geometry into questions about polytopes.For example, calculating the degree of a variety given a toric degeneration can be achievedby computing the volume of the moment polytope of the toric fiber.Toric degenerations have been studied extensively in the literature for flag varieties andtheir Schubert varieties, see e.g. [FFL17, GL96, SSBW19]. Closely related are toric degener-ations for Grassmannians, which have been widely studied, see e.g. [RW19, BFF +
18, CM20].For all of these varieties, one of the most well-known examples of toric degeneration is theGelfand-Tsetlin degeneration which is readily understood through standard monomial theoryand semi-standard Young tableaux [ACK18, KM05]. Natural questions to ask are; what arethe other possible toric degenerations of these varieties? And how are they related to eachother? For instance, it has been shown that plabic graphs, arising from the cluster algebrastructure of the Grassmannian, parametrize certain toric degenerations, see [BFF +
18, RW19].One approach to study toric degenerations of varieties is by way of Gr¨obner degeneration.Given a variety X , any weight vector w gives rise to a one-parameter family for X wherethe ideal of the special fiber is the initial ideal in w ( I ( X )) . Therefore, we search for weightvectors w such that the initial ideal in w ( I ( X )) is toric, i.e. a prime binomial ideal. Thetropicalization trop ( X ) , see [MS15], is the collection of weight vectors for which the initialideal in w ( I ( X )) does not contain any monomials and has the structure of a polyhedral fan. Sonatural candidates for weight vectors giving rise to toric degenerations are interior points oftop-dimensional cones of the tropicalization, see e.g. [KM19, MS19, BLMM17]. In the caseof Gr ( , n ) , it was shown in [SS04] that every such point gives rise to a toric degeneration ofGr ( , n ) . A combinatorial approach to finding such points in the tropicalization of Gr ( , n ) is taken in [MS19] in which the authors, following the work of [SZ93, FR15], study the so-called coherent matching fields . More precisely, the authors classify which matching fieldsgive rise to toric degenerations of Gr ( , ) and provide a family of matching fields called blockdiagonal matching fields that exhibit, up to isomorphism, all but one of the possible Gr¨obnerdegenerations of Gr ( , ) . In [CM19], it is shown that the weight vectors arising from blockdiagonal matching fields give rise to toric degenerations of the flag variety. Furthermore, by[Stu96], whenever a toric degeneration is obtained via a matching field, the Pl¨ucker variablesform a SAGBI basis for the corresponding Pl¨ucker algebra.2e consider a family of toric degenerations of Schubert varieties which are parametrizedby matching fields, in the sense of Sturmfels-Zelevinsky [SZ93]. It is shown in [CM19] thatall so-called block diagonal matching fields give rise to toric degenerations of the full flagvariety. The associated toric ideals can be directly read from the matching field and so arecalled matching field ideals. In this note, we extend the results of [CM19] by consideringhow these toric degenerations restrict to certain subvarieties of the flag variety, namely itsSchubert varieties. For each Schubert variety, indexed by some permutation w ∈ S n anda block diagonal matching field B (cid:96) , we define the restricted matching field ideal by settingsome variables of the matching field ideal to zero. Our main results are Theorems A, B andC. Theorems B and C give combinatorial conditions on w and B (cid:96) such that the restrictedmatching field ideal is monomial-free. Theorem A shows that a matching field B (cid:96) gives riseto a toric degeneration of the corresponding Schubert variety subject to the condition thatthe initial ideal is generated in degree two, which we show for some particular matchingfields. Our methods use combinatorial properties of the permutations, which parametrizethe Schubert varieties, and properties of the generating sets of matching field ideals. More-over, we use semi-standard Young tableaux to construct monomial bases for each restrictedmatching field ideals. As a result, we obtain new families of monomial bases for the fullflag variety that are compatible with its Schubert varieties. Moreover, we obtain minimalgenerating sets of the ideals arising from Gr¨obner degenerations of Schubert varieties. Structure of the paper. In § § § § § § § § § § § § § § Acknowledgement.
We thank Narasimha Chary and J¨urgen Herzog for many helpfulconversations. We are grateful to the anonymous referees for very helpful comments onearlier versions of this paper. FM was partially supported by a BOF Starting Grant ofGhent University and EPSRC Early Career Fellowship EP/R023379/1. OC is supported byEPSRC Doctoral Training Partnership (DTP) award EP/N509619/1.3 . Preliminaries
Throughout we fix a field K with char ( K ) =
0. We are mainly interested in the case when K = C . We let [ n ] be the set { , . . . , n } and by S n we denote the symmetric group on [ n ] .A permutation w ∈ S n , unless stated otherwise, is written w = ( w , . . . , w n ) where w i = w ( i ) for each 1 ≤ i ≤ n , which is often called single line notation. It will be convenient for us tohave the elements of a set be in increasing order so we write J = { j < · · · < j s } for the setwith elements j , . . . , j s in increasing order. However, unless otherwise stated, sets are notordered. A full flag is a sequence of vector subspaces of K n : { } = V ⊂ V ⊂ · · · ⊂ V n − ⊂ V n = K n where dim K ( V i ) = i . The set of all full flags is called the flag variety denoted by Fl n , which isnaturally embedded in a product of Grassmannians using the Pl¨ucker variables. Each pointin the flag variety can be represented by an n × n matrix X = ( x i , j ) whose first k rows span V k . Each V k corresponds to a point in the Grassmannian Gr ( k , n ) . The ideal of Fl n , denotedby I n is the kernel of the polynomial map ϕ n : K [ P J : (cid:156) (cid:44) J (cid:40) { , . . . , n }] → K [ x i , j : 1 ≤ i ≤ n − , ≤ j ≤ n ] sending each Pl¨ucker variable P J to the determinant of the submatrix of X with row indices1 , . . . , | J | and column indices in J . We refer to [MS05, § Remark . By abuse of notation we use P J to denote both the variable in the ring K [ P J ] and also for the image of P J under the map ϕ n . Later we will introduce the notion of weightson variables in both rings K [ P J ] and K [ x i , j ] . However, the weight on P J will be induced bythe weights on x i , j so this abuse of notation will not cause problems for weights. Let SL ( n , C ) be the set of n × n matrices with determinant 1, and let B be its subgroupconsisting of upper triangular matrices. There is a natural transitive action of SL ( n , C ) onthe flag variety Fl n which identifies Fl n with the set of left cosets SL ( n , C )/ B , since B is thestabilizer of the standard flag 0 ⊂ (cid:104) e (cid:105) ⊂ · · · ⊂ (cid:104) e , . . . , e n (cid:105) = C n . Given a permutation w ∈ S n , we denote by σ w the n × n permutation matrix with 1’s in the positions ( w ( i ) , i ) for all i . By the Bruhat decomposition, we can write the aforementioned set of cosets asSL ( n , C )/ B = (cid:221) w ∈ S n B σ w B / B . Given a permutation w , its Schubert variety is X ( w ) = B σ w B / B ⊆ Fl n which is the Zariski closure of the corresponding cell in the Bruhat decomposition. The idealof the Schubert variety X ( w ) is obtained from I n by setting P J to zero for each J ∈ S w where S w = { J : J ⊂ [ n ] with J (cid:2) { w , w , . . . , w | J | }} . Where { a < · · · < a m } ≤ { b < · · · < b m } means that a i ≤ b i for each 1 ≤ i ≤ m .4 xample 2.2. Suppose n = w = ( , , , ) ∈ S n is a permutation written in single linenotation. To calculate S w we take each subset of [ n ] , for example { , } ⊆ [ ] , and compareit to { w , w } . In this case { , } ≤ { , } = { w , w } and so { , } (cid:60) S w . Continuing thisprocess for all other subsets we obtain S w = { , , , , , , } . The ideal of X ( w ) is obtained from I n by setting P J to zero for each J ∈ S w : I ( X ( w )) = (cid:104) P P − P P + P P (cid:105) ⊂ K [ P J ] . Definition 2.3. A matching field is a map Λ n : { J : (cid:156) (cid:44) J (cid:40) [ n ]} → S n . For ease of notationwe write Λ for Λ n if there is no ambiguity. Suppose J = { j < · · · < j k } ⊂ [ n ] , we think ofthe permutation σ = Λ ( J ) as inducing an ordering on the elements of J , where the positionof j s is σ ( s ) .Given a matching field Λ and a k -subset J = { j , . . . , j k } ⊂ [ n ] with j < · · · < j k , let σ = Λ ( J ) . We represent the Pl¨ucker form P J as a k × ( σ ( (cid:96) ) , ) is j (cid:96) for each 1 ≤ (cid:96) ≤ k . Let X = ( x i , j ) be an n × n matrix of indeterminates. Toeach subset J ⊂ [ n ] as before, we associate the monomial x Λ ( J ) : = x σ ( ) j x σ ( ) j · · · x σ ( k ) j k . A matching field ideal J Λ is defined as the kernel of the monomial map φ Λ : K [ P J ] → K [ x i j ] with P J (cid:55)→ sgn ( Λ ( J )) x Λ ( J ) , (2.1)where sgn denotes the sign of the permutation Λ ( J ) . We define the algebra associated to Λ to be K [ P J ]/ ker ( φ Λ ) . Definition 2.4.
A matching field Λ is coherent if there exists an n × n matrix M with entriesin R such that for every proper non-empty subset (cid:156) (cid:44) J (cid:40) [ n ] the initial form of the Pl¨uckerform P J ∈ K [ x i j ] , in M ( P J ) is sgn ( Λ ( J )) x Λ ( J ) . Where in M ( P J ) is the sum of all terms in P J with the lowest weight with respect to M . In this case, we say that the matrix M inducesthe matching field Λ . Example 2.5.
Let us see an example of a non-coherent matching field. Suppose that n ≥ Λ such that Λ ({ , , }) = id and Λ ({ , , }) = ( , ) is thetransposition which swaps 1 and 2. Suppose by contradiction that Λ is a coherent matchingfield, then there exists an n × n matrix M which induces Λ . Let us consider the submatrix M (cid:48) of M which consists of the first two rows and first two columns. M (cid:48) = (cid:20) m , m , m , m , (cid:21) . Since Λ ({ , , }) = id , this implies that m , + m , < m , + m , . However Λ ({ , , }) = ( , ) implies that m , + m , < m , + m , , a contradiction.5 efinition 2.6. Let Λ be a coherent matching field induced by the matrix M . We define w M to be the weight vector induced by M on the Pl¨ucker variables. That is, the entry of the vector w M corresponding to the variable P J ∈ K [ P J ] is the minimum weight of monomials appearingin ϕ n ( P J ) with respect to M . The weight of a monomial is the sum of the correspondingterms in the weight matrix M . For ease of notation we write P α for the monomial P α J . . . P α s J s where α = ( α , . . . , α s ) . And so the weight of P α is simply α · w M . Definition 2.7.
Let Λ be a coherent matching field induced by M . We denote the initialideal of I n with respect to w M by in w M ( I n ) . The ideal in w M ( I n ) is generated by polynomialsin w M ( f ) for all f ∈ I n , wherein w M ( f ) = (cid:213) α j · w M = d c α j P α j for f = t (cid:213) i = c α i P α i and d = min { α i · w M : i = , . . . , t } . Example 2.8.
Consider the matching field Λ induced by the matrix M = . The single column tableaux arising from the matching field are:1 2 3 4 12 31 41 32 42 34 123 124 314 324 . So the Pl¨ucker variables are P , P , P , P , P , P , P , P , P , P , P , P , P , P and the corresponding weight vector is w M = ( , , , , , , , , , , , , , ) . Performing thecalculation in Macaulay2 [GS], we obtain the following generating set for in w M ( I ) :in w M ( I ) = (cid:104) P P − P P , P P − P P , P P − P P , P P − P P , P P − P P , P P + P P , P P − P P , P P − P P , P P + P P , P P + P P (cid:105) . Example 2.9.
Let Λ be the matching field induced by the matrix M = . So the Pl¨ucker variables are P , P , P , P , P , P , P , P , P , P , P , P , P , P andthe corresponding weight vector is w M = ( , , , , , , , , , , , , , ) . Performing the cal-culation in Macaulay2 [GS], we obtain the following generating set for in w M ( I ) :in w M ( I ) = (cid:104) P P − P P , P P − P P , P P − P P , P P − P P , P P − P P , P P − P P , P P − P P , P P − P P , P P − P P , P P − P P , (cid:105) . Note that the entries in each tableau are strictly increasing. We call such matching fields diagonal and we denote them by D n , or D when there is no confusion. Their correspondingdegenerations are called Gelfand-Tsetlin degenerations in [KM05]. Definition 2.10.
Given n and 0 ≤ (cid:96) ≤ n , we define the block diagonal matching field denotedby B (cid:96) = ( · · · (cid:96) | (cid:96) + · · · n ) as a map from the power set of [ n ] = { , . . . , n } to S n such that B (cid:96) ( J ) = (cid:26) id : if | J | = | J ∩ { , . . . , (cid:96) }| ≥ ( ) : otherwiseThe matching field B (cid:96) is induced by the weight matrix: M (cid:96) = · · · · · · (cid:96) (cid:96) − · · · n n − · · · (cid:96) + n ( n − ) · · · ( n − (cid:96) + ) ( n − (cid:96) ) ( n − (cid:96) − ) · · · ... ... . . . ... ... ... . . . ... ( n − ) n ( n − ) · · · ( n − )( n − (cid:96) + ) ( n − )( n − (cid:96) ) ( n − )( n − (cid:96) − ) · · · n − . Therefore, all block diagonal matching fields are coherent. We denote w (cid:96) for the weightvector induced by M (cid:96) on the Pl¨ucker variables. The case (cid:96) = n corresponds to thediagonal matching field. The weight vector w (cid:96) is explicitly given as follows. For each J = { j < · · · < j s } ⊂ [ n ] the component of w (cid:96) corresponding to P J is given by w (cid:96) ( P J ) =
0, if s = ( n + (cid:96) + − j ) + (cid:205) sk = ( k − )( n + − j k ) , if s ≥ | J ∩ { , . . . , (cid:96) }| = ( (cid:96) + − j ) + (cid:205) sk = ( k − )( n + − j k ) , if s ≥ | J ∩ { , . . . , (cid:96) }| = ( (cid:96) + − j ) + (cid:205) sk = ( k − )( n + − j k ) , if | J ∩ { , . . . , (cid:96) }| ≥ F n ,(cid:96) for the matching ideal of B (cid:96) which is the kernelof the monomial map φ (cid:96) : K [ P J ] → K [ x i j ] with P J (cid:55)→ sgn ( B (cid:96) ( J )) in w (cid:96) ( P J ) . (2.2)The following result highlights the motivation and the importance of the understudied familyof degenerations induced by matching fields. Corollary 2.11 ([CM19, Corollary 4.13 and Theorem 3.3]) . Each block diagonal matchingfield produces a toric degeneration of Fl n . Equivalently, in w (cid:96) ( I n ) is toric for all n and ≤ (cid:96) ≤ n , and it equals to F n ,(cid:96) . Moreover, the ideal F n ,(cid:96) is generated by quadratic binomials. .4. Initial ideals of Schubert varieties inside Fl n Here we introduce the family of ideals F n ,(cid:96), w that are closely related to the initial idealsin w (cid:96) ( I ( X ( w ))) where I ( X ( w )) is the ideal of the corresponding Schubert variety. In general,the initial ideals of Schubert varieties are difficult to calculate, see Remark 3.5. Howeverthe ideals F n ,(cid:96), w , which arise from matching fields, have a canonical generating set which weexploit in order to generalize Corollary 2.11 to Schubert varieties. Definition 2.12 (Restricted matching field ideals) . Given a block diagonal matching field B (cid:96) and a permutation w in S n , we define the ideal F n ,(cid:96), w = ( F n ,(cid:96) + (cid:104) P J : J ∈ S w (cid:105)) ∩ K [ P J : J ⊆ [ n ] , J (cid:60) S w ] , (2.3)which can be computed in Macaulay2 [GS] as an elimination ideal as follows F n ,(cid:96), w = eliminate ( in w (cid:96) ( I n ) + (cid:104) P J : J ∈ S w (cid:105) , { P J : J ∈ S w }) . We may think of F n ,(cid:96), w as the ideal obtained from F n ,(cid:96) = in w (cid:96) ( I n ) by setting the variables { P J : J ∈ S w } to be zero. And so we say that the variable P J vanishes in F n ,(cid:96), w if J ∈ S w . If P J does not vanish we write P J (cid:44)
0. More generally, we say that a polynomial g ∈ K [ P I ] vanishes in F n ,(cid:96), w if g ∈ (cid:104) P I : I ∈ S w (cid:105) ⊆ K [ P I ] . We will often use the language of vanishingpolynomials when determining which terms of generators in F n ,(cid:96) vanish in F n ,(cid:96), w . Example 2.13.
Let us continue Example 2.2 where n = w = ( , , , ) . Consider thematching field from Example 2.8 which is the block diagonal matching field B . We beginby calculating the initial ideal in w ( I ) which is (cid:104) P P − P P , P P − P P , P P − P P , P P − P P , P P + P P , P P − P P , P P + P P , P P − P P , P P + P P , P P − P P (cid:105) . So we can now calculate the ideal F , , w which is F , , w = (cid:104) P P − P P (cid:105) . Note that this isthe same as finding in w (cid:96) ( I ( X ( w ))) where I ( X ( w )) is the ideal of the Schubert variety whichwe found in Example 2.2. Also note that the resulting ideal F n ,(cid:96), w is binomial. This alsofollows from Theorem C and in particular the other matching fields B (cid:96) and permutations w which give rise to binomial ideas, where n =
4, can be found in Table 1.One of our main results is Theorem A, which shows that if a restricted matching fieldideal is monomial-free then it coincides with initial ideals of the corresponding Schubertvariety. To prove this result, we show that semi-standard Young tableaux are in bijectionwith a set of standard monomials for F n ,(cid:96), w . Definition 2.14 (semi-standard Young tableaux) . A tableau T = [ I I . . . I k ] is an orderedcollection of columns where each column is an ordered subset I j ⊆ [ n ] for each j ∈ [ k ] . Ifthe order of the entries in each column coincides with the order induced by a fixed matchingfield B , then we say the tableau is a matching field tableau for B . Write I j = { i , j , i , j , . . . i t j , j } for each j ∈ [ k ] . We say T is a semi-standard Young tableau if the following hold.8 The size of the columns weakly decreasing, i.e. if 1 ≤ i < j ≤ k then | I i | ≥ | I j | . • The entries in each column are increasing, i.e. I j = { i , j < i , j < · · · < i t j , j } for each j ∈ [ k ] . • The entries in each row are weakly increasing, i.e. i j , ≤ i j , ≤ . . . i j , m j for each j ∈{ , . . . , | I |} where m j = max { i ∈ [ k ] : | I i | ≥ j } . Example 2.15.
Let n = T = , T = . The tableau T is a semi-standard Young tableau. The monomial represented by T is theimage of P P under the diagonal matching field map: φ B ( P P ) = x , x , x , x , x , .The tableau T is not a semi-standard Young tableau. However the columns of T are orderedby the matching field B , see Example 2.8, and so T is called a matching field tableau. Thetableau T represents the image of P P under the block diagonal matching field map: φ B ( P P ) = x , x , x , x , x , .In order to characterize permutations w ∈ S n for which the ideals F n ,(cid:96), w are monomialfree, see Theorem 3.13, we require the following definitions about permutations. Definition 2.16 (Permutation avoidance) . We say that two finite sequences w = ( w , . . . , w s ) and v = ( v , . . . , v s ) have the same type if their respective entries satisfy all the same pairwisecomparisons, i.e. w i < w j if and only if v i < v j for all i , j ∈ [ s ] . We say that a permutation w = ( w , . . . , w n ) ∈ S n avoids another permutation v ∈ S m where m ≤ n if every subsequence ( w i , . . . , w i m ) of w has a different type to v . If w avoids v then we also say that w is v -free. Example 2.17.
The sequences ( , , , ) and ( , , , ) have the same type but neither hasthe same type as ( , , , ) . The permutation ( , , , , ) does not avoid ( , , , ) because thesubsequence ( , , , ) has the same type as ( , , , ) . However, the permutation ( , , , , ) does avoid ( , , ) . Definition 2.18.
Let w = ( w , . . . , w n ) ∈ S n be a permutation and m ≤ n be a naturalnumber. The restriction of w to [ m ] is the permutation w | m ∈ S m obtained from w byremoving the values m + , . . . , n . Example 2.19.
Let w = ( , , , ) then the restrictions of w are as follows. w | = ( , , , ) , w | = ( , , ) , w | = ( , ) , w | = ( ) . . Schubert varieties inside flag varieties This section aims to answer the following question on Schubert varieties; this is a refor-mulation of
Degeneration Problem posed by Caldero [Cal02] in our setting.
Question 3.1.
Characterize toric initial ideals of the Pl¨ucker ideals of Schubert varietiesinside flag varieties. In other words, determine the toric ideals of form in w (cid:96) ( I ( X ( w ))) . In § w (cid:96) ( I ( X ( w ))) and F n ,(cid:96), w by way ofstandard monomial theory and prove the following result. Theorem A.
