Colorful combinatorics and Macdonald polynomials
aa r X i v : . [ m a t h . C O ] O c t Colorful combinatorics and Macdonald polynomials
Ryan Kaliszewski and Jennifer Morse Lehigh UniversityDepartment of MathematicsBethlehem, PA 19104University of VirginiaDepartment of MathematicsCharlottesville, VA 22903
Abstract
The non-negative integer cocharge statistic on words was introduced in the 1970’s by Lascoux and Sch¨utzenbergerto combinatorially characterize the Hall-Littlewood polynomials. Cocharge has since been used to explain phe-nomena ranging from the graded decomposition of Garsia-Procesi modules to the cohomology structure of theGrassman variety. Although its application to contemporary variations of these problems had been deemedintractable, we prove that the two-parameter, symmetric Macdonald polynomials are generating functions ofa distinguished family of colored words. Cocharge adorns one parameter and the second measure its devia-tion from cocharge on words without color. We use the same framework to expand the plactic monoid, applyKashiwara’s crystal theory to various Garsia-Haiman modules, and to address problems in K -theoretic Schubertcalculus. Dedicated to Alain Lascoux
Kostka-Foulkes polynomials, K λµ ( t ) ∈ N [ t ], describe the connections between characters of GL n ( F q ) and the Hall-Steinitz algebra [Gre55], give characters of cohomology rings of Springer fibers for GL n [Spr78, HS77], and aregraded multiplicities of modules for the general linear group obtained by twisting functions on the nullcone bya line bundle [Bry89]. Lusztig [Lus83, Lus81] showed they are the t -analog of the weight multiplicities in theirreducible representations of the classical Lie algebras, K λµ ( t ) = X σ ∈ W ( − ℓ ( σ ) P t ( σ ( λ + ρ )) − ( µ + ρ )) , obtained from a t -deformation of Kostant’s partition function P defined by Y positive roots α − tx α ) = X β P t ( β ) x β . Algebraically, Kostka-Foulkes polynomials are the entries in transition matrices between the Schur and theHall-Littlewood { H µ ( x ; t ) } bases for the algebra Λ of symmetric functions in variables x = x , x , . . . , over thefield Q ( t ). In fact, this reflects the graded decomposition of a simple quotient of the coinvariant ring viewed as Partially supported by the NSF grant DMS-1600953. S n -module [GP92]; each irreducible submodule of polynomials with homogeneous degree r corresponds to aSchur function with coe ffi cient t r , and the sum over all irreducibles corresponds to a Hall-Littlewood polynomial.Kostka-Foulkes polynomials are wrapped in the most fundamental combinatorial ideas. Namely, the set ofwords forms a monoid under the operations of RSK-insertion and jeu-de-taquin. The monoid structure was firstmotivated by Sch¨utzenberger [Sch77] in his proof that the Schubert structure constants for the cohomology of theGrassmann variety are enumerated by Young tableaux with a distinguished Yamanouchi property. The structureis compatible with the assignment of each word to a non-negative integer (statistic) called cocharge . Lascouxand Sch