Suppose that in w (cid:96) ( I ( X ( w ))) is generated in degree two. If F n ,(cid:96), w is monomial-free then F n ,(cid:96), w = in w (cid:96) ( I ( X ( w ))) . Moreover in w (cid:96) ( I ( X ( w )) is the kernel of a monomial map,hence it is a toric (prime binomial) ideal.As an immediate corollary of Theorem A and [Stu96, Theorem 11.4] we have that: Corollary 3.2.
The block diagonal matching fields give rise to a family of toric degenerationsof the Schubert varieties inside the full flag variety. Moreover, the Pl¨ucker variables P I forma finite Khovanskii basis for the corresponding Pl¨ucker algebras.Remark . In the forthcoming paper [CHM20], we study the polytopes arising from toricvarieties in Corollary 3.2. In particular, we show that such polytopes are related by sequencesof combinatorial mutations.Our computational results lead us to the following conjecture.
Conjecture 3.4.
The ideal in w (cid:96) ( I ( X ( w ))) is generated in degree two. In § (cid:96) = F n ,(cid:96), w is monomial-free. We havealso verified this conjecture for all ideals in w (cid:96) ( I ( X ( w ))) where n ∈ { , , } . If this conjectureholds then the conclusion of Theorem A holds for all block diagonal matching fields. Remark . We use
Macaulay2 to calculate the ideals F n ,(cid:96), w and check whether they aretoric, i.e. they are non-zero prime binomial ideals. The code is available on Github: https://github.com/ollieclarke8787/toric degenerations schubert flag We verify inclusions of the ideals F n ,(cid:96), w with the ideals in w (cid:96) ( I ( X ( w ))) where I ( X ( w )) is theideal obtained from I n by setting the variables { P J : J ∈ S w } to be zero. We perform allcalculations for Fl and Fl . We also include documentation which allows users to producesimilar code for different flag varieties. For Fl our computations did not terminate on astandard desktop computer. In all cases for which computations terminated, we see thatin w (cid:96) ( I ( X ( w ))) is generated in degree two, verifying Conjecture 3.4 in those cases.To answer Question 3.1, in light of Theorem A, we provide a complete characterizationof ideals of type F n ,(cid:96), w introduced in Definition 2.12 into the categories: zero or non-zeroand binomial or non-binomial. An ideal F n ,(cid:96), w is monomial-free, hence toric and equal toin w (cid:96) ( I ( X ( w ))) , if and only if F n ,(cid:96), w is either zero or binomial. In particular, Theorem B10etermines which ideals F n ,(cid:96), w are zero and Theorem C determines which ideals F n ,(cid:96), w arenon-zero and binomial. We illustrate our main results in Figure 1 by providing a pictorialsurvey. Notation . Before stating further results, we fix the following notation. • From this section and on, I and J will denote subsets of [ n ] that index variables P I and P J . This should not be confused with the Pl¨ucker ideal I n . If the ideal does appear,then it will be made clear. • Given a block diagonal matching field B = ( E | E ) on [ n − ] we denote by B thematching field ( E | E ∪ n ) on [ n ] . Similarly, given a block diagonal matching field B = ( E | E ) on [ n ] for n ≥ E (cid:44) (cid:156) , we denote by B the block diagonal matchingfield ( E | E \ n ) . In which case we say B is the restriction of B to [ n − ] . • Given a permutation w = ( w , . . . , w n − ) on [ n − ] and t ∈ { , , . . . , n − } , we de-note by w for the permutation ( w , . . . , w t , n , w t + , . . . , w n − ) on [ n ] . Similarly, givena permutation w = ( w , . . . , w n ) on [ n ] with w s = n , we denote by w the permuta-tion ( w , . . . , w s − , w s + , . . . , w n ) on [ n − ] . Note w = w | n − is a special example of arestriction. • If B is the diagonal matching field on [ n − ] , we can regard this either as the blockdiagonal matching field ((cid:156) | , . . . , n − ) or ( , . . . , n − | (cid:156)) . This gives B to be ((cid:156) | , . . . , n ) , i.e. the diagonal matching field on [ n ] , or ( , . . . , n − | n ) , a non-diagonal matching field. Where necessary we distinguish between these, otherwise ifleft unstated all results apply to both cases.Here, we state our main results on Schubert varieties. Theorem B (Theorem 4.1) . For each (cid:96) , F n ,(cid:96), w = w ∈ Z n , where Z n = { s i . . . s i p ∈ S n : | i k − i (cid:96) | ≥ , for all k , (cid:96) } . Here, s i = ( i , i + ) ∈ S n is the transposition interchanging i and i + Definition 3.6.
For each block diagonal matching field B (cid:96) , we let T n ,(cid:96) = { w ∈ S n : F n ,(cid:96), w is binomial } and Z n = { w ∈ S n : F n , n , w = } , along with N n ,(cid:96) = S n \( T n ,(cid:96) ∪ Z n ) for the set of permutations for which F n ,(cid:96), w is non-binomial.Note that B n is the diagonal matching field denoted by D . Definition 3.7.
We say that a permutation w ∈ S n has the descending property if for w t = n we have that n = w t > w t + > · · · > w n . We denote S > n for the set of permutations in S n withdescending property. Definition 3.8.
For each block diagonal matching field B (cid:96) , we let11 Z n = { w ∈ S n : w ∈ Z n − } , • T n ,(cid:96) = { w ∈ S n : w ∈ T n − ,(cid:96) } , • A = Z n ∩ { w ∈ S n : w n = n − { w n − , w n − } = { n − , n }} , • A = T n ,(cid:96) ∩ { w ∈ S n : w ∈ S > n − and if w s = n − , w t = n then t ≥ s − } , • A = T n ,(cid:96) ∩ { w ∈ S n : w ∈ S n − \ S > n − and if w s = n − , w t = n then t ≥ s + } , • A (cid:48) = A \{( n − , n , n − , n − , . . . , )} , • ˜ A = A (cid:48) ∩ T n , n − , where (cid:96) = n − A , • ˜ A = ( T n , n − \ T n , n − ) ∩ { w ∈ S n : if w s = n − , w t = n then t ≥ s + } . In the following theorem, we classify all binomial ideals arising from block diagonalmatching fields inductively, i.e., in terms of the sets defined above which are themselveswritten in terms of T n − ,(cid:96) and Z n − . Note that for n = n = Z n = S n sothere are no non-zero ideals of the form F n ,(cid:96), w . The toric ideals of the form F ,(cid:96), w appear inTable 1 as the binomial ideals. Note that all the ideals are principal so it is straightforwardto determine when these binomial ideals are prime, hence toric. Theorem C (Theorems 5.5, 5.10 and 5.18) . Let n ≥
4. With the notation above, we have: C . . T n , n = A ∪ A , where (cid:96) = n , C . . T n , n − = A ∪ ˜ A ∪ ˜ A , where (cid:96) = n − C . . T n ,(cid:96) = A ∪ A (cid:48) ∪ A ∪ {( n , (cid:96), n − , n − , . . . , (cid:96) + , (cid:96) − , . . . , )} for 1 ≤ (cid:96) ≤ n − Remark . Note that (cid:209) n (cid:96) = T n ,(cid:96) ⊃ A . For n =
4, this indicates that the permutations1342 , Example 3.10.
For n = Z = { , , } and Z = { , , , , } . Table 1 shows all non-zero ideals F ,(cid:96), w and all permutations w for which F ,(cid:96), w is binomial. Ineach case we have verified that all binomial ideals are in fact prime, hence toric. In addition,we can calculate the ideals F n ,(cid:96), w for each 3 ≤ n ≤
6, 0 ≤ (cid:96) ≤ n − w ∈ S n . Table 2 showsthe number of permutations w ∈ S n , such that F n ,(cid:96), w is binomial for each given n and (cid:96) . Forthese examples we have also verified that all binomial ideals are prime, and so toric, when n = many of the permutations in T n ,(cid:96) .12 ies in both is not equal to is not equal tow has the descending property.Sn-1 Theorem Bdoes not lie in Key
Yellow arrows passbetween Sn-1 and SnCondition onpermutation. Snw does not have thedescending property.
Figure 1: The above diagram depicts Theorem B and C and shows how each permutation in T n ,(cid:96) and Z n isobtained from T n − ,(cid:96) and Z n − . The starting and ending boxes are shown in darker blue for T n ,(cid:96) and red for Z n . In each starting box we fix a permutation w . We move to adjacent boxes along arrows until reaching anending box. Purple boxes are conditions for permutations. A permutation passes through a purple box onlyif the condition is satisfied. The yellow arrows indicate a transition from w in S n − to w in S n . The boxesbefore and after a yellow arrow indicate the position in which n is added to w to obtain w . w F ,(cid:96), w (cid:104) P P − P P (cid:105) (cid:104) P P (cid:105) (cid:104) P P − P P (cid:105) (cid:104) P P (cid:105) (cid:104) P P − P P (cid:105) (cid:104) P P − P P (cid:105) (cid:104) P P (cid:105) (cid:104) P P (cid:105) (cid:104) P P − P P (cid:105) (cid:96) Toric Permutations0 1342 1432 2314 2341 2431 3214 3241 3421 43211 1342 1432 3124 3142 3214 3241 4132 43212 1342 1432 3214 3241 4231 43213 1342 1432 2314 2341 3214 3241 4321
Table 1: The ideals F ,(cid:96), w where w (cid:60) Z and the list of all w (cid:60) Z for which F ,(cid:96), w is binomial and prime(toric). Binomial (cid:96) n Table 2: For each 3 ≤ n ≤ ≤ (cid:96) ≤ n − w ∈ S n for which F n ,(cid:96), w is binomial. For each row of this table where n ≤ Corollary 3.11.
For each block diagonal matching field B (cid:96) , there is at most one permutation w for which F n ,(cid:96), w is binomial and w does not have the descending property. More precisely,the only exceptions are for ≤ (cid:96) ≤ n − and w = ( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , ) .Proof. We take cases on B (cid:96) . Case 1.
Let (cid:96) ∈ { n , n − } . By Lemmas 5.6 and 5.11 we have T n ,(cid:96) ⊂ S > n and so T n ,(cid:96) \ S > n = (cid:156) . Case 2.
Let (cid:96) ∈ { , . . . , n − } . By Corollary 5.21 we have that T n ,(cid:96) \ S > n = {( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , )} . (cid:3) Using the language of permutation avoidance, we give a simple description of the permu-tations w for which F n ,(cid:96), w is monomial-free. Definition 3.12.
Fix (cid:96) ∈ { , . . . , n − } . We define P (cid:96) ⊆ S n to be the collection of permuta-tions w ∈ S n such that the following hold. • If w is not 312-free then w > w = (cid:96) and w \ (cid:96) is 312-free.14 If w | m = ( m − , m , m − , . . . , ) for some 3 ≤ m ≤ n then w < w ≤ (cid:96) and w | w = ( w , w , w − , . . . , w + , w − , . . . , ) .We define P n ⊆ S n to be the collection of 312-free permutations. Theorem 3.13.
The ideal F n ,(cid:96), w is monomial-free if and only if w ∈ P (cid:96) . Note that Theorem C gives an inductive description of these permutations. Showing thatthe sets of permutations defined in Theorem C and P (cid:96) coincide is non-trivial and the proofis given in §
4. Zero initial ideals
In this section, we examine the permutations w for which F n ,(cid:96), w is the zero ideal. We showthat the statement F n ,(cid:96), w = (cid:96) and so we need to only checkthe permutation w to decide if F n ,(cid:96), w is zero. In particular, this means that Definition 3.6 for Z n is well-defined. The main result of this section is the following. Theorem 4.1.
For each (cid:96) , F n ,(cid:96), w = if and only if w ∈ Z n , where Z n = { s i . . . s i p ∈ S n : | i k − i (cid:96) | ≥ , for all k , (cid:96) } . Here, s i = ( i , i + ) ∈ S n is the transposition interchanging i and i + .Proof. The result follows from the following claims.
Claim 1. If F n − ,(cid:96), w =
0, then F n ,(cid:96), w = w = ( w , . . . , w n − , n ) .Suppose F n − ,(cid:96), w = w = ( w , . . . , w n − ) . We will take n ≥ n = , n = F n ,(cid:96), w (cid:44) w = ( w , . . . , w n − , n ) . Then there exists a product of variables P I P J which appears in F n ,(cid:96), w as a monomial or part of a relation P I P J − P I (cid:48) P J (cid:48) . Since P I P J does not vanish in F n ,(cid:96), w , then | I | , | J | ≤ n − n (cid:60) I ∪ J . However, w , w are identical on w , . . . , w n − so P I P J must also appear in F n ,(cid:96), w , a contradiction. So F n ,(cid:96), w = Claim 2. If F n − ,(cid:96), w = w n − = n −
1, then F n ,(cid:96), w = w = ( w , . . . , w n − , n , n − ) .Suppose by contradiction that F n ,(cid:96), w (cid:44) P I P J either as a monomialor as part of a relation, P I P J − P I (cid:48) P J (cid:48) from in w (cid:96) ( I n ) . Now if n (cid:60) I ∪ J then by the sameargument as before we have that P I P J appears in F n − ,(cid:96), w , a contradiction. So without lossof generality let us assume that n ∈ I and | I | , | J | ≤ n −
1. Since P I (cid:44) F n ,(cid:96), w , therefore I ≤ ( w , . . . , w | I | ) . Since n ∈ I we must have n ∈ ( w , . . . , w | I | ) . Since w n − = n we deducethat | I | = n −
1. Now if we also have that n ∈ J then P I \ n P J \ n (cid:44) F n − ,(cid:96), w and belongsto the non-trivial relation P I \ n P J \ n − P I (cid:48) \ n P J (cid:48) \ n in in w (cid:96) ( I n − ) . Note that this is a true relationamong the variables regardless of the block diagonal matching field B (cid:96) since by assumption n ≥ | I | ≥
3. So F n − ,(cid:96), w (cid:44)
0, a contradiction. So we deduce that n (cid:60) J . Since n is contained in exactly one of the subsets I (cid:48) , J (cid:48) , we may assume n ∈ I (cid:48) .Now consider P I \ n P J − P I (cid:48) \ n P J (cid:48) . This is a (possibly trivial) relation with P I \ n P J (cid:44) F n − ,(cid:96), w . Again note that this is a true relation among the variables regardless of B (cid:96) since15 ≥
4. However F n − ,(cid:96), w =
0. Thus this relation must be trivial, otherwise P I \ n P J would becontained in F n − ,(cid:96), w . By assumption the relation P I P J − P I (cid:48) P J (cid:48) is non-trivial so I (cid:44) I (cid:48) and J (cid:44) J (cid:48) . Therefore we must have I \ n = J (cid:48) and J = I (cid:48) \ n . We deduce that | I \ n | = | J | = n − P I \ n P J (cid:44) I \ n and J ≤ ( w , . . . , w n − ) = { , . . . , n − } . However, fromthis we deduce that I \ n = J = { , . . . , n − } and so I = I (cid:48) , a contradiction. Therefore F n ,(cid:96), w = Claim 3. If F n ,(cid:96), w =
0, then either w = ( w , . . . , w n − , n ) or w = ( w , . . . , w n − , n , n − ) .First we show that w n ≥ n −
1. So suppose by contradiction w n < n −
1. Then we haveeither w = ( α, n , β, n − , γ, w n ) or w = ( α, n − , β, n , γ, w n ) for some ordered subsets α, β, γ of [ n ] . Now suppose | α ∪ β | ≥ P α,β, n − P α, w n ,β, n − P α, w n ,β P α,β, n − , n . We must justify that this is indeed a relation for non-diagonal matching field cases. Since | α ∪ β | ≥ P M P N , n − P N P M , n where | M | , | N | ≥
2. It followsimmediately from the definition of B (cid:96) that B (cid:96) ( M ) = B (cid:96) ( M ∪ { n }) and B (cid:96) ( N ) = B (cid:96) ( N ∪ { n }) for any (cid:96) . Hence this is a true relation among the variables.Observe that none of the variables in the above relation vanishes in F n ,(cid:96), w for w = ( α, n , β, n − , γ, w n ) and w = ( α, n − , β, n , γ, w n ) . So F n ,(cid:96), w (cid:44)
0, a contradiction.Next suppose α ∪ β = (cid:156) . Then w either has the form ( n − , n , γ, w n ) or ( n , n − , γ, w n ) .Now we take cases on B (cid:96) , the block diagonal matching field. We have that either (cid:96) = ≤ (cid:96) ≤ n − (cid:96) = n − Case 1.
Let (cid:96) =
0, the diagonal matching field. Then it is easy to check that the relation P n − P , n − P P n − , n does not vanish in F n ,(cid:96), w . Case 2.
Let 1 ≤ (cid:96) ≤ n −
2. Then we have the relation P n − P n , − P n P n − , . Note that P n − P n , does not vanish in F n ,(cid:96), w . Case 3.
Let (cid:96) = n −
1. Then consider the relation P P n , n − − P n P , n − . Note that P P n , n − does not vanish in F n ,(cid:96), w .Therefore, we have shown that F n ,(cid:96), w (cid:44) w n ≥ n − w n = n − w n − = n . So suppose by contradiction that w is of the form w = ( α, n , β, n − ) for some ordered subsets α, β of [ n ] , where | β | ≥
1. Let b ∈ β be an arbitrary element. Now we take two cases on the matching field B (cid:96) . Case 3a. B (cid:96) is the diagonal matching field or α ∪ β \ b (cid:44) (cid:156) . Consider the followingrelation P α,β \ b , n − P α,β, n − P α,β P α,β \ b , n − , n . We must justify that this is indeed a relation for non-diagonal matching field cases. Since | α ∪ β \ b | ≥ P M P N , n − P N P M , n where | M | , | N | ≥
2. So, asabove, this is a true relation among the variables. It is easy to check that P M P N , n does notvanish in F n ,(cid:96), w and so F n ,(cid:96), w (cid:44)
0, a contradiction.
Case 3b. B (cid:96) is not a diagonal matching field and α ∪ β \ b = (cid:156) . Then w = ( , , ) . Wenow refer to Example 3.10 where we observe that F n ,(cid:96), w is non-zero for each (cid:96) . Hence we havea contradiction, so if w n = n − w n − = n . Therefore w must be of the desired form.16n the above series of claims we have shown that for each (cid:96) , F n ,(cid:96), w = F n − ,(cid:96), w = w = ( w , . . . , w n − , n , n − ) or w = ( w , . . . , w n − , n ) . We now proceedby induction on n . If n = F n ,(cid:96), w = w = ( ) . We observe thatthe set Z n satisfies the inductive relation: Z n = { w ∈ S n : w ∈ Z n − , and either w = ( w , . . . , w n − , n ) or w = ( w , . . . , w n − , n , n − )} . This is the same inductive relation shown in the claims which completes the proof. (cid:3)
As an immediate corollary of the above theorem we have:
Corollary 4.2. | Z n | = | Z n − | + | Z n − | for all n .Proof. Using the formulation of Z n in the proof of Theorem B, we can verify that Z n = { w ∈ Z n : w n = n } (cid:116) { w ∈ Z n : w n = n − , w n − = n } . But if w n = n , then w is determined by its first n − | Z n − | . And if w n = n − w n − = n , then w is determined by its first n − | Z n − | , as desired. (cid:3)
5. Binomial initial ideals
In this section, we present the main ingredients required for the proof of Theorem C. Wewill prove results that connect key properties of permutations w , matching fields B (cid:96) and theideal F n ,(cid:96), w . We begin by showing that A ⊂ T n ,(cid:96) for all n and (cid:96) . In the following, we dividethe results into three parts. Firstly the diagonal case with (cid:96) = n , secondly the semi-diagonalcase, i.e. (cid:96) = n −
1, and finally all remaining cases. Figure 2 shows the dependencies amongthe results required for the proof of Theorem C. The different colours in the diagram indicatethe different sections in which the results can be found.Many results of this section are inductive in nature. In the next example, we explicitlycalculate the ideals F ,(cid:96), w for each matching field B (cid:96) and each permutation w ∈ S . Therefore,we will assume n > Example 5.1.
Let n =
4. For each w ∈ A = { , } and matching field B (cid:96) we calculatethe ideal F ,(cid:96), w . In particular, we note that each such ideal is principal and toric, i.e. binomialand prime. (cid:96) w F ,(cid:96), w (cid:104) P P − P P (cid:105) (cid:104) P P − P P (cid:105) (cid:104) P P − P P (cid:105) (cid:104) P P − P P (cid:105) (cid:104) P P − P P (cid:105) (cid:104) P P − P P (cid:105) (cid:104) P P − P P (cid:105) (cid:104) P P − P P (cid:105) ependency Chart Corollary 5.21Theorem CTheorem 5.5 (C1)* Theorem 5.10 (C2)* Theorem 5.18 (C3)*Lemma 5.12
A depends onboth B and C.
ABC Lemma 5.6Proposition 5.7Lemma 5.8 Lemma 5.11Lemma 5.13Lemma 5.15Lemma 5.16Lemma 5.17Lemma 5.14 Lemma 5.22Lemma 5.23Lemma 5.24Lemma 5.20Lemma 5.19*Proposition 5.4Section 5.1Section 5.2Section 5.3
Sections:
Section 5
Figure 2: The dependency chart above shows the key steps in the proof of Theorem C which classifiesthe permutations in T n ,(cid:96) . We split Theorem C into three cases: C1, C2 and C3 corresponding to (cid:96) = n ,1 ≤ (cid:96) ≤ n − (cid:96) = n − A ⊂ T n ,(cid:96) for each (cid:96) . Definition 5.2.
Let w = ( w , . . . , w n ) . Recall that S w = { I : I (cid:54)≤ w ( I ) } . We denote itscomplement by S c w = { I : (cid:156) (cid:44) I (cid:40) [ n ]} \ S w = { I : I ≤ { w , . . . , w | I | }} and for 1 ≤ t ≤ n wedefine its projection as S c ( w ,..., w t ) = { I : | I | ≤ t , I ∈ S c w } . Note that S c w is the collection of subsets I ⊂ [ n ] for which P I do not vanish in F n ,(cid:96), w forany (cid:96) . From Definition 5.2 we obtain the following description of S c w for specific cases. Corollary 5.3.
Let w ∈ S n with w t = n for t ∈ { , . . . , n } . Then S c w = S c ( w ,..., w t − ) ∪ { I : I = { i < · · · < i (cid:96) } , (cid:96) ≥ t , I \ i (cid:96) ∈ S c w } . Moreover, for w ∈ S > n we have that S c w = S c ( w ,..., w t − ) ∪ { I : I = { i < · · · < i (cid:96) } , (cid:96) ≥ t , ( i , . . . , i t − ) ∈ S c ( w ,..., w t − ) } . For each block diagonal matching field B (cid:96) we have: Proposition 5.4.
For each w ∈ A and ≤ (cid:96) ≤ n , F n ,(cid:96), w is a principal toric ideal. Inparticular, A ⊂ T n ,(cid:96) . roof. Assume that w ∈ S n has the form w = ( w , . . . , w n − , n , n − , n − ) or w = ( w , . . . , w n − , n − , n , n − ) . We prove that if w is in Z n − , then F n ,(cid:96), w is toric and principal, i.e. generated by a singlepolynomial. For n =
3, we have that A = { } and the result follows from the calculationin Example 3.10. Similarly for n =
4, the result follows from Example 5.1. Now we assumethat n > F n ,(cid:96), w (cid:44)
0. Let α = { w , . . . , w n − } = { , . . . , n − } . Note that | α | ≥ F n ,(cid:96), w : P α, n − P α, n − , n − P α, n − P α, n − , n . Notice that none of these variables vanish in F n ,(cid:96), w for either w above. We must check thatthis relation does not depend on the block diagonal matching field B (cid:96) . This follows from twobasic properties of the matching field B (cid:96) . Firstly, the matching field permutes only entriesof α and fixes all others. And secondly, if β ⊂ [ n ] is disjoint from α then B (cid:96) ( α ) = B (cid:96) ( α ∪ β ) .Now suppose that we have two variables P I and P J which do not vanish in F n ,(cid:96), w and belongto a relation P I P J − P I (cid:48) P J (cid:48) ∈ in w (cid:96) ( I n ) . We will show that | I | , | J | > n − | I | ≤ | J | and | I | ≤ n −
3. Additionally, we mayassume that | I | = | I (cid:48) | , | J | = | J (cid:48) | . We proceed by taking cases on | J | . Case 1.
Let | J | ≤ n −
3. Since neither P I nor P J vanish, we have that I ≤ ( w , . . . , w | I | ) and J ≤ ( w , . . . , w | J | ) . Since w i < n − ≤ i ≤ n − P I P J − P I (cid:48) P J (cid:48) appears in F n − ,(cid:96), w . However w ∈ Z n − so F n − ,(cid:96), w =
0, a contradiction.
Case 2.
Let | J | > n −
3. We have that ( w , . . . , w n − ) = { , . . . , n − } . Since I ≤( w , . . . , w | I | ) clearly we must have I ⊆ { , . . . , n − } . Similarly, J ≤ ( w , . . . , w | J | ) and sowe deduce that J = { , . . . , n − , j n − , . . . , j | J | } . Hence I ⊂ J and it is easy to check that P I P J − P I (cid:48) P J (cid:48) is a trivial relation, a contradiction.So | I | , | J | > n −
3. Note that n ≥
5. Since ( w , . . . , w n − ) = ( , . . . , n − ) we have I = { < · · · < n − < i n − < · · · < i | I | } and similarly J = { < · · · < n − < j n − < · · · < j | J | } .The relation P I P J − P I (cid:48) P J (cid:48) can be seen to arise from a relation in in w D ( I ) under the diagonalmatching field. This relation is obtained by removing { , . . . , n − } from I , J , I (cid:48) , J (cid:48) , and thensubtracting n − B (cid:96) because n ≥ B (cid:96) permutes only the entries in { , . . . , n − } and fixes all others. However, in w D ( I ) is principal and generated by P P − P P . So therelation P I P J − P I (cid:48) P J (cid:48) must be P α, n − P α, n − , n − P α, n − P α, n − , n where α = { , . . . , n − } . It isclear that this relation is contained in F n ,(cid:96), w for each w above. Since P I P J was arbitrary, itfollows that F n ,(cid:96), w = (cid:104) P α, n − P α, n − , n − P α, n − P α, n − , n (cid:105) is a principal ideal. (cid:3) Recall that the set T n , n is the collection of all permutations w ∈ S n such that F n , n , w is anon-zero binomial ideal. The sets of permutations A and A are defined inductively from Z n − and T n − , n − respectively by ‘inserting’ n into the permutation in allowed places.19ere, we state our main theorem for the diagonal matching fields. Note that the clas-sification of F n ,(cid:96), w is simpler for the diagonal case than for the other matching fields andserves as a good template for the proofs in the following sections. In particular, we will seeanalogues for Lemmas 5.6, 5.8 and Proposition 5.7 for other matching fields in later sections. Theorem 5.5. T n , n = A ∪ A . Proof.
By Propositions 5.4 and 5.7 we have that A ∪ A ⊆ T D , n . To prove the other directionsuppose that F n , n , w is binomial and write the permutation w = ( w , . . . , w t , n , w t + , . . . , w n − ) for some t ∈ { , , . . . , n − } . Now by Lemma 5.8 F n − , n − , w is either zero or binomial.Firstly, suppose that F n − , n − , w =
0. Theorem B implies that w is of form w = ( w , . . . , w n − , n − ) or w = ( w , . . . , w n − , n − , n − ) . Since F n , n , w is binomial, by Lemma 5.6 we have n > w t + > · · · > w n − . So if w = ( w , . . . , w n − , n − ) then we have that w = ( w , . . . , w n − , n − , n ) or w = ( w , . . . , w n − , n , n − ) . However, in both cases we have that F n , n , w = w = ( w , . . . , w n − , n − , n − ) . Now by Lemma 5.6 we have that w is one of thefollowing permutations: • w = ( w , . . . , w n − , n − , n − , n ) , • w = ( w , . . . , w n − , n − , n , n − ) , • w = ( w , . . . , w n − , n , n − , n − ) .However, if w = ( w , . . . , w n − , n − , n − , n ) then by Theorem B we have that F n , n , w =
0, acontradiction. The remaining cases are of the desired form.Secondly, suppose that F n − , n − , w is binomial and w s = n −
1. By Lemma 5.6 we havethat w s > w s + > · · · > w n − and n > w t + > · · · > w n − . Thus we must have that t ≥ s − n > w s − and w s − < w s which contradicts Lemma 5.6. (cid:3) Lemma 5.6. If F n , n , w is binomial, then w ∈ S > n .Proof. Let w = ( w , . . . , w n ) with w t = n . Suppose by contradiction that there exists k > t such that w k < w k + . Without loss of generality, take k to be the minimum such index. Wewill show that F n , n , w contains a monomial. Let I = { w , . . . , w k − , w k + } and I (cid:48) = { w , . . . , w k } . By this construction P I vanishes and P I (cid:48) does not vanish in F n , n , w because w k + > w k andso I > { w , . . . , w k } = I (cid:48) . Now we write I ∪ I (cid:48) as an ordered set as ( α, w k , β, w k + , γ, n ) forsome ordered subsets α, β, γ of [ n ] , so I = ( α, β, w k + , γ, n ) and I (cid:48) = ( α, w k , β, γ, n ) . Now define J = ( α, w k , β, γ ) and J (cid:48) = ( α, β, w k + , γ ) . By construction, we have P I P J − P I (cid:48) P J (cid:48) is a relation20n in w (cid:96) ( I n ) where P I vanishes and P I (cid:48) does not vanish in F n , n , w . However P J (cid:48) does not vanishbecause { w , . . . , w k − } = ( α, β, γ, n ) ≥ ( α, β, w k + , γ ) = J (cid:48) . And so, we have shown that the monomial P I (cid:48) P J (cid:48) appears in the binomial ideal F n , n , w , whichis a contradiction. (cid:3) Proposition 5.7.
Suppose that F n − , n − , w is binomial, where w = ( w , . . . , w n − ) and w s = n − . Then F n , n , w is binomial for w = ( w , . . . , w t , n , w t + , . . . , w n − ) , where t ≥ s − .Proof. Suppose that P I P J − P I (cid:48) P J (cid:48) ∈ in w (cid:96) ( I n ) and P I P J (cid:44) F n , n , w . We show that P I (cid:48) P J (cid:48) (cid:44) | I | and | J | . Without loss of generality we assume | I | ≤ | J | so we havethat either | I | , | J | ≤ t or | I | ≤ t and | J | > t or | I | , | J | > t . Case 1.
Let | I | , | J | ≤ t . Since w , w are identical from w to w t , we deduce that P I P J (cid:44) F n − , n − , w . Since F n − , n − , w is binomial we have that P I (cid:48) P J (cid:48) (cid:44) F n − , n − , w and so it isnon-zero in F n , n , w . Case 2.
Let | I | ≤ t , | J | > t . Since P I P J (cid:44) F n , n , w , by Corollary 5.3 we have P I (cid:44) P J \ j | J | (cid:44) F n − , n − , w where J = { j < · · · < j | J | } . Now, P I P J \ j | J | − P I (cid:48) P J (cid:48) \ j | J | is a valid(possibly trivial) relation among the variables in F n − , n − , w . Since this ideal is binomial wehave that P I (cid:48) (cid:44) P J (cid:48) \ j | J | (cid:44)
0. By Corollary 5.3, P I (cid:48) (cid:44) P J (cid:48) (cid:44) F n , n , w . Case 3.
Let | I | , | J | > t . Write I = { i < · · · < i p } and J = { j < · · · < j q } . By Lemma 5.6we have that n > w t + > · · · > w n − and so we may apply Corollary 5.3 as follows. P I (cid:44) F n , n , w if and only if ( i , . . . , i t ) ≤ ( w , . . . , w t ) . Similarly, P J (cid:44) F n , n , w if and only if ( j , . . . , j t ) ≤ ( w , . . . , w t ) . Next let us write I (cid:48) = { i (cid:48) < · · · < i (cid:48) p } and J (cid:48) = { j (cid:48) < · · · < j (cid:48) q } .Now suppose without loss of generality that p ≤ q . Since we are working with the diagonalmatching field, for each 1 ≤ e ≤ p we have that i (cid:48) e , j (cid:48) e ∈ { i e , j e } . Hence ( i (cid:48) , . . . , i (cid:48) t ) ≤ ( w , . . . , w t ) and ( j (cid:48) , . . . , j (cid:48) t ) ≤ ( w , . . . , w t ) . By Corollary 5.3, we have P I (cid:48) (cid:44) P J (cid:48) (cid:44) F n , n , w . Hence F n , n , w is binomial. (cid:3) Lemma 5.8. If F n − , n − , w is non-binomial, then F n , n , w is non-binomial.Proof. Suppose F n − , n − , w is non-binomial. Then there exists a monomial P I P J ∈ F n − , n − , w .Suppose this monomial arises from the relation P I P J − P I (cid:48) P J (cid:48) ∈ in w (cid:96) ( I n − ) such that P I (cid:48) P J (cid:48) = F n − , n − , w . Without loss of generality assume that | I | = | I (cid:48) | ≤ | J | = | J (cid:48) | . Write w = ( w , . . . , w n − ) and w = ( w , . . . , w t , n , w t + , . . . , w n − ) for some t ∈ { , , . . . , n − } . We takecases based on t , | I | and | J | . In particular, we must either have | J | ≤ t or | I | ≤ t < | J | or t < | I | . Case 1.
Let | J | ≤ t . In this case we have that P I P J is a monomial in F n , n , w because ( w , . . . , w t ) determines whether the variables in the above relation vanish in F n , n , w and w , w are identical on ( w , . . . , w t ) . Case 2.
Let | I | ≤ t < | J | . Consider the relation P I P J ∪{ n } − P I (cid:48) P J (cid:48) ∪{ n } in in w (cid:96) ( I n ) . ByCorollary 5.3 we have that P I P J (cid:44) P I (cid:48) P J (cid:48) = F n − , n − , w if and only if P I P J ∪{ n } (cid:44) P I (cid:48) P J (cid:48) ∪{ n } = F n , n , w . 21 ase 3. Let t < | I | . Consider the relation P I ∪{ n } P J ∪{ n } − P I (cid:48) ∪{ n } P J (cid:48) ∪{ n } ∈ in w (cid:96) ( I n ) .Applying Corollary 5.3 we have that P I P J (cid:44) P I (cid:48) P J (cid:48) = F n − , n − , w if and only if P I ∪{ n } P J ∪{ n } (cid:44) P I (cid:48) ∪{ n } P J (cid:48) ∪{ n } = F n , n , w .In each case we have shown that F n , n , w is non-binomial, as desired. (cid:3) Remark . Note that the converse to Lemma 5.8 is false. For example if w = ( , , , ) then F , , w is non-binomial, however F , , w is binomial for w = ( , , ) . Below, we state and prove the main result for (cid:96) = n −
1, which decomposes the collectionof permutations T n , n − into three parts: A , ˜ A and ˜ A , see Definition 3.8. The proof is splitup into five steps. Each step is written with the claim at the beginning, followed by theproof of that claim. Steps a, b and c are very similar to the diagonal case, since we haveseen that A ⊂ T n , n . Steps d and e are particular to the semi-diagonal case and show howthe subsets ˜ A and ˜ A arise in the decomposition of T n , n − . Theorem 5.10. T n , n − = A ∪ ˜ A ∪ ˜ A .Proof. We will use Lemmas 5.11, 5.12, 5.15 and 5.16. We will break down the proof into thefollowing steps.
Step a. A ∪ ˜ A ∪ ˜ A ⊂ T n , n − and so RHS ⊆ LHS .First by Proposition 5.4, F n ,(cid:96), w is binomial for each w ∈ A , so A ⊂ T n , n − . Next ˜ A ⊂ T n , n − and ˜ A ⊂ T n , n − by Lemma 5.15 and Lemma 5.16, respectively. So we have shown A ∪ ˜ A ∪ ˜ A ⊆ T n , n − . Step b.
For any w ∈ T n , n − , w ∈ Z n − ∪ T n − , n − .Now take w ∈ T n , n − . By Lemma 5.17, w ∈ T n − , n − ∪ Z n − . By Lemma 5.11 we have that w has the descending property. We denote w t = n and w s = n − Step c. If w ∈ Z n − then w ∈ A .First suppose w ∈ Z n − . By Theorem B, either w = ( w , . . . , w n − , n − ) or w = ( w , . . . , w n − , n − , n − ) . If w = ( w , . . . , w n − , n − ) then w = ( w , . . . , w n − , n , n − ) or w = ( w , . . . , w n − , n − , n ) since w has the descending property. However F n , n − , w = w = ( w , . . . , w n − , n − , n − ) . If w = ( w , . . . , w n − , n − , n − , n ) then F n , n − , w =
0, so w = ( w , . . . , w n − , n − , n , n − ) or w = ( w , . . . , w n − , n , n − , n − ) .Therefore w ∈ A . Step d. If w ∈ T n − , n − ∩ T n − , n − then w ∈ ˜ A .Next suppose w ∈ T n − , n − ∩ T n − , n − . Since w has the descending property, we have t ≥ s −
1. By Lemma 5.14 we have w (cid:44) ( n − , n , n − , . . . , ) . Therefore w ∈ ˜ A . Step e. If w ∈ T n − , n − ∩ N n − , n − then w ∈ ˜ A .Finally suppose w ∈ T n − , n − \ T n − , n − = T n − , n − ∩ N n − , n − . Since w has the descendingproperty, t ≥ s −
1. We show that t ≥ s + t = s . Suppose t = s −
1. Then by Lemma 5.13 we have w ∈ N n ,(cid:96) , a contradiction. Therefore w ∈ ˜ A . (cid:3) emma 5.11. If w ∈ T n , n − then w ∈ S > n .Proof. Let w = ( w , . . . , w n ) with w t = n . Suppose by contradiction that there exists k > t such that w k < w k + . Let I = { w , . . . , w k } and I (cid:48) = { w , . . . , w k − , w k + } . Note that P I does not vanish in F n , n − , w but P I (cid:48) does vanish. Let us write in ascending order I ∪ I (cid:48) = { α, w k , β, w k + , γ, n } for some subsets α, β, γ of [ n ] . Note that I = { α, w k , β, γ, n } and I (cid:48) = { α, β, w k + , γ, n } . Now we take cases on | α ∪ β ∪ γ | . Case 1.
Let | α ∪ β ∪ γ | ≥
1. Let J = { α, β, w k + , γ } and J (cid:48) = { α, w k , β, γ } . Then it iseasy to check that P I P J − P I (cid:48) P J (cid:48) is a valid relation in in w n − ( I n ) . This is because for each L ∈ { I , J , I (cid:48) , J (cid:48) } , we have B n − ( L ) = id . However, P I P J does not vanish in F n , n − , w , so F n , n − , w contains the monomial P I P J since P I (cid:48) vanishes in F n , n − , w . Therefore F n , n − , w is non-binomial,a contradiction. Case 2.
Let | α ∪ β ∪ γ | =
0. So w = ( n , w , w , . . . , w n ) with k =
2. Consider therelation P n P w w − P w P n w . in in w n − ( I n ) . This relation holds because B n − ({ w , w }) = id and B n − ({ w , n }) = ( ) is a transposition. However P n vanishes in F n , n − , w , so F n , n − , w contains the monomial P w P n w and so is non-binomial, a contradiction. (cid:3) Lemma 5.12.
Let w ∈ S n and B n − = ( . . . n − | n ) be a block diagonal matching field. Sup-pose that P I P J is a monomial in F n , n − , w arising from the relation P I P J − P I (cid:48) P J (cid:48) in in w n − ( I n ) .If B n − ( I ) = B n − ( J ) = id then B n − ( I (cid:48) ) = B n − ( J (cid:48) ) = id .Proof. We show the result by contradiction. Suppose without loss of generality that B n − ( I (cid:48) ) (cid:44) id . So I (cid:48) = { i , n } for some 1 ≤ i ≤ n −
1. We have either n ∈ I or n ∈ J . Without loss ofgenerality suppose n ∈ J . After ordering I (cid:48) according to the matching field B n − we see that n is the first element. So n is the first element of J . However B n − ( J ) = id so we deduce that J = { n } . Since | J | (cid:44) | I (cid:48) | , we have | I | = | I (cid:48) | and | J | = | J (cid:48) | . Write I = { a , i } for some a < i .Then the relation is given by P a , i P n − P n , i P a . We have that P n does not vanish in F n , n − , w so w = ( n , w , . . . w n ) . Since P a , i does not vanishwe have i ≤ w . On the other hand P a does not vanish but P n , i P a vanishes so P n , i must vanishin F n , n − , w . Therefore { n , i } (cid:54)≤ { n , w } so i > w , a contradiction. (cid:3) Lemma 5.13.
Let w ∈ T n − , n − ∩ N n − , n − with w s = n − and w t = n . If t = s − then w ∈ N n , n − .Proof. Let P I P J be a monomial appearing in F n − , n − , w which arises from the relation P I P J − P I (cid:48) P J (cid:48) in in w n − ( I n − ) .Note that by definition of B n − = ( , . . . , n − | n − ) , the only subsets L ⊆ [ n − ] forwhich B n − ( L ) (cid:44) id are those with | L | = n − ∈ L . If B n − ( I ) = B n − ( J ) = B n − ( I (cid:48) ) = B n − ( J (cid:48) ) = id then P I P J − P I (cid:48) P J (cid:48) is a relation in in w D ( I n − ) . This relation gives rise to amonomial in F D , w , n − but by assumption w ∈ T n − , n − , a contradiction. So by Lemma 5.12we may assume without loss of generality that B n − ( I ) (cid:44) id . We write I = { i , n − } for some1 ≤ i ≤ n −
2. We take cases on | J | . 23 ase 1. Let | J | =
1. Let us write J = { j } . The relation is given by P n − , i P j − P j , i P n − . It follows that j < i . Now consider w = ( w , w , . . . , w n − ) . Since P I P J does not vanishin F n − , n − , w we have j ≤ w and { i , n − } ≤ { w , w } . Therefore either w = n − w = n −
1. Since j ≤ n − { j , i } ≤ { w , w } . By assumption P j , i P n − vanishesso P n − vanishes in F n − , n − , w . We deduce that w (cid:44) n − w = n −
1. So by ourassumption w = ( w , n , n − , . . . ) . Now consider P n , i P j − P j , i P n . This is a relation in in w n − ( I n ) . Note that P n , i P j does not vanish but P j , i P n does vanish in F n , n − , w . So we have shown that w ∈ N n , n − . Case 2.
Let | J | ≥
2. First we show that B n − ( J ) = id . Suppose by contradiction that B n − ( J ) (cid:44) id . By definition of B n − = ( , . . . , n − | n − ) , we have B n − ( J ) (cid:44) id impliesthat | J ∩ { , . . . , n − }| =
1, and so | J | = n − ∈ J . Hence, J = { j , n − } forsome 1 ≤ j ≤ n −
2. The relation is given by P n − , i P n − , j − P n − , j P n − , i which is trivial, acontradiction. So B n − ( J ) = id . Let us write J = { j < j < · · · < j | J | } . The relation is givenby P n − , i P j , j , j ,..., j | J | − P n − , j P j , i , j ,..., j | J | . Next we show that i < j by contradiction. Note that i (cid:44) j otherwise the aboverelation is trivial. Suppose that i > j . Since P n − , i does not vanish, P n − , j does not vanishin F n − , n − , w . So P j , i , j ,..., j | J | vanishes in F n − , n − , w . By Lemma 5.6, w has the descendingproperty since w ∈ T n − , n − . So if w = n − w = ( n − , n − , . . . , , ) and F B (cid:48) , w , n − isbinomial, a contradiction. Since P n − , i does not vanish in F B (cid:48) , w , n − and w (cid:44) n −
1, it followsthat w = n −
1. Now by applying Corollary 5.3 to w we have that P j , i , j ,..., j | J | vanishes in F n − , n − , w if and only if { j , i } (cid:54)≤ { w , w } . On the other hand, { j , i } (cid:54)≤ { w , w } implies that I = { n − , i } (cid:54)≤ { w , w } , a contradiction.So we have i < j . We deduce that P j , i , j ,..., j | J | does not vanish in F n − , n − , w so P n − , j doesvanish and j > w . Since P n − , i does not vanish, w = n −
1. Now w = ( w , n , n − , . . . ) hasthe descending property because w does. Consider the relation P n , i P j , j , j ,..., j | J | − P n , j P j , i , j ,..., j | J | . This is a valid relation in in w n − ( I n ) . By Corollary 5.3 we see that P n , i P j , j , j ,..., j | J | does notvanish in F n , n − , w but P n , j does vanish in F n , n − , w . So F n , n − , w is non-binomial and w ∈ N n , n − . (cid:3) Lemma 5.14.
Fix (cid:96) ∈ { , . . . , n − } . Then F n ,(cid:96), w is non-binomial for w = ( n − , n , n − , . . . , ) .Proof. If (cid:96) (cid:44) n −
1, then consider the following relation in in w (cid:96) ( I n ) : P n − P n , − P n P n − , . P n − P n , does not vanish in F n ,(cid:96), w whereas P n vanishes in F n ,(cid:96), w . So F n ,(cid:96), w contains the monomial P n − P n , , hence F n ,(cid:96), w is non-binomial.If (cid:96) = n −
1, then we consider the relation P P n , n − − P n P , n − in in w (cid:96) ( I n ) . The term P P n , n − does not vanish in F n ,(cid:96), w whereas P n does vanish in F n ,(cid:96), w . So F n ,(cid:96), w contains the monomial P P n , n − and hence is non-binomial. (cid:3) Lemma 5.15.
We have that ˜ A ⊂ T n , n − .Proof. Let w ∈ ˜ A . Note that ˜ A ⊆ A = T n , n − ∩ { w ∈ S n : w ∈ S > n − and if w s = n − , w t = n then t ≥ s − } , so by Theorem C part C1, we have that w ∈ T n , n . Hence, by Lemma 5.6,we conclude that w ∈ S > n . By contradiction suppose that F n , n − , w is non-binomial. So thereexists a monomial P I P J ∈ F n , n − , w arising from a relation P I P J − P I (cid:48) P J (cid:48) in in w (cid:96) ( I n ) . We assumewithout loss of generality that | I | = | I (cid:48) | and | J | = | J (cid:48) | . If B n − ( I ) = B n − ( J ) = id then byLemma 5.12 we have B n − ( I (cid:48) ) = B n − ( J (cid:48) ) = id so P I P J would be a monomial in F n , n , w , acontradiction. So without loss of generality suppose that B n − ( I ) (cid:44) id . Write I = { i , n } forsome 1 ≤ i ≤ n −
1. If B n − ( J ) (cid:44) id then the relation would be trivial since we would have J = { j , n } for some j . Therefore B n − ( J ) = id .We have that P n , i does not vanish in F n , n − , w so { n , i } ≤ { w , w } and so either w = n or w = n . If w = n then w = ( n , n − , . . . , ) since w ∈ S > n . In this case, for all L ⊆ [ n ] , L ∈ S c w so in particular P I (cid:48) P J (cid:48) does not vanish in F n , n − , w , a contradiction. So w = n . Note that w (cid:44) n − w = ( n − , n , n − , . . . , ) contradicting our assumption. So we havededuced that w = ( w , n , n − , . . . ) where w ≤ n −
2. Now we take cases on | J | .If | J | = J = { j } , then the relation is given by P n , i P j − P n P j , i . We show that i = n − i (cid:44) n − i < n −
1. Considerthe relation P n − , i P j − P n − P j , i in in w n − ( I n − ) . Since w = ( w , n , n − , . . . ) we have that w = ( w , n − , . . . ) . It is easy to check that P n − , i P j does not vanish but P n − does vanish in F n − , n − , w . This contradicts the assumption that w ∈ T n − , n − .So i = n − I = { n , n − } . Since P I does not vanish in F n , n − , w we have { n − , n } ≤{ w , w } . Therefore w = n − w = n . Since w ∈ S > n we deduce that w = ( n − , n , n − , . . . , ) , a contradiction.If | J | ≥ J = { j < j < · · · < j | J | } . The relation is given by P n , i P j , j , j ,..., j | J | = P n , j P j , i , j ,..., j | J | . If P n , j does not vanish in F n , n − , w then P j , i , j ,..., j | J | must vanish. By Corollary 5.3, j > w .But w < j < i ≤ w , a contradiction. So P n , j vanishes in F n , n − , w . Since P I does not vanishand w (cid:44) ( n − , n , n − , . . . , ) we have that i < n − j < n −
1. If j = n then | J | = B n − ( J ) (cid:44) id , a contradiction.If j = n − P n , i P j , n − − P n , n − P j , i . Since i < n − < n −
2. Now consider the relation P n − , i P j , n − − P n − , n − P j , i in in w n − ( I n − ) . Note that w = ( w , n − , . . . ) so P n − , i P j , n − does not vanish in F n − , n − , w . Since w ∈ T n − , n − it followsthat P n − , n − P j , i does not vanish. In particular P n − , n − does not vanish. Therefore w = n − w = ( n − , n − , n − , . . . , ) , however this contradicts Lemma 5.14. And so j < n − P n − , i P j , j − P n − , j P j , i in in w n − ( I n − ) . Clearly P n − , i P j , j doesnot vanish in F n − , n − , w . Since P n , j vanishes in F n , n − , w it follows that P n − , j vanishes in F n − , n − , w . And so we have shown F n − , n − , w is non-binomial, a contradiction. (cid:3) Lemma 5.16.
We have that ˜ A ⊂ T n , n − .Proof. Take w ∈ ˜ A . Since ˜ A ⊂ T n , n − ∩ { w ∈ S n : if w s = n − , w t = n then t ≥ s + } itfollows that w ∈ S > n . So by Theorem C part C1 we have w ∈ T n , n . Suppose P I P J − P I (cid:48) P J (cid:48) is a relation in in w n − ( I n ) with P I P J non-vanishing in F n , n − , w . We show that P I (cid:48) P J (cid:48) is non-vanishing by taking cases on | I | and | J | . We assume without loss of generality that | J | ≤ | I | and so we must have that either | I | , | J | < t or | I | ≥ t and | J | < t or | I | , | J | ≥ t . Case 1.
Let | I | , | J | < t . Since I ≤ { w , . . . , w | I | } , J ≤ { w , . . . , w | J | } and w t = n , wededuce that n (cid:60) I and n (cid:60) J . Therefore B n − ( I ) = B n − ( J ) = id so B n − ( I (cid:48) ) = B n − ( J (cid:48) ) = id .Since w ∈ T n , n we have that P I (cid:48) P J (cid:48) does not vanish in F n , n , w and so does not vanish in F n , n − , w . Case 2.
Let | I | ≥ t , | J | < t . Note that B n − ( J ) = id . We show that B n − ( I ) = id bycontradiction. Suppose B n − ( I ) (cid:44) id then I = { i , n } for some 1 ≤ i ≤ n −
1. Since | J | < | I | then J = { j } for some 1 ≤ j ≤ n . So the relation is given by P n , i P j − P n P j , i . Since P n , i (cid:44) w n − ( I n ) , we have that { i , n } ≤ { w , w } . Hence, n ∈ { w , w } . Notethat w ∈ T n , n has the descending property by Lemma 5.6. This together with the assumptionthat t ≥ s + w = ( n − , n , n − , . . . , ) . However w = ( n − , n − , . . . , ) and so F n − , n − , w = in w n − ( I n − ) is binomial, a contradiction.So B n − ( I ) = id . Then similarly to Case 1, we deduce that P I (cid:48) P J (cid:48) does not vanish in F n , n − , w . Case 3.
Let | I | , | J | ≥ t . We show that B n − ( I ) = B n − ( J ) = id by contradiction. Suppose B n − ( I ) (cid:44) id so I = { i , n } for some 1 ≤ i ≤ n −
1. Since P I does not vanish in F n , n − , w , wehave that n ∈ { w , w } . Since t ≥ s + w has the descending property, we deduce that w = ( n − , n , n − , . . . , ) . So w = ( n − , n − . . . , ) ∈ T n − , n − , a contradiction.So we have B n − ( I ) = B n − ( J ) = id , hence P I (cid:48) P J (cid:48) does not vanish in F n , n − , w . (cid:3) Lemma 5.17.
Let w ∈ S n . If w ∈ N n − , n − then w ∈ N n , n − .Proof. We write w = ( w , . . . , w n ) and w t = n for some t ∈ { , . . . , n } . Suppose P I P J is amonomial in F n , n − , w arising from a relation P I P J − P I (cid:48) P J (cid:48) in in w n − ( I n − ) . Without loss ofgenerality, | I | = | I (cid:48) | ≥ | J | = | J (cid:48) | . We take cases on | I | , | J | and t . In particular we must havethat either | I | , | J | < t or | I | ≥ t and | J | < t or | I | , | J | ≥ t . Case 1.
Let | I | , | J | < t . We have that P I P J − P I (cid:48) P J (cid:48) is a relation in in w n − ( I n ) . Since w and w agree on w , . . . , w t − we have that P I P J is a monomial in F n , n − , w and so w ∈ N n , n − .26 ase 2. Let | I | ≥ t , | J | < t . Note that we have | I | ≥ I = I ∪{ n } and ˜ I (cid:48) = I (cid:48) ∪{ n } .By Corollary 5.3, P ˜ I does not vanish in F n , n − , w . Since | I | ≥ B n − ( ˜ I ) = B n − ( ˜ I (cid:48) ) = id . Hencewe have the following is a relation in in w n − ( I n ) : P ˜ I P J − P ˜ I (cid:48) P J (cid:48) . By Corollary 5.3, P ˜ I (cid:48) P J (cid:48) vanishes in F n , n − , w . So w ∈ N n , n − . Case 3.
Let | I | , | J | ≥ t . We write ˜ L = L ∪ { n } for each L ∈ { I , J , I (cid:48) , J (cid:48) } . Suppose | I | , | J | ≥ B n − ( ˜ I ) = B n − ( ˜ J ) = B n − ( ˜ I (cid:48) ) = B n − ( ˜ J (cid:48) ) = id and so we have thefollowing relation in in w n − ( I n ) : P ˜ I P ˜ J − P ˜ I (cid:48) P ˜ J (cid:48) . By Corollary 5.3 we have that P ˜ I P ˜ J does not vanish and P ˜ I (cid:48) P ˜ J (cid:48) does vanish in F n , n − , w . So w ∈ N n , n − .Now suppose | I | = t =
1. Since the relation P I P J − P I (cid:48) P J (cid:48) is non-trivial, | J | ≥ B n − ( ˜ J ) = B n − ( ˜ J (cid:48) ) = id . We write I = { i } and J = { j , j , . . . , j | J | } . The relation isgiven by: P i P j , j ,..., j | J | = P j P i , j ,..., j | J | . Since P J does not vanish in F n − , n − , w we have that j ≤ min { w , . . . , w | J | } ≤ w and so P j does not vanish. We deduce that P i , j ,..., j | J | vanishes in F n − , n − , w . Consider the relation P I P ˜ J − P I (cid:48) P ˜ J (cid:48) in in w n − ( I n ) given by: P i P j , j ,..., j | J | , n = P j P i , j ,..., j | J | , n . By Corollary 5.3, P ˜ J does not vanish in F n , n − , w and P ˜ J (cid:48) does vanish. And so w ∈ N n , n − . (cid:3) Throughout this section, unless otherwise stated, we assume that (cid:96) ∈ { , . . . , n − } . Werecall the definitions of the sets A (cid:48) and A from Definition 3.8. Below, we state and provethe main result of this section. Similarly to the semi-diagonal case, the main result of thissection decomposes T n ,(cid:96) into three main parts: A , A (cid:48) and A along with the exceptionalpermutation ( n , (cid:96), n − , n − , . . . , (cid:96) + , (cid:96) − , . . . , ) . The proof is similar to the semi-diagonalcase, in fact steps a, b and c follow the same structure. Steps d and e carefully use thestructure of the matching field to show how the sets A (cid:48) , A and the exceptional permutationarises in the decomposition of T n ,(cid:96) . Theorem 5.18. T n ,(cid:96) = A ∪ A (cid:48) ∪ A ∪ {( n , (cid:96), n − , n − , . . . , (cid:96) + , (cid:96) − , . . . , )} for ≤ (cid:96) ≤ n − .Proof. Before stating the proof we first note that for every block diagonal matching field B = ( , . . . , (cid:96) | (cid:96) + , . . . , n ) for 1 ≤ (cid:96) ≤ n − I = { i < i < · · · < i | I | } of [ n ] we have the following cases: • If B (cid:96) ( I ) = id , then either i , i ∈ { (cid:96) + , . . . , n } , or i , i ∈ { , . . . , (cid:96) } or | I | = • If B (cid:96) ( I ) (cid:44) id then i ∈ { , . . . , (cid:96) } and i ∈ { (cid:96) + , . . . , n } .27e will now break down the proof into the following steps. Step a. A ∪ A (cid:48) ∪ A ⊂ T n ,(cid:96) and so RHS ⊆ LHS .First we show A ⊂ T n ,(cid:96) . Suppose w = ( w , . . . , w n − , n − , n − ) ∈ Z n − . Then for w = ( w , . . . , w n − , n , n − , n − ) and w = ( w , . . . , w n − , n − , n , n − ) , F n ,(cid:96), w is binomial byProposition 5.4 so A ⊂ T n ,(cid:96) . Next A (cid:48) ⊂ T n ,(cid:96) and A ⊂ T n ,(cid:96) by Lemma 5.22 and Lemma 5.23respectively. By Lemma 5.19 we have ( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , ) ∈ T n ,(cid:96) . So we haveshown A ∪ A (cid:48) ∪ A ∪ {( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , )} ⊆ T n ,(cid:96) . Step b.
For any w ∈ T n ,(cid:96) with w (cid:44) ( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , ) we have that w ∈ Z n − ∪ T n − ,(cid:96) .Now take w ∈ T n ,(cid:96) with w (cid:44) ( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , ) . By Lemma 5.24, w ∈ T n − ,(cid:96) ∪ Z n − . By Lemma 5.20 we have that w has the descending property. We denote w t = n and w s = n − Step c. If w ∈ Z n − then w ∈ A .First suppose w ∈ Z n − . By Theorem B, either w = ( w , . . . , w n − , n − ) or w = ( w , . . . , w n − , n − , n − ) . If w = ( w , . . . , w n − , n − ) then w = ( w , . . . , w n − , n , n − ) or w = ( w , . . . , w n − , n − , n ) since w has the descending property. However F n ,(cid:96), w = w = ( w , . . . , w n − , n − , n − ) . If w = ( w , . . . , w n − , n − , n − , n ) then F n ,(cid:96), w =
0, so w = ( w , . . . , w n − , n − , n , n − ) or w = ( w , . . . , w n − , n , n − , n − ) .Therefore w ∈ A . Step d. If w ∈ T n − ,(cid:96) and has descending property then w ∈ A (cid:48) .Next suppose w ∈ T n − ,(cid:96) and w has the descending property. Since w has the descendingproperty we must have t ≥ s − w ∈ A (cid:48) . Step e. If w ∈ T n − ,(cid:96) and does not have descending property then w ∈ A .If w ∈ T n − ,(cid:96) and w does not have the descending property then by Corollary 5.21, w = ( n − , (cid:96), n − , . . . , ) . Since w has the descending property, we must have t ≥ s +
2. Andso we have shown w ∈ A (cid:48) . (cid:3) Lemma 5.19.
We have w = ( n , (cid:96), n − , n − , . . . , ) ∈ T n ,(cid:96) .Proof. Suppose P I P J − P I (cid:48) P J (cid:48) is a relation in in w (cid:96) ( I n ) and P I P J does not vanish in F n ,(cid:96), w .We show that P I (cid:48) P J (cid:48) does not vanish either and hence F n ,(cid:96), w contains no monomials. Write I = { i < · · · < i | I | } and J = { j < · · · < j | J | } and assume without loss of generality that | I | = | I (cid:48) | and | J | = | J (cid:48) | . We take cases on B (cid:96) ( I ) and B (cid:96) ( J ) . In particular, we may assumethat either B (cid:96) ( I ) = B (cid:96) ( J ) = id or B (cid:96) ( I ) (cid:44) id . Case 1.
Let B (cid:96) ( I ) = B (cid:96) ( J ) = id . If | I | = | J | ≥ P i P j , j ,..., j | J | − P j P i , j ,..., j | J | . Since B (cid:96) ( J ) = id then either j , j ∈ { , . . . , (cid:96) } or j , j ∈ { (cid:96) + , . . . , n } . However, if j , j ∈ { (cid:96) + , . . . , n } then it follows that P J vanishes in F n ,(cid:96), w , a contradiction. Suppose j , j ∈ { , . . . , (cid:96) } .The fact that J (cid:48) (cid:60) S w follows from the following two observations:28i) if L ⊆ [ n ] , | L | ≥ L ∈ S w ⇐⇒ L ∈ S v , where v = ( (cid:96), n , n − , . . . , , ) ,(ii) Corollary 5.3 can be applied to S v because v has the descending property.Since j ∈ { , . . . , (cid:96) } we have that P i , j ,..., j | J | does not vanish in F n ,(cid:96), w by Corollary 5.3. It isclear that P j does not vanish and so we have P j P i , j ,..., j | J | − P I (cid:48) P J (cid:48) does not vanish in F n ,(cid:96), w .If | I | , | J | ≥ i , j ≤ (cid:96) because I ≤ { n , (cid:96), n − , . . . } and J ≤ { n , (cid:96), n − , . . . } . Since B (cid:96) ( I ) = B (cid:96) ( J ) = id we must have i , i , j , j ∈ { , . . . , (cid:96) } and so I (cid:48) ∩ { , . . . , (cid:96) } (cid:44) (cid:156) and J (cid:48) ∩ { , . . . , (cid:96) } (cid:44) (cid:156) . It is easy to show that I (cid:48) ≤ { n , (cid:96), n − , . . . } and J (cid:48) ≤ { n , (cid:96), n − . . . } .Therefore P I (cid:48) P J (cid:48) does not vanish in F n ,(cid:96), w . Case 2.
Let B (cid:96) ( I ) (cid:44) id . We have i ∈ { , . . . , (cid:96) } and i ∈ { (cid:96) + , . . . , n } . If | J | = P i , i , i ,..., i | I | P j − P j , i , i ,..., i | I | P i . Note that we have i ∈ I (cid:48) and so P I (cid:48) does not vanish in F n ,(cid:96), w . Therefore P I (cid:48) P J (cid:48) does notvanish in F n ,(cid:96), w .If | I | , | J | ≥
2, suppose B (cid:96) ( J ) (cid:44) id . Since i , j ∈ { , . . . , (cid:96) } appear at the same index in I and J respectively, it is easy to check that I (cid:48) ∩ { , . . . , (cid:96) } (cid:44) (cid:156) and J (cid:48) ∩ { , . . . , (cid:96) } (cid:44) (cid:156) . And so P I (cid:48) P J (cid:48) does not vanish in F n ,(cid:96), w . On the other hand if B (cid:96) ( J ) = id , since J ≤ { n , (cid:96), n − , . . . } ,then j ∈ { , . . . , (cid:96) } . So j ∈ { , . . . , (cid:96) } as well, because B (cid:96) ( J ) = id . We have i , j , j ∈{ , . . . , (cid:96) } , it follows that I (cid:48) ∩ { , . . . , (cid:96) } (cid:44) (cid:156) and J (cid:48) ∩ { , . . . , (cid:96) } (cid:44) (cid:156) . So P I (cid:48) P J (cid:48) does not vanishin F n ,(cid:96), w . (cid:3) Lemma 5.20. If w ∈ T n ,(cid:96) and w (cid:44) ( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , ) then w ∈ S > n .Proof. Let w = ( w , . . . , w n ) ∈ T n ,(cid:96) \{( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , )} and w t = n . Supposeby contradiction there exists k > t such that w k < w k + . Let I = { w , . . . , w k } and I (cid:48) = { w , . . . , w k − , w k + } . Since w k < w k + we have that P I (cid:48) vanishes in F n ,(cid:96), w whereas P I doesnot. Let us write in ascending order I ∪ I (cid:48) = { α, w k , β, w k + , γ, n } for some ordered subsets α, β and γ of [ n ] . Note that I = { α, w k , β, γ, n } and I (cid:48) = { α, β, w k + , γ, n } . Let J = { α, β, w k + , γ } and J (cid:48) = { α, w k , β, γ } and note that both P J and P J (cid:48) do not vanish in F n ,(cid:96), w . We take caseson p = |( α ∪ β ∪ γ ) ∩ { , . . . , k }| . In particular, we either have that p ≥ p = p = B (cid:96) ( I ) which is either theidentity or the transposition ( , ) . Case 1.
Let p ≥
2. We have B (cid:96) ( I ) = B (cid:96) ( J ) = B (cid:96) ( I (cid:48) ) = B (cid:96) ( J (cid:48) ) = id hence P I P J − P I (cid:48) P J (cid:48) isa relation in in w (cid:96) ( I n ) . And so P I P J is a monomial in F n ,(cid:96), w hence w (cid:60) T n ,(cid:96) , a contradiction. Case 2.
Let p =
1. We now consider cases for B (cid:96) ( I ) . Case 2a.
Let B (cid:96) ( I ) = id . It follows that w k ∈ { , . . . , (cid:96) } . If w k + ∈ { , . . . (cid:96) } then B (cid:96) ( I ) = B (cid:96) ( J ) = B (cid:96) ( I (cid:48) ) = B (cid:96) ( J (cid:48) ) = id and so P I P J − P I (cid:48) P J (cid:48) is a relation in in w (cid:96) ( I n ) . Therefore F n ,(cid:96), w contains the monomial P I P J , a contradiction.If w k + ∈ { (cid:96) + , . . . , n } then B (cid:96) ( I (cid:48) ) (cid:44) id . We see that P I P J − P I (cid:48) P J (cid:48) is a valid relation in F n ,(cid:96), w as follows. Let M = { α, w k , β, γ } and N = { α, β, w k + , γ } . The relation can be writtenas P M ∪{ n } P N − P N ∪{ n } P M . | M | = | N | ≥
2, it follows that B (cid:96) ( M ) = B (cid:96) ( M ∪ { n }) and B (cid:96) ( N ) = B (cid:96) ( N ∪ { n }) and sothis is a relation in in w (cid:96) ( I n ) . Hence P I P J is a monomial in F n ,(cid:96), w , a contradiction. Case 2b.
Let B (cid:96) ( I ) (cid:44) id . We have w k ∈ { (cid:96) + , . . . , n } and so w k + ∈ { (cid:96) + , . . . , n } hence B (cid:96) ( I (cid:48) ) (cid:44) id . It follows that B (cid:96) ( J ) (cid:44) id and B (cid:96) ( J (cid:48) ) (cid:44) id . We deduce that P I P J − P I (cid:48) P J (cid:48) is arelation in in w (cid:96) ( I n ) and so P I P J is a monomial in F n ,(cid:96), w , a contradiction. Case 3.
Let p =
0. We consider cases for B (cid:96) ( I ) . Case 3a.
Let B (cid:96) ( I ) = id . So w k ∈ { (cid:96) + , . . . , n } and so w k + ∈ { (cid:96) + , . . . , n } hence B (cid:96) ( I (cid:48) ) = id . It follows that B (cid:96) ( J ) = B (cid:96) ( J (cid:48) ) = id and so P I P J − P I (cid:48) P J (cid:48) is a valid relation inin w (cid:96) ( I n ) . Therefore F n ,(cid:96), w contains the monomial P I P J , a contradiction. Case 3b.
Let B (cid:96) ( I ) (cid:44) id . So w k ∈ { (cid:96) + , . . . , n } . We take cases on w k + . Case 3b.i.
Let w k + ∈ { , . . . , (cid:96) } . So B (cid:96) ( I (cid:48) ) (cid:44) id . If | α ∪ β ∪ γ | > B (cid:96) ( J ) (cid:44) id and B (cid:96) ( J (cid:48) ) (cid:44) id and so P I P J − P I (cid:48) P J (cid:48) is a relation in in w (cid:96) ( I n ) . Therefore P I P J is amonomial in F n ,(cid:96), w , a contradiction.If | α ∪ β ∪ γ | = w = ( n , w , w , . . . , w n ) with k =
2. Consider the relationin in w (cid:96) ( I n ) P n P w w − P w P n w . This is indeed a valid relation which gives rise to a monomial P n P w w in F n ,(cid:96), w , a contradiction. Case 3b.ii.
Let w k + ∈ { (cid:96) + , . . . , n } . So B (cid:96) ( I (cid:48) ) = id . If | α ∪ β ∪ γ | > P I P J − P I (cid:48) P J (cid:48) is a relation in in w (cid:96) ( I n ) where B (cid:96) ( J ) = id and B (cid:96) ( J (cid:48) ) (cid:44) id . So P I P J is a monomial in F n ,(cid:96), w , a contradiction.If | α ∪ β ∪ γ | = w = ( n , w , w , . . . , w n ) with k =
2. Now without loss of generalitywe may assume that w > w > · · · > w n otherwise we may use one of the previous cases.So w = ( n , w , n − , n − , . . . , w + , w − , . . . , ) . Also by assumption we have w (cid:44) (cid:96) so w ≤ (cid:96) −
1. Consider the relation P n P w ,(cid:96) − P w P n ,(cid:96) . Clearly this is a relation in in w (cid:96) ( I n ) . Themonomial P n P w ,(cid:96) does not vanish in F n ,(cid:96), w but P n ,(cid:96) does vanish and so F n ,(cid:96), w is non-binomial,a contradiction. (cid:3) As an immediate corollary of Lemma 5.19 and Lemma 5.20 we have that:
Corollary 5.21. If w ∈ T n ,(cid:96) \ S > n , then w = ( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , ) . Lemma 5.22.
We have A (cid:48) ⊂ T n ,(cid:96) .Proof. Let w ∈ A (cid:48) . Let P I P J − P I (cid:48) P J (cid:48) be a relation in in w (cid:96) ( I n ) where P I P J does not vanishin F n ,(cid:96), w . We show that P I (cid:48) P J (cid:48) does not vanish in F n ,(cid:96), w by taking cases on | I | , | J | and t . Wemay assume that | I | ≥ | J | and so we must either have | I | , | J | < t or | I | ≥ t and | J | < t or | I | , | J | ≥ t . Case 1.
Let | I | , | J | < t . Since w and w agree on w , . . . , w t − , we deduce that P I P J − P I (cid:48) P J (cid:48) is a relation in F n − ,(cid:96), w . Since F n − ,(cid:96), w is binomial, we conclude that P I (cid:48) P J (cid:48) does not vanish in F n ,(cid:96), w . Case 2.
Let | I | ≥ t , | J | < t . Write I = { i , . . . , i | I | } . If B (cid:96) ( I ) = B (cid:96) ( I \{ i | I | }) and B (cid:96) ( I (cid:48) ) = B (cid:96) ( I (cid:48) \{ i | I | }) , then we have the following relation in in w (cid:96) ( I n − ) : P I \{ i | I | } P J − P I (cid:48) \{ i | I | } P J (cid:48) . P I \{ i | I | } P J does not vanish in F n − ,(cid:96), w . Since F n − ,(cid:96), w is binomial we have that P I (cid:48) \{ i | I | } P J (cid:48) does not vanish in F n − ,(cid:96), w . So by Corollary 5.3, P I (cid:48) P J (cid:48) does not vanish in F n ,(cid:96), w .If B (cid:96) ( I ) (cid:44) B (cid:96) ( I \{ i | I | }) or B (cid:96) ( I (cid:48) ) (cid:44) B (cid:96) ( I (cid:48) \{ i | I | }) , then | I | = t = | J | =
1. Wewrite J = { j } . Since w has the descending property and w = n we deduce w = ( w , n , n − , . . . , w + , w − , . . . , ) . First suppose B (cid:96) ( I ) (cid:44) B (cid:96) ( I \{ i | I | }) , so B (cid:96) ( I ) (cid:44) id . The relation isgiven by P i , i P j − P j , i P i . Note that j ≤ w so P j , i does not vanish in F n ,(cid:96), w . We show that P i does not vanish by contradiction. Suppose i > w . We have that w < n − w (cid:44) ( n − , n , n − , . . . , ) . Note that i , j ≤ w since P I P J does not vanish in F n ,(cid:96), w . Considerthe following relation in in w (cid:96) ( I n − ) : P n − , i P j − P j , i P n − . Clearly P n − , i P j does not vanish in F n − ,(cid:96), w however P n − does vanish and so F n − ,(cid:96), w is non-binomial, a contradiction.Secondly suppose B (cid:96) ( I ) = B (cid:96) ( I \{ i | I | }) = id . Then the relation is given by P i , i P j − P j , i P i . Since P i , i P j does not vanish in F n ,(cid:96), w we have i ≤ w and j ≤ w . And so P j , i P i does notvanish. Case 3.
Let | I | , | J | ≥ t . We write I = { i < · · · < i | I | } , J = { j < · · · < j | J | } , I (cid:48) = { i (cid:48) < · · · < i (cid:48)| I | } and J (cid:48) = { j (cid:48) < · · · < j (cid:48)| J | } . Suppose t ≥
2. By assumption w has the descendingproperty and t ≥ s −
1, hence w has the descending property. So by Corollary 5.3 we have { i , . . . , i t } ≤ { w , . . . , w t } and { j , . . . , j t } ≤ { w , . . . , w t } . For each (cid:96) ≥ { i (cid:48) (cid:96) , j (cid:48) (cid:96) } = { i (cid:96) , j (cid:96) } because B (cid:96) ( L ) does not permute any index (cid:96) ≥ B (cid:96) ( L )( (cid:96) ) = (cid:96) for any L ⊆ [ n ] and (cid:96) ≥
3. So { i (cid:48) , i (cid:48) , j (cid:48) , j (cid:48) } = { i , i , j , j } . It follows that { i (cid:48) , . . . , i (cid:48) t } ≤ { w , . . . , w t } and { j (cid:48) , . . . , j (cid:48) t } ≤ { w , . . . , w t } . So by Corollary 5.3, P I (cid:48) P J (cid:48) does not vanish in F n ,(cid:96), w .Suppose t =
1. Since w has the descending property we have w = ( n , n − , . . . , ) . Clearly P I (cid:48) P J (cid:48) does not vanish in F n ,(cid:96), w since F n ,(cid:96), w = in w (cid:96) ( I n ) and no variable vanishes. (cid:3) Lemma 5.23.
We have A ⊂ T n ,(cid:96) .Proof. Suppose w ∈ S n with w ∈ T n − ,(cid:96) and w does not have the descending property. Let w s = n − w t = n and suppose t ≥ s +
2. By Corollary 5.21, w = ( n − , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , ) .We deduce that w has the descending property.Let P I P J − P I (cid:48) P J (cid:48) be a relation in in w (cid:96) ( I n ) and suppose P I P J does not vanish in F n ,(cid:96), w .We show that P I (cid:48) P J (cid:48) does not vanish by taking cases on | I | , | J | and t . We may assume that | I | ≥ | J | and so we must have that either | I | , | J | < t or | I | ≥ t and | J | < t or | I | , | J | ≥ t . Case 1.
Let | I | , | J | < t . Since w and w agree on w , . . . , w t − , P I P J − P I (cid:48) P J (cid:48) is a relationin F n − ,(cid:96), w which is binomial. So P I (cid:48) P J (cid:48) does not vanish in F n ,(cid:96), w . Case 2.
Let | I | ≥ t , | J | < t . We write I = { i , . . . , i | I | } . We show that we cannot have B (cid:96) ( I ) (cid:44) B (cid:96) ( I \{ i | I | }) or B (cid:96) ( I (cid:48) ) (cid:44) B (cid:96) ( I (cid:48) \{ i | I | }) . Otherwise we would have | I | = | I (cid:48) | = t =
2. But by assumption t ≥ s + ≥
3, a contradiction. So we have B (cid:96) ( I ) = B (cid:96) ( I \{ i | I | }) and B (cid:96) ( I (cid:48) ) = B (cid:96) ( I (cid:48) \{ i | I | }) . We have the following relation in F n − ,(cid:96), w : P I \{ i | I | } P J − P I (cid:48) \{ i | I | } P J (cid:48) . Since F n − ,(cid:96), w is binomial we deduce that P I (cid:48) \{ i | I | } P J (cid:48) does not vanish in F n − ,(cid:96), w . So byCorollary 5.3, P I (cid:48) P J (cid:48) does not vanish in F n ,(cid:96), w . Case 3.
Let | I | , | J | > t . Write I = { i < · · · < i | I | } , J = { j < · · · < j | J | } , I (cid:48) = { i (cid:48) < · · · < i (cid:48)| I | } and J (cid:48) = { j (cid:48) < · · · < j (cid:48)| J | } . By assumption, t ≥ w has the descending property. So31y Corollary 5.3, { i , . . . , i t } ≤ { w , . . . , w t } and { j , . . . , j t } ≤ { w , . . . , w t } . For each (cid:96) ≥ { i (cid:48) (cid:96) , j (cid:48) (cid:96) } = { i (cid:96) , j (cid:96) } and { i (cid:48) , i (cid:48) , j (cid:48) , j (cid:48) } = { i , i , j , j } . It follows that { i (cid:48) , . . . , i (cid:48) t } ≤ { w , . . . , w t } and { j (cid:48) , . . . , j (cid:48) t } ≤ { w , . . . , w t } . So by Corollary 5.3, P I (cid:48) P J (cid:48) does not vanish in F n ,(cid:96), w . (cid:3) Lemma 5.24.
Let w ∈ S n with w ∈ N n − ,(cid:96) and w (cid:44) ( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , ) . Then w ∈ N n , n − .Proof. Write w = ( w , . . . , w n ) with w t = n . Suppose P I P J − P I (cid:48) P J (cid:48) is a relation in in w (cid:96) ( I n − ) giving rise to the monomial P I P J ∈ F n − ,(cid:96), w . Without loss of generality we assume | I | = | I (cid:48) | and | J | = | J (cid:48) | . For each L ∈ { I , I (cid:48) , J , J (cid:48) } , let ˜ L = L ∪ { n } . We take cases on | I | , | J | and t . Wemay assume that | I | ≥ | J | and so we must have that either | I | , | J | < t or | I | ≥ t and | J | < t or | I | , | J | ≥ t . Case 1.
Let | I | , | J | < t . Since w and w agree on w , . . . , w t − we have that P I P J is amonomial in F n ,(cid:96), w via the same relation and so w ∈ N n ,(cid:96) . Case 2.
Let | I | ≥ t , | J | < t . We have | I | ≥
2. By Corollary 5.3, P ˜ I does not vanish and P ˜ I (cid:48) P J (cid:48) does vanish in F n ,(cid:96), w . Since | I | ≥ B (cid:96) ( I ) = B (cid:96) ( ˜ I ) and B (cid:96) ( I (cid:48) ) = B (cid:96) ( ˜ I (cid:48) ) . So we have thefollowing relation in in w (cid:96) ( I n ) : P ˜ I P J − P ˜ I (cid:48) P J (cid:48) . Therefore F n ,(cid:96), w contains the monomial P ˜ I P J andso w ∈ N B , w . Case 3.
Let | I | , | J | ≥ t . If | I | , | J | ≥ B (cid:96) ( I ) = B (cid:96) ( ˜ I ) , B (cid:96) ( I (cid:48) ) = B (cid:96) ( ˜ I (cid:48) ) , B (cid:96) ( J ) = B (cid:96) ( ˜ J ) and B (cid:96) ( J (cid:48) ) = B (cid:96) ( ˜ J (cid:48) ) . And so we have the following relation in in w (cid:96) ( I n ) : P ˜ I P ˜ J − P ˜ I (cid:48) P ˜ J (cid:48) . ByCorollary 5.3, P ˜ I P ˜ J does not vanish and P ˜ I (cid:48) P ˜ J (cid:48) does vanish in F n ,(cid:96), w . So F n ,(cid:96), w is non-binomialand w ∈ N n ,(cid:96) .If | I | = | J | = t =
1. So w = ( n , w , . . . , w n ) . Assume by contradiction that w has the descending property. Then w = ( n , n − , . . . , ) so w = ( n − , . . . , ) . Clearly w ∈ T n − ,(cid:96) as no variable vanishes, a contradiction. So w does not have the descending property. Byassumption w (cid:44) ( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , ) so by Corollary 5.21, F n ,(cid:96), w is non-binomial.And so we have shown w ∈ N n ,(cid:96) . (cid:3) Remark . Finally, we would like to remark that the results of this section can be gen-eralized to Richardson varieties [CCM20]. Moreover, for Grassmannian varieties, there areother combinatorial constructions leading to toric degenerations [RW19, BMC20]. Althoughmost of these degenerations can be realized as Gr¨obner degenerations, this is not true ingeneral; See e.g. [KMS15] for a family of toric degnerations that cannot be identified as aGr¨obner degeneration.
6. Standard monomial theory for Schubert varieties
In this section we study monomial bases for the ideals F n ,(cid:96), w and in w (cid:96) ( I ( X ( w ))) . We showfor the diagonal matching field, (cid:96) =
0, that if F n ,(cid:96), w is monomial-free then in w (cid:96) ( I ( X ( w ))) and F n ,(cid:96), w coincide and, moreover, these ideals are toric. We also show that for the otherblock diagonal matching fields, (cid:96) ∈ { , . . . , n − } , the same results hold if the initial idealin w (cid:96) ( I ( X ( w ))) is generated in degree two.We begin by defining the monomial map whose kernel will coincide with in w (cid:96) ( I ( X ( w ))) when F n ,(cid:96), w is monomial-free. 32 efinition 6.1 (Restricted monomial map) . Fix natural numbers n , (cid:96) and let w be a per-mutation in S n . Let R = K [ P I : I ⊆ [ n ] , | I | ∈ { , . . . , n − } , I (cid:60) S w ] and S = K [ x i , j : i ∈{ , . . . , n − } , j ∈ { , . . . , n }] be polynomial rings. We define the map φ (cid:96), w : R → S to be therestriction of the monomial map φ (cid:96) defined in (2.2) to the ring R . Notation.
Fix n , (cid:96) natural numbers and w a permutation. We use the following shorthandnotation for ideals of R throughout this section. • J : = F n ,(cid:96), w , the restricted matching field ideal defined in (2.3). • J : = in w (cid:96) ( I ( X ( w ))) , the initial ideal of the ideal of the Schubert variety. • J : = ker ( φ (cid:96), w ) , the kernel of the restricted monomial map. By studying the generating sets of J and J , we will show that they coincide if and onlyif J is monomial-free and J ⊆ J . To prove the remaining containments, we will considermonomial bases in the subsequent subsection. Recall that the matching field ideal F n ,(cid:96) isquadratically generated and is the kernel of a monomial map, see Corollary 2.11. Startingwith a quadratic generating set for F n ,(cid:96) , we explicitly construct a generating set for F n ,(cid:96), w . Definition 6.2.
Let G ⊂ K [ x , . . . , x n ] be a collection of homogeneous quadratic polynomialsand S ⊆ { x , . . . , x n } be a collection of variables. We identify S with its characteristic vector,i.e. S i = x i ∈ S otherwise S i =
0. For each g ∈ G we write g = (cid:205) α c α x α and defineˆ g = (cid:213) S · α = c α x α . We define G S = { ˆ g : g ∈ G } to be the collection of all such polynomials.By definition, we have F n ,(cid:96), w = ( F n ,(cid:96) + (cid:104) S (cid:105)) ∩ R where R is the ring given in Definition 6.1and S = { P I : I ∈ S w } is the set of variables that vanish in F n ,(cid:96), w . We show that if G is aquadratic generating set for F n ,(cid:96) then G S is a quadratic generating set for F n ,(cid:96), w . Lemma 6.3.
Let G ⊆ K [ x , . . . , x n ] be a set of quadratic polynomials and S ⊆ { x , . . . , x n } asubset of variables. Then (cid:104) G S (cid:105) = (cid:104) G ∪ S (cid:105) ∩ K [{ x , . . . , x n }\ S ] .Proof. To show that G S ∪ S and G ∪ S generate the same ideal, for each g ∈ G we write g = (cid:205) α c α x α , for some c α ∈ K . We have that g − ˆ g = (cid:213) S · α ≥ c α x α . Each term appearing in the above sum is divisible by some variable in S , hence ˆ g ∈ (cid:104) G ∪ S (cid:105) and g ∈ (cid:104) G S ∪ S (cid:105) . For any polynomial f ∈ (cid:104) G ∪ S (cid:105) ∩ K [{ x , . . . , x n }\ S ] we have that33 = (cid:205) g ∈ g c g h g g + (cid:205) x i ∈ S c i h i x i for some c g , c i ∈ K and h g , h i ∈ K [ x , . . . , x n ] . For each h g we define ˆ h g similarly to ˆ g and rewrite this polynomial as f = (cid:213) g ∈ G c g ˆ h g ˆ g + (cid:32)(cid:213) g ∈ G c g ( h g g − ˆ h g ˆ g ) + (cid:213) x i ∈ S c i p i x i (cid:33) . All monomials appearing in (cid:205) g ∈ G c g ˆ h g ˆ g are not divisible by any monomials that lie in S .However each monomial appearing in the expressions (cid:205) g ∈ G c g ( h g g − ˆ h g ˆ g ) and (cid:205) x i ∈ S c i p i x i isdivisible by some x i ∈ S . Since f ∈ K [{ x , . . . , x n }\ S ] it follows that the large bracketedexpression above is zero and so f = (cid:205) g ∈ G c g ˆ h g ˆ g ∈ (cid:104) G S (cid:105) . (cid:3) Using the generating sets of F n ,(cid:96), w constructed above, we now consider the ideals J , J and J . Lemma 6.4.
The ideals J and J coincide if and only if J is monomial-free.Proof. Note that J is the kernel of a monomial map that does not send any variables tozero. Therefore J does not contain any monomials. If J contains a monomial, then J (cid:44) J .For the converse, suppose J does not contain any monomials. Let G be a quadraticgenerating set for F n ,(cid:96) and let S = { P I : I ∈ S w } be the collection of variables that vanish in F n ,(cid:96), w . By definition J = (cid:104) G ∪ S (cid:105) ∩ K [ P I : I (cid:60) S w ] . So by Lemma 6.3 we have J is generatedby G S . Since J is monomial-free, we have that G S does not contain any monomials. ByCorollary 2.11, the ideal F n ,(cid:96) is the kernel of the monomial map φ (cid:96) and by definition J isthe kernel of the restriction φ (cid:96), w . Since all binomials m − m ∈ G S lie in F n ,(cid:96) and containonly the non-vanishing Pl¨ucker variables P J for J (cid:60) S w , therefore m − m ∈ J . And so wehave J ⊆ J . Also, for any polynomial f ∈ J we have that f ∈ F n ,(cid:96) . Since f contains onlythe non-vanishing Pl¨ucker variables, therefore f ∈ J . (cid:3) Lemma 6.5. J ⊆ J .Proof. Let G be a quadratic binomial generating set for F n ,(cid:96) and S = { P I : I ∈ S vw } . Letˆ f ∈ G S ⊂ J be any polynomial. By the definition of G S , there exists f ∈ G such that ˆ f isobtained from f by setting some variables to zero. Recall F n ,(cid:96) = in w (cid:96) ( F n ) , so there exists apolynomial g ∈ F n such that f = in w (cid:96) ( g ) . Since the leading term of g is not set to zero in I ( X ( w )) , it follows that ˆ f ∈ in w (cid:96) ( I ( X ( w ))) . (cid:3) We begin by recalling a description of a collection of standard monomials for the Schubertvariety X ( w ) . Definition 6.6 (Definition V.5. in [Kim15]) . Let T be a semi-standard Young tableauwith columns I , . . . , I t . Let w = ( w , . . . , w t ) be a sequence of permutations and write w k = ( w k , , . . . , w k , n ) ∈ S n for each k ∈ [ t ] . We say that w is a defining chain for T if thepermutations are monotonically increasing w ≤ w ≤ · · · ≤ w t with respect to the Bruhatorder and for each k ∈ [ t ] we have I k = { w k , , . . . , w k , | I k | } .34et w = ( w , . . . , w t ) and v = ( v , . . . , v t ) be two defining chains for a fixed semi-standardYoung tableau. There is a natural partial order on the set of defining chains given by v ≤ w if v k ≤ w k for all k ∈ [ t ] . It turns out there exists a unique minimum defining chain.The standard monomials for Schubert varieties can be determined by the minimum definingchain. In the following theorem we summarise these results. Theorem 6.7 (Lemma V.9, Proposition V.13 and Theorem V.14 in [Kim15]) . Let w be apermutation. The collection of monomials corresponding to tableau T with d columns suchthat w − d ≤ w forms a monomial basis for X ( w ) , where w − ( T ) = ( w − , . . . , w − d ) is the uniqueminimum defining chain for T . If the tableau is not clear from the context, we write w − i ( T ) for w − i , where i ∈ { , . . . , d } .In this section we will show that the following. Theorem 6.8. If w is -free then the semi-standard Young tableaux whose columns I satisfy I ≤ w form a monomial basis for the Schubert variety X ( w ) .Proof. We show that each semi-standard Young tableau T is standard for the Schubertvariety X ( w ) . Note that the ideal of the Schubert variety X ( w ) is generated in degree two soit suffices to check all tableaux with at most two columns. If T has a single column I thenthe minimum defining sequence for T has a single permutation v which is the Grassmannianpermutation defined by I . Hence v ≤ w and T is standard for X ( w ) . By Lemma 6.13, allsemi-standard Young tableaux with two columns are standard for X ( w ) and we are done. (cid:3) We prove Theorem 6.8 in two steps. We begin with tableaux that have exactly onecolumn of size one. We then use this to show the general case.
Notation.
Let Q = ( Q , Q , . . . , Q k ) be a partition of [ n ] where each Q i ⊆ [ n ] is non-empty and disjoint. Write Q i = { Q i , < q i , < · · · < q i , | Q i | } for each i ∈ [ k ] . We define thepermutation ( Q , Q , . . . , Q k ) = ( q , , q , , . . . , q , | Q | , q , , . . . , q , | Q | , q , , . . . , q k , | Q k | ) . In particular if I ⊆ [ n ] is a subset then ( I , [ n ]\ I ) is the Grassmannian permutation definedby I . Lemma 6.9.
Let T be a semi-standard Young tableau with two columns I = { i < · · · < i t } and J = { j } . The minimum defining sequence for T is ( v , v ) where v = ( I , [ n ]\ I ) , v = ( j , I \ i s , ([ n ]\( I ∪ j )) ∪ i s ) and s = max { k ∈ [ t ] : i k ≤ j } .Proof. Let ( w , w ) be a defining sequence for T . By definition of defining sequence we have w ≥ v = ( I , [ n ]\ I ) . Since v ≤ w ≤ w , therefore the smallest possible permutation for w with respect to the Bruhat order is v . (cid:3) Lemma 6.10.
Let Q , Q , Q be a partition of [ n ] . Let w be a permutation such that ( Q , Q , Q ) (cid:2) w . If Q ≤ w then Q ∪ Q (cid:2) w . roof. If Q ≤ w and Q ∪ P ≤ w then it follows that ( Q , Q , Q ) ≤ w . (cid:3) Lemma 6.11.
Let T be a semi-standard Young tableau with two columns I and J . Let ( v , v ) be the minimum defining sequence for T . We have v = ( I , [ n ]\ I ) and v = ( J , ˜ I , [ n ]\( J ∪ ˜ I )) for some subset ˜ I ⊆ I .Proof. It is clear that ( I , [ n ]\ I ) is the smallest permutation u = ( u , . . . , u n ) such that { u , . . . , u | I | } = I . And so v is the smallest permutation u = ( u , . . . , u n ) such that v ≤ u and { u , . . . , u | J | } = J . It follows easily that the smallest such permutation has the form ( J , ˜ I , [ n ]\( J ∪ ˜ I )) for some˜ I ⊆ I . (cid:3) Lemma 6.12.
Suppose T is a semi-standard Young tableau with columns I and J where | J | = . If w is -free and I , J ≤ w then T is standard for X ( w ) .Proof. Write I = { i < · · · < i t } and J = { j } for the columns of T and write w = ( w , . . . , w n ) for the permutation. We define ( (cid:98) w , . . . , (cid:98) w t ) = { w , . . . , w t } ↑ . By assumption we have i k ≤ (cid:98) w k for each k ∈ [ t ] and j ≤ w . Let w − = ( v , v ) be the minimum defining sequence for T .By Lemma 6.9 we have that v = ( j , I \ i s , ([ n ]\ I ) ∪ i s ) where s = max { k ∈ [ t ] : i s ≤ j } .If s = t then v = (( I \ i t ) ∪ j , [ n ]\(( I \ i t ) ∪ j )) ≤ w and we are done. Suppose that s < t and, in this case, we assume by contradiction that v (cid:2) w . By Lemma 6.10, we have that ( I \ i s ) ∪ j = { i < . . . i s − < j < i s + < · · · < i t } (cid:2) w . Since i k ≤ (cid:98) w k for each k ∈ [ t ] , it followsthat j > (cid:98) w s . Since s < t , therefore (cid:98) w s < j < i s + ≤ (cid:98) w s + . And so there exist unique values p ∈ { , . . . , t } and q ∈ { t + , . . . , n } such that w p = (cid:98) w s and w q = j . And so w , w p , w q hastype 312 a contradiction. (cid:3) Lemma 6.13.
Suppose T is a semi-standard Young tableau with two columns I and J . If w is -free and I , J ≤ w then T is standard for X ( w ) .Proof. Write I = { i < · · · < i t } and J = { j < · · · < j s } . We assume that I is the leftmostcolumn of T and so s ≤ t . Let w − = ( v , v ) be the minimum defining sequence for T . If s = t then we have that v = ( I , [ n ]\ I ) and v = ( J , [ n ]\ J ) . In particular we have v ≤ w and so T is standard for X ( w ) . So from now on, we assume that s < t .We proceed by induction on s . Note that if s = s >
1. Without loss of generality we may assume that w < · · · < w s . Let usassume by contradiction that v (cid:2) w . We define T (cid:48) to be the tableau with columns I = { i < · · · < i t } and J (cid:48) = { j < · · · < j s − } . We have J (cid:48) ≤ w and so by induction T (cid:48) is a standardtableau for X ( w ) . Let ( v (cid:48) , v (cid:48) ) be the minimum defining sequence for T (cid:48) . By Lemma 6.11 wehave v (cid:48) = ( J (cid:48) , ˜ I (cid:48) , [ n ]\( J (cid:48) ∪ ˜ I (cid:48) )) for some subset ˜ I (cid:48) ⊆ I . Write ˜ I (cid:48) = { r < r < · · · < r t − s + } .Let p = max { k ∈ [ t − s + ] : r k ≤ j s } . By Lemma 6.11 we have v = ( J , ˜ I , [ n ]\( J ∪ ˜ I )) forsome ˜ I ⊆ I . It is easy to show that ˜ I = ˜ I (cid:48) \ r p . By assumption J ≤ w and v (cid:2) w and so byLemma 6.10 we have J ∪ ˜ I (cid:2) w . Since v (cid:48) ≤ w , we have J (cid:48) ∪ ˜ I (cid:48) ≤ w . Write J (cid:48) ∪ ˜ I (cid:48) = { u < · · · < u q − < u q = r p < u q + < · · · < u t } , J ∪ ˜ I = { u < · · · < u q − < j s < u q + < · · · < u t } , { w , . . . , w s } ↑ = { (cid:101) w < · · · < (cid:101) w s } and { w , . . . , w t } ↑ = { (cid:98) w < · · · < (cid:98) w t } . J ≤ w we have that j s ≤ (cid:101) w s . Since J (cid:48) ∪ ˜ I (cid:48) ≤ w we have that u k ≤ (cid:98) w k for all k ∈ [ t ] . Since J ∪ ˜ I (cid:2) w we must have that (cid:98) w q < j s . Since j s ≤ (cid:101) w s therefore q < t and so (cid:98) w q < j s < u q + ≤ (cid:98) w q + . If w k ≥ j s for all k ∈ { s + , . . . , t } then |{ k ∈ [ t ] : w k < j s }| = |{ k ∈ [ s ] : w k < j s }| < s because w s ≥ j s . However we have j < · · · < j s and so q ≥ s . Since (cid:98) w < · · · < (cid:98) w q < j s so |{ k ∈ [ t ] : w k < j s }| ≥ s , a contradiction. Therefore there exists a ∈ { s + , . . . , t } such that w a < j s . Since (cid:98) w q < j s < u q + ≤ (cid:98) w q + , there exists b ∈ { t + , . . . , n } such that w b = j s . Notethat j s < (cid:98) w q + ≤ w s . And so w s , w a , w b is a subsequence of type 312 in w , a contradiction. In this section we prove Theorem A by considering a collection of standard monomials forthe ideals F n ,(cid:96), w and in w (cid:96) ( I ( X ( w ))) . We begin by stating the proof that relies on Lemma 6.16,which we show following a proof of Theorem 3.13. Proof of Theorem A.
Suppose that J is monomial free. We will show that J = J = J andin particular we have that J = J , hence the initial ideal of the Schubert variety is toric.By Lemmas 6.4 and 6.5 we have J = J ⊆ J . So for all d ≥
1, any collection of standardmonomials for J of degree d is linearly independent in R / J . Since J = in w (cid:96) ( I ( X ( w ))) is aninitial ideal of a homogeneous ideal, the number of standard monomials of degree d coincideswith the number of standard monomials of degree d of I ( X ( w )) . By Theorem 6.8, the semi-standard Young tableaux with d -columns, such that each column I satisfies I ≤ w , are incanonical bijection with a collection of standard monomials of I ( X ( w )) of degree d .Suppose (cid:96) =
0. We have that two monomials are equal in R / J if and only if theircorresponding tableaux are row-wise equal. Therefore, the semi-standard Young tableauxare in bijection with standard monomials for J . And so we have J = J = J .Suppose that J is generated in degree two. By Lemma 6.16 we have that J and J havethe same number of standard monomials in degree two. This, together with the fact that J and J are both generated in degree two and J ⊆ J , implies that J = J . (cid:3) We now give an alternative description of the permutations w ∈ T n ,(cid:96) ∪ Z n such that J is monomial-free. We will write this set P (cid:96) and prove that P (cid:96) = T n ,(cid:96) ∪ Z n . In the proofs tofollow, we write w ∈ P (cid:96) to indicate that w satisfies Definition 3.12. Remark . Recall the definition of the collection of permutations P (cid:96) . Most permutationsin this set are 312-free. If w ∈ S n is 312-free then w has the descending property. Moreoverall restrictions of w also have the descending property. It is easy to show that a permutation w is 312-free if and only if all restrictions of w have the descending property. Remark . The set P n is defined to be the collection of 312-free permutations. It isstraightforward to show that w ∈ S n is 312-free if and only if it satisfies the two bulletpointed conditions in Definition 3.12 where (cid:96) = n . Proof of Theorem 3.13.
Throughout this proof we use Theorem C which shows the following. • T n , n = A ∪ A , • T n , n − = A ∪ ˜ A ∪ ˜ A , 37 T n ,(cid:96) = A ∪ A (cid:48) ∪ A ∪ {( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , )} .We show that P (cid:96) = T n ,(cid:96) ∪ Z n . We will write P n (cid:96) ⊆ S n for the set P (cid:96) to distinguish n . Given apermutation w ∈ S n we write w = ( w , . . . , w n ) and write w = ( w , . . . , w n − ) for the restrictionof w to [ n − ] .We show P n (cid:96) = T n ,(cid:96) ∪ Z n by induction on n . For the base case we consider n =
3. Thepermutations in P (cid:96) , T ,(cid:96) and Z are shown below. (cid:96) P (cid:96) T ,(cid:96) Z , , , ,
321 231 ,
321 123 , , , , , ,
321 312 ,
321 123 , , , , ,
321 321 123 , , P (cid:96) = T ,(cid:96) ∪ Z .Let n ≥ P n − (cid:96) = T n − ,(cid:96) ∪ Z n − . We take cases on values of (cid:96) . Case 1.
Assume (cid:96) = n . Let w ∈ P nn . It follows that w is 312-free and so w ∈ P n − n − = T n − , n − ∪ Z n − . Note that w is 312-free and so has the descending property. • If w ∈ Z n − then w ∈ Z n . By definition of Z n we have that either w = ( . . . , n − ) or w = ( . . . , n − , n − ) . Note that w has the descending property. If w = ( . . . , n − ) then w = ( . . . , n − , n ) or w = ( . . . , n , n − ) and so w ∈ Z n . If w = ( . . . , n − , n − ) then either w = ( . . . , n − , n − , n ) in which case w ∈ Z n , or w = ( . . . , n − , n , n − ) or ( . . . , n , n − , n − ) in which case w ∈ A and so w ∈ T n , n . • If w ∈ T n − , n − then w ∈ T n , n . Since w is 312-free, w has the descending property andso w ∈ A hence w ∈ T n , n .Conversely take w ∈ T n , n ∪ Z n . If w ∈ Z n then it is easy to check that w is 312-free and so w ∈ P nn . Suppose w ∈ T n , n = A ∪ A . • If w ∈ A then we have w ∈ Z n and so w is 312-free since w ∈ Z n − . So it suffices toshow that w contains no 312-type subsets of the form w i , w j , w k where i < j < k and w i = n . However by construction, if w ∈ A then either w = ( . . . , n , n − , n − ) or w = ( . . . , n − , n , n − ) . And so w is 312-free. • If w ∈ A then we have that w ∈ T n , n and so by induction w ∈ T n − , n − ⊆ P n − n − is312-free. Let t , s ∈ { , . . . , n } such that w s = n − w t = n . Since w is 312-free,it suffices to show that w contains no 312-type subsets of the form w t , w j , w k where t < j < k . Suppose there exists such a subset. Since w ∈ A we have t ≥ s − w s , w j , w k is also of type 312 but lies in w , a contradiction. Therefore so w is 312-free.And so we have shown that w ∈ P nn . Case 2.
Assume 1 ≤ (cid:96) < n −
1. Let w ∈ P n (cid:96) . Suppose that w is 312-free. Then w is312-free and if w | m = ( m − , m , m − , . . . , ) for some 3 ≤ m ≤ n then we have w | m = w | m so w < w ≤ (cid:96) < n so w = w < w = w ≤ (cid:96) . Hence w ∈ P n − (cid:96) = T n − ,(cid:96) ∪ Z n .38 If w ∈ Z n − then, similarly to the diagonal case, we have w ∈ Z n ∪ A ⊆ T n ,(cid:96) ∪ Z n . • If w ∈ T n − ,(cid:96) then w ∈ T n ,(cid:96) . Since w is 312-free, w has the descending property and so w ∈ A . By definition we have ( n − , n , n − , . . . , ) (cid:60) P n (cid:96) since (cid:96) < n . So w ∈ A (cid:48) ⊆ T n ,(cid:96) .Suppose w is not 312-free. By definition we have w > w = (cid:96) . • If w = n then by Lemma 6.23 we have w = ( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , ) and so w ∈ T n ,(cid:96) . • If w = n − w = ( n − , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , ) ∈ T n − ,(cid:96) .Let t ∈ { , . . . , n } such that w t = n . Since w \ (cid:96) is 312-free it follows that t ≥
3. Hence w ∈ A ⊆ T n ,(cid:96) . • If w < n − w \ (cid:96) is 312-free it follows that w has the descending property. Let s , t ∈ { , . . . , n } such that w s = n − w t = n . If t < s − i ∈ { t + , . . . , s − } .We have w t , w i , w s is of type 312 that lies in w \ (cid:96) , a contradiction. Therefore t ≥ s − w ∈ A hence w ∈ T n ,(cid:96) .And so we have shown that w ∈ T n ,(cid:96) ∪ Z n .Conversely let w ∈ T n ,(cid:96) ∪ Z n . If w ∈ Z n ∪ A then it is straightforward to check that w is 312-free and for all 3 ≤ m ≤ n we have w | m (cid:44) ( m − , m , m − , . . . , ) . Also note that ( n , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , ) ∈ P n (cid:96) . So let w ∈ T n ,(cid:96) = A (cid:48) ∪ A . • If w ∈ A (cid:48) then we have w ∈ T n − ,(cid:96) ⊆ P n − (cid:96) . Suppose w is 312-free. Since w has thedescending property we have w is also 312-free. If, for some 3 ≤ m ≤ n , we have w | m = ( m − , m , m − , . . . , ) then ( n − , n , n − , . . . , ) (cid:60) A (cid:48) hence m < n . Therefore w | m = w | m . Since w ∈ P n − (cid:96) , we have w = w < w = w ≤ (cid:96) . Since w ≤ (cid:96) < n − w | w = w | w . And so w ∈ P n (cid:96) .Suppose w is not 312-free then by definition of P n − (cid:96) we have w > w = (cid:96) . Since w ∈ A (cid:48) we have that w has the descending property. If w = n − w = ( n − , n − , . . . , ) , which is 312-free, a contradiction. So w < n −
1. Let s , t ∈ { , . . . , n } such that w s = n − w t = n . Since w , w < n − s ≥
3. Since P n − (cid:96) , we have w \ (cid:96) is 312-free. By definition of A (cid:48) we have t ≥ s − w \ (cid:96) is 312-free, otherwise if w t , w i , w j is of type 312 then so is w s w i w j . Hence w ∈ P n (cid:96) . • If w ∈ A then we have w ∈ T n − ,(cid:96) ⊆ P n (cid:96) . Also w does not have the descending propertyand so w is not 312-free. By definition of P n − (cid:96) we have w > w = (cid:96) and w \ (cid:96) is312-free. If w (cid:44) n − w \ (cid:96) is 312-free and w = (cid:96) , it follows that w has thedescending property, a contradiction. Therefore w = n −
1. By Lemma 6.23 we have w = ( n − , (cid:96), n − , . . . , (cid:96) + , (cid:96) − , . . . , ) . Let w t = n for some t . By definition of A wehave t ≥ w = w and w = w = (cid:96) . And so have shown that w is not 312-freeand w > w = (cid:96) . We have w \ (cid:96) = ( n − , n − , . . . , ) and so for any t ≥
3, it followsthat w \ (cid:96) is 312-free. Hence w ∈ P n (cid:96) . 39nd so we have shown that w ∈ P n (cid:96) . Case 3.
Assume (cid:96) = n −
1. Let w ∈ P nn − . Suppose that w is not 312-free. Then wehave w > w = (cid:96) = n −
1. Therefore w = n . Note that w \ (cid:96) is 312-free and so w \ (cid:96) has thedescending property. Therefore w \ (cid:96) = ( n , n − , . . . , ) and so w = ( n , n − , . . . , ) . However w is 312-free a contradiction. Therefore w is 312-free. It follows that w is 312-free and so w ∈ P n − n − = T n − , n − ∪ Z n − . • If w ∈ Z n − then, similarly to the diagonal case, we have w ∈ Z n ∪ A ⊆ T n , n − ∪ Z n • If w ∈ T n − , n − then w ∈ T n , n − . Since w is 312-free, w has the descending property.Also w has the descending property so if w s = n − w t = n then t ≥ s −
1, otherwisethere exists i ∈ { t + , . . . , s − } and so w t , w i , w s has type 312. Hence w ∈ A . Bydefinition it follows that ( n − , n , n − , . . . , ) (cid:60) P nn − and so w ∈ A (cid:48) .Suppose that w ∈ A (cid:48) \ T n , n − . We will show that w ∈ ˜ A as follows. By assumptionwe have w ∈ ( T n − , n − \ T n − , n − ) . By induction we have P n − n − = T n − , n − ∪ Z n − and P n − n − = T n − , n − ∪ Z n − . And so we have w ∈ P n − n − \ P n − n − . By assumption w is 312-free,so suppose w | m = ( m − , m , m − , . . . , ) for some 3 ≤ m ≤ n −
1. Since w (cid:60) P n − n − itfollows that m = n − w = ( n − , n − , n − , . . . , ) . Let w t = n for some t . Since w is 312-free it follows that t ≥
2, otherwise if t = w , w , w has type 312. If t = w = ( n − , n , n − , n − , . . . , ) (cid:60) P nn − . So we must have t ≥
3. Hence w ∈ ˜ A .And so we have shown that w ∈ Z n ∪ A ∪ ˜ A ∪ ˜ A = T n , n − ∪ Z n .Conversely let w ∈ T n , n − ∪ Z n . If w ∈ Z n ∪ A then it is straightforward to check that w is 312-free and for all 3 ≤ m ≤ n we have w | m (cid:44) ( m − , m , m − , . . . , ) . Let w ∈ ˜ A ∪ ˜ A . • If w ∈ ˜ A then w ∈ T n − , n − ⊆ P n − n − so w is 312-free. Since w ∈ A we have that w has thedescending property and so w is also 312-free. Suppose that w | m = ( m − , m , m − , . . . , ) for some 3 ≤ m ≤ n . Since w ∈ A (cid:48) we have w (cid:44) ( n − , n , n − , . . . , ) so m ≤ n − w | m = w | m . Since w ∈ P n − n − it follows that w < w ≤ n − w | w = ( w , w , w − , . . . , w + , w − , . . . , ) . Let w t = n for some t . Since w ≤ n − t ≥
3. And so w = w , w = w and w | w = w | w . And so we have shownthat w ∈ P nn − . • If w ∈ ˜ A then w ∈ T n − , n − \ T n − , n − = P n − n − \ P n − n − so w is 312-free. Suppose w | m = ( m − , m , m − , . . . , ) for some 3 ≤ m ≤ n −
1. Since w (cid:60) P n − n − it follows that m = n − w = ( n − , n − , n − , . . . , ) . Let w t = n for some t . Since w ∈ ˜ A , we have that t ≥
2. And so w = w = n − w = w = n −
1. And so we have w ∈ P nn − .Using the particularly nice description of P (cid:96) we can now prove the following for w ∈ P (cid:96) . Lemma 6.16.
The number of standard monomials for J in degree two is | SSYT ( w )| .
40o prove this, we construct a bijection between the semi-standard Young tableaux
SSYT d ( w ) to some matching field tableaux whose image forms a monomial basis for J . Definition 6.17.
We define Γ (cid:96) be the map taking the SSYT tableau T , with columns I , J ,often written T = [ I J ] , to the matching field tableau T (cid:48) = [ I (cid:48) J (cid:48) ] , for B (cid:96) , by the followingcases. Write I = { i < i < · · · < i t } and J = { j < j < · · · < j s } where t ≥ s . • If 2 ≤ s ≤ t then T (cid:48) is obtained by applying Γ (cid:96) from the Grassmannian case to therectangular part of T : { i , . . . , i s ; j , . . . , j s } , and fix the other entries. • If s = t ≥ – If i ∈ { , . . . , (cid:96) } , i , i , . . . , i t , j ∈ { (cid:96) + , . . . , n } and i < j . If t ≥ j < i . Then we define I (cid:48) = { i , j , i , . . . , i t } and J (cid:48) = { i } . – If i ∈ { , . . . , (cid:96) } , i , i , . . . , i t , j ∈ { (cid:96) + , . . . , n } and j < i then we define I (cid:48) = { j , i , i , . . . , i t } and J (cid:48) = { i } . – Otherwise we define I (cid:48) = I and J (cid:48) = J . • If s = t = T (cid:48) = T . Example 6.18.
If a semi-standard Young tableau has at least two rows in each column thenwe use the definition of the Γ (cid:96) from the Grassmannian case. For example Γ (cid:169)(cid:173)(cid:171) (cid:170)(cid:174)(cid:172) = , Γ (cid:169)(cid:173)(cid:171) (cid:170)(cid:174)(cid:172) = . If a semi-standard Young tableau has exactly one column which with a single row then wecheck the entries in the first two or three rows to determine the image of Γ (cid:96) . For example Γ (cid:169)(cid:173)(cid:171) (cid:170)(cid:174)(cid:172) = , Γ (cid:169)(cid:173)(cid:171) (cid:170)(cid:174)(cid:172) = , Γ (cid:169)(cid:173)(cid:171) (cid:170)(cid:174)(cid:172) = . Lemma 6.19.
Let T , T (cid:48) be semi-standard Young tableaux. If Γ (cid:96) ( T ) and Γ (cid:96) ( T (cid:48) ) are row-wiseequal then T = T (cid:48) .Proof. Let us write T = [ I J ] and T (cid:48) = [ I (cid:48) J (cid:48) ] where I = { i , . . . , i t } , J = { j , . . . , j s } and s ≤ t . We note that Γ (cid:96) does not alter the shape of the tableau and keeps the row-wisecontents of the third row and all rows below. Therefore we write I (cid:48) = { i (cid:48) , i (cid:48) , i , . . . , i t } and J (cid:48) = { j (cid:48) , j (cid:48) , j , . . . , j s } . We also note that Γ (cid:96) , does not change the contents of the tableau asa multi-set so we have { i , i , j , j } = { i (cid:48) , i (cid:48) , j (cid:48) , j (cid:48) } . If s = | J | ≥ ( , n ) case. So let us assume that s =
1. We proceed by taking cases on r = |{ i , i , j } ∩ { , . . . , (cid:96) }| ∈ [ ] . Case 1.
Assume r = Γ (cid:96) ( T ) = T and Γ (cid:96) ( T (cid:48) ) = T (cid:48) . So T and T (cid:48) are row-wise equal semi-standard Young tableau, hence T = T (cid:48) .41 ase 2. Assume r =
1. It follows that i = i (cid:48) ∈ { , . . . , (cid:96) } . Let us assume, by contradic-tion, that T (cid:44) T (cid:48) . Since { i , i , j } = { i (cid:48) , i (cid:48) , j } and i = i (cid:48) , we may assume without loss ofgenerality that i = j (cid:48) < j = i (cid:48) . By definition of Γ (cid:96) we have Γ (cid:96) ( T ) = Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:171) i j i ... (cid:170)(cid:174)(cid:174)(cid:172) = j i i ... , Γ (cid:96) ( T (cid:48) ) = Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:171) i i j ... (cid:170)(cid:174)(cid:174)(cid:172) = i i j ... . However Γ (cid:96) ( T ) and Γ (cid:96) ( T (cid:48) ) are not row-wise equal, a contradiction. Case 3.
Assume r =
2. Since T and T (cid:48) are semi-standard Young tableau and { i , i , j } = { i (cid:48) , i (cid:48) , j (cid:48) } , we have that i = i (cid:48) ∈ { , . . . , (cid:96) } . Assume, by contradiction, that T (cid:44) T (cid:48) . Withoutloss of generality we have that i = j (cid:48) ∈ { , . . . , (cid:96) } and j = i (cid:48) ∈ { (cid:96) + , . . . , n } . Therefore Γ (cid:96) ( T ) = Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:171) i j i ... (cid:170)(cid:174)(cid:174)(cid:172) = i j i ... , Γ (cid:96) ( T (cid:48) ) = Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:171) i i j ... (cid:170)(cid:174)(cid:174)(cid:172) = j i i ... . However Γ (cid:96) ( T ) and Γ (cid:96) ( T (cid:48) ) are not row-wise equal, a contradiction. (cid:3) Lemma 6.20.
Let T be a matching field tableau for B (cid:96) with two columns. Then there existsa semi-standard Young tableau T (cid:48) such that T and Γ (cid:96) ( T (cid:48) ) are row-wise equal.Proof. Write T = [ I J ] a matching field tableau for B (cid:96) with columns I = { i , . . . , i t } and J = { j , . . . , j s } where s ≤ t .If 2 ≤ s ≤ t then consider the tableau T = [ I J ] where I = { i , . . . , i s } and J = { j , . . . , j s } .By the Grassmannian case, there exists a semi-standard Young tableau T (cid:48) = [ I (cid:48) J (cid:48) ] such that Γ (cid:96) ( T (cid:48) ) = T . Let T (cid:48) = [ I (cid:48) J (cid:48) ] be the tableau with I (cid:48) = { i (cid:48) , . . . , i (cid:48) s , i s + , . . . , i t } and J (cid:48) = J (cid:48) . Itfollows that T (cid:48) is a semi-standard Young tableau and Γ (cid:96) ( S ) = T .Let us now assume that | J | =
1. If | I | = | I | ≥
2. We proceed by taking cases on r = |{ i , i , j } ∩ { , . . . , (cid:96) }| . We count elements withmultiplicity in case j = i or j = i . Case 1.
Assume that r = T toput the tableau in semi-standard form, call this tableau T (cid:48) . In this case we have Γ (cid:96) ( T (cid:48) ) = T (cid:48) . Case 2.
Assume that r =
1. If i ∈ { , . . . , (cid:96) } let α = min { i , j } and β = max { i , j } .Then we have Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:171) i αβ... (cid:170)(cid:174)(cid:174)(cid:172) = β α i ... or α β i ... If j ∈ { , . . . , (cid:96) } then we have Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:171) j i i ... (cid:170)(cid:174)(cid:174)(cid:172) = i j i ... . ase 3. Assume r =
2. If i , i ∈ { , . . . , (cid:96) } then Γ (cid:96) ( T ) = T . If i , j ∈ { , . . . , (cid:96) } then Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:171) i j i ... (cid:170)(cid:174)(cid:174)(cid:172) = i j i ... . (cid:3) Lemma 6.21.
Let w ∈ S n be any permutation and T = [ I J ] be a semi-standard Young tableauwith two columns. Write T (cid:48) = [ I (cid:48) J (cid:48) ] = Γ (cid:96) ( T ) . If I (cid:48) , J (cid:48) ≤ w then I , J ≤ w .Proof. We write I = { i < · · · < i t } and J = { j < · · · < j s } for the columns of T . If T and T (cid:48) have the same column-wise contents then the result holds trivially. So let us assume T and T (cid:48) have different column-wise contents. If | I | = | J | then we are done by the Grassmanniancase. And so we assume s < t and we take cases on the contents of the first two rows of I , J . Case 1.
Assume 2 ≤ s , j < i , i ∈ { , . . . , (cid:96) } and i , j , j ∈ { (cid:96) + , . . . , n } . So thetableaux are Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) I Ji j i j ... ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j j i i ... ... . Since I (cid:48) ≤ w we have j ≤ (cid:98) w and i ≤ (cid:98) w . Since J (cid:48) ≤ w we have i ≤ (cid:101) w and j ≤ (cid:101) w . Sowe have that i < j ≤ (cid:98) w and i ≤ (cid:98) w hence I ≤ w . We also have that j ≤ (cid:98) w ≤ (cid:101) w and j ≤ (cid:101) w , hence J ≤ w . Case 2.
Assume 2 ≤ s , j < i , i , i , j ∈ { , . . . , (cid:96) } and j ∈ { (cid:96) + , . . . , n } . So thetableaux are Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) I Ji j i j ... ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j j i i ... ... . The result follows by the same argument as Case 1.
Case 3.
Assume s = , t ≥ j < i , i ∈ { , . . . , (cid:96) } and j , i ∈ { (cid:96) + , . . . , n } . Thetableaux are Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) I Ji j i ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j i i ... . Since J (cid:48) ≤ w , we have i ≤ w . Since I (cid:48) ≤ w , we have j ≤ (cid:98) w and i ≤ (cid:98) w . And so we have i < j ≤ (cid:98) w and i ≤ (cid:98) w , hence I ≤ w . We also have j ≤ (cid:98) w ≤ w , hence J ≤ w .43 ase 4. Assume s = , t ≥ i < j , i ∈ { , . . . , (cid:96) } and i , j ∈ { (cid:96) + , . . . , n } . If t ≥ j < i . The tableaux are Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) I Ji j i ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j i i ... . Since I (cid:48) ≤ w we have i ≤ (cid:98) w and j ≤ (cid:98) w . Since J (cid:48) ≤ w we have i ≤ w . So i ≤ (cid:98) w and i < j ≤ (cid:98) w hence I ≤ w . We also have j ≤ (cid:98) w ≤ w hence J ≤ w . (cid:3) We now give some important properties for permutations w ∈ P (cid:96) . Lemma 6.22.
Let w ∈ P (cid:96) . If i < j < k and w i , w j , w k is of type , then w i = w and w j = (cid:96) .Proof. By definition w \ (cid:96) is 312-free, so we have that (cid:96) ∈ { w i , w j , w k } . Also by definition, wehave w > w = (cid:96) . So w k (cid:44) (cid:96) because k ≥
3. We have w i (cid:44) (cid:96) otherwise w , w j , w k would beof type 312 lying in w \ (cid:96) . So we have w j = (cid:96) = w . Since i < j =
2, we have w i = w . (cid:3) Lemma 6.23. If w ∈ P (cid:96) and w is not -free then w | w = ( w , (cid:96), w − , . . . , (cid:96) + , (cid:96) − , . . . , ) .Proof. Let 3 ≤ i < j ≤ n . If w i , w j < w then we have w i > w j otherwise w , w i , w j is a 312in w that also appears in w \ (cid:96) . So w | w = ( w , (cid:96), w − , . . . , (cid:96) + , (cid:96) − , . . . , ) . (cid:3) Lemma 6.24. If w ∈ P (cid:96) and w | m = ( m − , m , m − , . . . , ) for some ≤ m ≤ n then w = m − .Proof. By definition w < w ≤ (cid:96) and w | w = ( w , w , w − , . . . , w + , w − , . . . , ) . If m < w then w | m = ( w | w )| m = ( m , m − , . . . , ) (cid:44) ( m − , m , m − , . . . , ) , a contradiction. So m ≥ w . Therefore w | m = ( w , . . . ) = ( m − , . . . ) hence w = m − (cid:3) Lemma 6.25.
Let w ∈ P (cid:96) . If there exists ≤ m ≤ n such that w | m = ( m − , m , . . . ) theneither m − = w < w ≤ (cid:96) or w > w = (cid:96) = m − .Proof. If w is 312-free then write w | m = ( w , w , . . . , w m ) = ( m − , m , . . . ) . For any 3 ≤ i < j ≤ n we have w i > w j , otherwise w , w i , w j is of type 312. Hence w | m = ( m − , m , m − , . . . , ) so by definition of P (cid:96) we have w < w ≤ (cid:96) . And by Lemma 6.24 we have w = m − w is not 312-free then write w = ( w , . . . , w n ) . By definition of P (cid:96) , we have w > w = (cid:96) .If m ≥ w then we have w | m = ( w , (cid:96), . . . ) (cid:44) ( m − , m , . . . ) , a contradiction. If m < w then by Lemma 6.23 we either have w | m = ( m , m − , . . . ) (cid:44) ( m − , m , . . . ) a contradiction or w | m = ( (cid:96), m , m − , . . . , (cid:96) + , (cid:96) − , . . . , ) and so we have (cid:96) = m − (cid:3) Lemma 6.26.
Let w ∈ P (cid:96) and t ≥ . Write (cid:98) w = min { w , . . . , w t } and (cid:98) w = min ({ w , . . . , w t }\ (cid:98) w ) .If (cid:98) w ≥ and (cid:98) w > (cid:98) w + then either (cid:98) w = w < (cid:98) w ≤ w ≤ (cid:96) or w > w = (cid:96) = (cid:98) w . roof. Let w d = (cid:98) w +
1. Since (cid:98) w < w d < (cid:98) w , we have d ∈ { t + , . . . , n } . If there exists i ∈ { t + , . . . , d − } such that w i < w d − (cid:98) w , w i , w d is of type 312. So by Lemma 6.22we have w i = (cid:96) however w i < min { w , . . . , w t } ≤ w = (cid:96) , a contradiction. So for all i ∈{ t + , . . . , d − } we have w i > w d . And so w | w d = ( w d − , w d , . . . ) . By Lemma 6.25 wehave either w d − = w < w ≤ (cid:96) or w > w = (cid:96) = w d −
1. In the former case note that (cid:98) w ≤ w ≤ (cid:96) . (cid:3) Lemma 6.27.
Fix w ∈ P (cid:96) . Let [ I J ] be a semi-standard Young tableau and write [ I (cid:48) J (cid:48) ] = Γ (cid:96) ([ I J ]) . If I , J ≤ w then I (cid:48) , J (cid:48) ≤ w .Proof. We write I = { i < · · · < i t } and J = { j < · · · < j s } where s ≤ t . We use the followingnotation to denote ordered initial segments of w . We let { w , . . . w t } = { (cid:98) w < · · · < (cid:98) w t } and { w , . . . , w s } = { (cid:101) w < · · · < (cid:101) w s } . We have that i ≤ (cid:98) w , . . . , i t ≤ (cid:98) w and j ≤ (cid:101) w , . . . , j s ≤ (cid:101) w s .Since Γ (cid:96) fixes all entries in the third row and below it remains to check the entries in thefirst two rows of Γ (cid:96) ( T ) . Note that I ≤ w implies i ≤ (cid:98) w and i ≤ (cid:98) w . Also J ≤ w implies j ≤ (cid:101) w and j ≤ (cid:101) w .If s = t , then the result holds by the Grassmannian case so we assume that s < t . If Γ (cid:96) fixes the entries in each column then the result trivially holds. So we take cases on thetableau T whose columns are not fixed by Γ (cid:96) . Case 1.
Assume 2 ≤ s , j < i , i ∈ { , . . . , (cid:96) } and i , j , j ∈ { (cid:96) + , . . . , n } . So thetableaux are Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) I Ji j i j ... ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j j i i ... ... . We have that i ≤ j ≤ (cid:101) w and j ≤ (cid:101) w and so J (cid:48) ≤ w .Assume by contradiction that I (cid:48) (cid:54)≤ w . Since i ≤ (cid:98) w , we must have j > (cid:98) w . Therefore (cid:98) w < j < i ≤ (cid:98) w ≤ (cid:101) w and so (cid:98) w + ∈ { w t + , . . . , w n } . Since j ≤ (cid:101) w we must have that (cid:98) w ∈ { w s + , . . . , w t } . And so (cid:101) w , (cid:98) w , (cid:98) w + P (cid:96) if w is not312-free then w = (cid:96) . However j ≤ (cid:101) w = min { w , w . . . , w s } ≤ w = (cid:96) , a contradiction. Case 2.
Assume 2 ≤ s , j < i , i , i , j ∈ { , . . . , (cid:96) } and j ∈ { (cid:96) + , . . . , n } . So thetableaux are Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) I Ji j i j ... ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j j i i ... ... . We have that i ≤ j ≤ (cid:101) w and j ≤ (cid:101) w and so J (cid:48) ≤ w .Assume by contradiction that I (cid:48) (cid:54)≤ w . Since i ≤ (cid:98) w , we must have j > (cid:98) w . Therefore (cid:98) w < j < i ≤ (cid:98) w ≤ (cid:101) w and so (cid:98) w + ∈ { w t + , . . . , w n } . Since j ≤ (cid:101) w we must have that (cid:98) w ∈ { w s + , . . . , w t } . And so (cid:101) w , (cid:98) w , (cid:98) w + (cid:98) w = (cid:96) . However (cid:98) w < j and j ∈ { , . . . , (cid:96) } , a contradiction. 45 ase 3. Assume s = , t ≥ j < i , i ∈ { , . . . , (cid:96) } and j , i ∈ { (cid:96) + , . . . , n } . Thetableaux are Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) I Ji j i ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j i i ... . We have i < j ≤ w = (cid:101) w hence J (cid:48) ≤ w .Assume by contradiction that I (cid:48) (cid:54)≤ w . Since i ≤ (cid:98) w , we must have j > (cid:98) w . Thereforewe have (cid:98) w < j < i ≤ (cid:98) w . And so we have (cid:98) w + ∈ { w t + , . . . , w n } . Since j ≤ w therefore (cid:98) w ∈ { w , . . . , w t } . So w , (cid:98) w , (cid:98) w + (cid:98) w = (cid:96) . Andso j < i ≤ (cid:96) , a contradiction. Case 4.
Assume s = , t ≥ i < j , i ∈ { , . . . , (cid:96) } and i , j ∈ { (cid:96) + , . . . , n } . If t ≥ j < i . The tableaux are Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) I Ji j i ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j i i ... . We have i < j ≤ w and so J (cid:48) ≤ w .Assume by contradiction that I (cid:48) (cid:54)≤ w . We have i ≤ (cid:98) w and so we must have j > (cid:98) w .Therefore (cid:98) w < (cid:98) w < j ≤ w . If t = w ∈ { (cid:98) w , (cid:98) w } . Sowe have t ≥
3. Therefore (cid:98) w < j < i ≤ (cid:98) w , and so (cid:98) w + ∈ { w t + , . . . , w n } . Since w > (cid:98) w ,we have w ≥ (cid:98) w . And so w , (cid:98) w , (cid:98) w + (cid:98) w = (cid:96) .And so i ≤ (cid:98) w = (cid:96) but by assumption i ∈ { (cid:96) + , . . . , n } , a contradiction. (cid:3) Lemma 6.28.
Fix w ∈ P (cid:96) . If [ I J ] is a matching field tableau for B (cid:96) such that I , J ≤ w thenthere exists a semi-standard Young tableau [ ˜ I ˜ J ] such that ˜ I , ˜ J ≤ w and Γ (cid:96) ([ ˜ I ˜ J ]) is row-wiseequal to [ I J ] .Proof. By Lemma 6.20 we have for any T = [ I J ] a matching field tableau for B (cid:96) there existsa semi-standard Young tableau ˜ T such that Γ (cid:96) ( ˜ T ) = T (cid:48) = [ I (cid:48) J (cid:48) ] is row-wise equal to T . Weproceed to show that if I , J ≤ w then I (cid:48) , J (cid:48) ≤ w . Once we show this we have ˜ T ∈ SSYT ( w ) by Lemma 6.21.We write I = { i , i , . . . , i t } and J = { j , j , . . . , j s } where s < t . We use the followingnotation to denote ordered initial segments of w . Let { w , . . . w t } = { (cid:98) w < · · · < (cid:98) w t } and { w , . . . , w s } = { (cid:101) w < · · · < (cid:101) w s } . We proceed by taking cases on s and t . Note that if s = t then the result holds by the Grassmannian case. Also note that if T = T (cid:48) then there isnothing to prove so we will assume T (cid:44) T (cid:48) . Without loss of generality, we assume that thethird row and below of the tableau T is in semi-standard form, i.e. i ≤ j , . . . , i s ≤ j s so wewill focus on the first two rows of I (cid:48) and J (cid:48) . 46 ase 1. Assume s = t ≥
2. Since T and T (cid:48) are different we must have the tableaux T = I Ji j i ... , T (cid:48) = I (cid:48) J (cid:48) j i i ... . Since T and T (cid:48) are different we have i (cid:44) j . We proceed by taking cases on r = |{ i , i , j } ∩{ , . . . , (cid:96) }| . Case 1.1
Assume r = T (cid:48) are fixed by Γ (cid:96) . So T (cid:48) must bein semi-standard form, hence j < i . Since I ≤ w we have i ≤ (cid:98) w and i ≤ (cid:98) i . Since J ≤ w we have j ≤ w .We have j < i ≤ (cid:98) w and i ≤ (cid:98) w and I (cid:48) ≤ w . We also have i ≤ (cid:98) w ≤ w and so J (cid:48) ≤ w . Case 1.2
Assume r =
1. If either i or j lie in { , . . . , (cid:96) } then we have that I or I (cid:48) respectively is not a valid column with respect to the matching field B (cid:96) . So we must have i ∈ { , . . . , (cid:96) } . Since T (cid:48) lies in the image of Γ (cid:96) , we have that two cases for the values of i and j . Case 1.2.1
Assume i < j . So we have Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ˜ I ˜ Ji j i ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j i i ... . Since the contents of ˜ T = [ ˜ I ˜ J ] is column-wise equal to T , therefore ˜ I , ˜ J ≤ w . Case 1.2.2
Assume i > j . We note that Γ (cid:96) acts by permuting the entries of the firsttwo rows of a tableau. By permuting the entries of the first two rows of T (cid:48) , there are twoways to form a semi-standard Young tableau. However we have Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:171) i i j ... (cid:170)(cid:174)(cid:174)(cid:172) = i j i ... , Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:171) i j i ... (cid:170)(cid:174)(cid:174)(cid:172) = j i i ... . And so each such semi-standard Young tableau does not map to T (cid:48) . Case 1.3
Assume r =
2. If i , j ∈ { , . . . , (cid:96) } then T is not a valid tableau for B (cid:96) . Sothere are two remaining cases, either i , i ∈ { , . . . , (cid:96) } or j , i ∈ { , . . . , (cid:96) } . Case 1.3.1
Assume i , i ∈ { , . . . , (cid:96) } . We have that exactly two entries in the first tworows of T (cid:48) lie in { , . . . , (cid:96) } . The map Γ (cid:96) acts on any tableau with this property by fixing theentries of the tableau column-wise. Hence Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ˜ I ˜ Ji i j ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j i i ... . I ≤ w we have i ≤ (cid:98) w and i ≤ (cid:98) w . Since J ≤ w we have j ≤ w . Since j ∈{ (cid:96) + , . . . , n } and i ∈ { , . . . , (cid:96) } we have i < j ≤ w . Hence J (cid:48) ≤ w .Let us assume by contradiction that I (cid:48) (cid:54)≤ w . Since i ≤ (cid:98) w , we must have (cid:98) w < j . So wehave (cid:98) w < (cid:98) w < j ≤ w . If t = w ∈ { (cid:98) w , (cid:98) w } . So wehave t ≥
3. Since (cid:98) w < w , we have (cid:98) w ≤ w . Therefore (cid:98) w < j < i ≤ (cid:98) w ≤ w . And so wehave (cid:98) w + ∈ { w t + , . . . , w n } . However w , (cid:98) w , (cid:98) w + (cid:98) w = (cid:96) . So (cid:96) < j however by assumption j ∈ { , . . . , (cid:96) } , a contradiction. Case 1.3.2
Assume that j , i ∈ { , . . . , (cid:96) } . We have that T (cid:48) is a semi-standard Youngtableau and T (cid:48) = Γ (cid:96) ( T (cid:48) ) . Since I ≤ w we have i ≤ (cid:98) w and i ≤ (cid:98) w . Since J ≤ w we have j ≤ w . We have j < i ≤ (cid:98) w and i < i ≤ (cid:98) w so I (cid:48) ≤ w .Assume by contradiction that J (cid:48) (cid:54)≤ w . So we have w < i ≤ (cid:98) w . Since (cid:98) w ≤ w < (cid:98) w ,so (cid:98) w = w . Since j < i ≤ (cid:98) w we have (cid:98) w ≥
2. If (cid:98) w = (cid:98) w + w | w + = ( w , w + , . . . ) . By Lemma 6.25 we have either w < w ≤ (cid:96) or w > w = (cid:96) . If w < w ≤ (cid:96) then we have i ≤ (cid:98) w ≤ w ≤ (cid:96) but i ∈ { (cid:96) + , . . . , n } , a contradiction. If w > w = (cid:96) thenwe have an immediate contradiction because (cid:98) w = w < w .So (cid:98) w + < (cid:98) w . By Lemma 6.26 we have w < w ≤ (cid:96) or w > w = (cid:96) . If w < w ≤ (cid:96) then we have i ≤ (cid:98) w ≤ w ≤ (cid:96) but i ∈ { (cid:96) + , . . . , n } , a contradiction. If w > w = (cid:96) thenwe have an immediate contradiction because (cid:98) w = w < w . Case 2.
Assume s ≥
2. Since T and T (cid:48) are different, we either have that the entries ofthe first row are changed or the second row are changed. We will treat these as two differentcases. Case 2.1
Assume that the entries in the first row of T and T (cid:48) are different. So we have T = I Ji j i j ... ... T (cid:48) = I (cid:48) J (cid:48) j i i j ... ... . We proceed by taking cases on r = |{ i , i , j , j } ∩ { , . . . , (cid:96) }| . Case 2.1.1
Assume r = T (cid:48) is a semi-standard Young tableau and T (cid:48) = Γ (cid:96) ( T (cid:48) ) . Since I ≤ w we have i ≤ (cid:98) w and i ≤ (cid:98) w . Since J ≤ w we also have j ≤ (cid:101) w and j ≤ (cid:101) w . And so we have i ≤ (cid:98) w ≤ (cid:101) w and j ≤ (cid:101) w hence J (cid:48) ≤ w . We also have j ≤ i ≤ (cid:98) w and i ≤ (cid:98) w hence I (cid:48) ≤ w . Case 2.1.2
Assume r =
1. So we have that either i ∈ { , . . . , (cid:96) } or j ∈ { , . . . , (cid:96) } . Case 2.1.2.1
Assume i ∈ { , . . . , (cid:96) } . So, by definition of Γ (cid:96) , it follows that i ≥ j and Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ˜ I ˜ Ji i j j ... ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j i i j ... ... . Since T and T (cid:48) are different we have that i (cid:44) j hence i > j . Since I ≤ w we have i ≤ (cid:98) w and i ≤ (cid:98) w . Since J ≤ w we also have j ≤ (cid:101) w and j ≤ (cid:101) w . We have i ≤ (cid:98) w and j < i ≤ (cid:98) w and so I (cid:48) ≤ w . 48ssume by contradiction that J (cid:48) (cid:54)≤ w . Since j ≤ (cid:101) w , we must have i > (cid:101) w . So wehave (cid:101) w < i < j ≤ (cid:101) w , in particular (cid:101) w > (cid:101) w +
1. Since (cid:96) ≥
1, we have 2 ≤ j ≤ (cid:101) w . ByLemma 6.26 we either have (cid:101) w = w < w ≤ (cid:96) or w > w = (cid:96) = (cid:101) w . If (cid:101) w = w < w ≤ (cid:96) then i ≤ (cid:98) w ≤ w ≤ (cid:96) and i ∈ { (cid:96) + , . . . , n } , a contradiction. If w > w = (cid:96) = (cid:101) w then (cid:96) < j ≤ (cid:101) w ≤ (cid:96) , a contradiction. Case 2.1.2.2
Assume j ∈ { , . . . , (cid:96) } . Then we have Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ˜ I ˜ Jj j i i ... ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j i i j ... ... . Since ˜ T = [ ˜ I ˜ J ] is a semi-standard Young tableau we have i ≤ i . However in T we have i < i , a contradiction. Case 2.1.3
Assume r =
2. We must have that i , j ∈ { , . . . , (cid:96) } otherwise at least one of T or T (cid:48) is not a valid tableau for B (cid:96) . We have Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ˜ I ˜ Ji j j i ... ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j i i j ... ... . Since ˜ T is a semi-standard Young tableau we have i ≤ j and j ≤ i . Since I ≤ w we have i ≤ (cid:98) w and i ≤ (cid:98) w . Since J ≤ w we have j ≤ (cid:101) w and j ≤ (cid:101) w . And so we have i ≤ (cid:98) w and j ≤ i ≤ (cid:98) w therefore I (cid:48) ≤ w . We also have j ≤ (cid:101) w and i ≤ (cid:98) w ≤ (cid:101) w and so J (cid:48) ≤ w . Case 2.1.4
Assume r =
3. We have that either i ∈ { (cid:96) + , . . . , n } or j ∈ { (cid:96) + , . . . , n } . Case 2.1.4.1
Assume i ∈ { (cid:96) + , . . . , n } . Then we have Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ˜ I ˜ Jj j i i ... ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) j i i j ... ... . Since I ≤ w we have i ≤ (cid:98) w and i ≤ (cid:98) w . Since J ≤ w we have j ≤ (cid:101) w and j ≤ (cid:101) w . Andso j < i ≤ (cid:98) w and i ≤ (cid:98) w < (cid:98) w hence I (cid:48) ≤ w .Assume by contradiction that J (cid:48) (cid:54)≤ w . Since i ≤ (cid:98) w ≤ (cid:101) w we must have j > (cid:101) w . Andso (cid:101) w < j < i ≤ (cid:98) w hence (cid:101) w = (cid:98) w and (cid:98) w > (cid:98) w +
1. Since j < i ≤ (cid:98) w , we have (cid:98) w ≥
2. By Lemma 6.26 we have we either have (cid:98) w = w < w ≤ (cid:96) or w > w = (cid:96) = (cid:98) w .If (cid:98) w = w < w ≤ (cid:96) then (cid:96) < i ≤ (cid:98) w ≤ (cid:96) , a contradiction. If w > w = (cid:96) = (cid:98) w then (cid:96) = (cid:98) w < j ≤ (cid:96) , a contradiction. Case 2.1.4.2
Assume j ∈ { (cid:96) + , . . . , n } . There are no possible semi-standard Youngtableau ˜ T such that Γ (cid:96) ( ˜ T ) = T (cid:48) . Suppose by contradiction that ˜ T = [ ˜ I ˜ J ] is such a semi-standard Young tableau where ˜ I = { i (cid:48) < i (cid:48) < . . . } and J = { j (cid:48) < j (cid:48) < . . . } . Then we must49ave j (cid:48) = j ∈ { (cid:96) + , . . . , n } as it is the maximum value appearing in the first two rows of T (cid:48) . For any possible values { i (cid:48) , i (cid:48) , j (cid:48) } ⊆ { , . . . , (cid:96) } we have that j (cid:48) lies in the second columnof Γ (cid:96) ( ˜ T ) . However j (cid:48) = j is contained in the first column of T (cid:48) . Case 2.2
Assume that the entries of the second row of T and T (cid:48) are different. So wehave T = I Ji j i j ... ... T (cid:48) = I (cid:48) J (cid:48) i j j i ... ... . We proceed by taking cases on r = |{ i , i , j , j } ∩ { , . . . , (cid:96) }| . Case 2.2.1
Assume r = T (cid:48) is a semi-standard Young tableauand Γ (cid:96) ( T (cid:48) ) = T (cid:48) . So we have i ≤ j and j ≤ i . Since I ≤ w we have i ≤ (cid:98) w and i ≤ (cid:98) i .Since J ≤ w we also have j ≤ (cid:101) w and j ≤ (cid:101) w . So i ≤ (cid:98) w and j ≤ i ≤ (cid:98) w hence I (cid:48) ≤ w .Also j ≤ (cid:101) w and i ≤ (cid:98) w ≤ (cid:101) w hence J (cid:48) ≤ w . Case 2.2.2
Assume r =
1. We have that either j ∈ { , . . . , (cid:96) } or i ∈ { , . . . , (cid:96) } . Case 2.2.2.1
Assume j ∈ { , . . . , (cid:96) } . We have Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ˜ I ˜ Jj j i i ... ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) i j j i ... ... . Since I ≤ w we have i ≤ (cid:98) w and i ≤ (cid:98) w . Since J ≤ w we have j ≤ (cid:101) w and j ≤ (cid:101) w . So j < i ≤ (cid:98) w and i ≤ (cid:98) w < (cid:98) w so I (cid:48) ≤ w .Assume by contradiction that J (cid:48) (cid:54)≤ w . Since i ≤ (cid:98) w ≤ (cid:101) w therefore we must have j > (cid:101) w .And so (cid:101) w < j < i ≤ (cid:98) w ≤ (cid:101) w . Since (cid:101) w < (cid:98) w therefore (cid:98) w = (cid:101) w . Note that j < i ≤ (cid:98) w so (cid:98) w ≥
2. Let w d = (cid:101) w + ≥
3. Let w d = (cid:101) w +
1, note that we have d ∈ { t + , . . . , n } . Andso we have the restriction w | w d = ( w d − , w d , . . . ) . And so by Lemma 6.25 we either have w < w ≤ (cid:96) or w > w = (cid:96) . In each case (cid:96) < i ≤ (cid:98) w ≤ (cid:96) , a contradiction. Case 2.2.2.2
Assume i ∈ { , . . . , (cid:96) } . We have Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ˜ I ˜ Ji i j j ... ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) i j j i ... ... . Since ˜ T = [ ˜ I ˜ J ] is a semi-standard Young tableau we have j ≤ j . However in T we have j < j , a contradiction. Case 2.2.3
Assume r =
2. We must have that i , j ∈ { , . . . , (cid:96) } otherwise at least one of50 or T (cid:48) is not a valid tableau for B (cid:96) . We have Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ˜ I ˜ Jj i i j ... ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) i j j i ... ... . Since ˜ T is a semi-standard Young tableau we have j ≤ i and i ≤ j . Since I ≤ w we have i ≤ (cid:98) w and i ≤ (cid:98) w . Since J ≤ w we have j ≤ (cid:101) w and j ≤ (cid:101) w . And so we have j ≤ i ≤ (cid:98) w and i ≤ (cid:98) w therefore I (cid:48) ≤ w . Also we have j ≤ (cid:101) w and i ≤ (cid:98) w ≤ (cid:101) w and so J (cid:48) ≤ w . Case 2.2.4
Assume r =
3. We have that either i ∈ { (cid:96) + , . . . , n } or j ∈ { (cid:96) + , . . . , n } . Case 2.2.4.1
Assume i ∈ { (cid:96) + , . . . , n } . Then there are no possible semi-standard Youngtableau ˜ T such that Γ (cid:96) ( ˜ T ) = T (cid:48) , see Case 2.1.4.2. Case 2.2.4.2
Assume j ∈ { (cid:96) + , . . . , n } . We have i < i and i < j , so Γ (cid:96) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ˜ I ˜ Ji i j j ... ... (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = I (cid:48) J (cid:48) i j j i ... ... . Since I ≤ w we have i ≤ (cid:98) w and i ≤ (cid:98) w . Also since J ≤ w we have j ≤ (cid:101) w and j ≤ (cid:101) w . So i ≤ (cid:98) w and j ≤ i ≤ (cid:98) w hence I (cid:48) ≤ w .Assume by contradiction that J (cid:48) (cid:54)≤ w . Since j ≤ (cid:101) w we must have i > (cid:101) w . We have (cid:101) w < i < j ≤ (cid:101) w , in particular (cid:101) w > (cid:101) w +
1. Also we have i < j ≤ (cid:101) w hence (cid:101) w ≥
2. Soby Lemma 6.26 we either have (cid:101) w = w < w ≤ (cid:96) or w > w = (cid:96) = (cid:101) w . If (cid:101) w = w < w ≤ (cid:96) then (cid:96) < j ≤ (cid:101) w ≤ w ≤ (cid:96) , a contradiction. If w > w = (cid:96) = (cid:101) w then (cid:96) = (cid:101) w < i and i ∈ { , . . . , (cid:96) } , a contradiction.We have shown that any tableau T = [ I J ] , for the matching field B (cid:96) with columns I , J ≤ w ,is row-wise equal to a tableau T (cid:48) = [ I (cid:48) J (cid:48) ] which lies in the image Γ (cid:96) and I (cid:48) , J (cid:48) ≤ w . (cid:3) We have shown that the map Γ (cid:96) has the desired properties and so we can show that thenumber of standard monomials for J in degree two is | SSYT ( w )| . Proof of Lemma 6.16.
Let w ∈ P (cid:96) be a permutation. For each semi-standard Young tableau T = [ I J ] let T (cid:48) = [ I (cid:48) J (cid:48) ] = Γ (cid:96) ( T ) . By Lemmas 6.27 and 6.21 we have that I , J ≤ w if andonly if I (cid:48) , J (cid:48) ≤ w . We deduce that Γ (cid:96) ( SSYT ( w )) corresponds to a collection of standardmonomials for J in degree two. By Lemma 6.19 we have that the monomials correspondingto Γ (cid:96) ( SSYT ( w )) are linearly independent. By Lemma 6.28 we have for any T = [ I J ] amatching field tableau for B (cid:96) with I , J ≤ w there exists a semi-standard Young tableau ˜ T such that Γ (cid:96) ( ˜ T ) = T (cid:48) = [ I (cid:48) J (cid:48) ] is row-wise equal to T and I (cid:48) , J (cid:48) ≤ w . Therefore, by Lemma 6.21,˜ I , ˜ J ≤ w and so the monomials corresponding to Γ (cid:96) ( SSYT ( w )) are a spanning set. (cid:3) eferences [ACK18] B.H. An, Y. Cho, and J.S. Kim. On the f-vectors of Gelfand-Tsetlin polytopes. European Journal of Combinatorics , 67:61–77, 2018.[And13] D. Anderson. Okounkov bodies and toric degenerations.
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Authors’ addresses:
University of Bristol, School of Mathematics, BS8 1TW, Bristol, UKE-mail addresses: [email protected]
Department of Mathematics: Algebra and Geometry, Ghent University, 9000 Gent, BelgiumDepartment of Mathematics and Statistics, UiT – The Arctic University of Norway, 9037 Tromsø, NorwayE-mail address: [email protected]@ugent.be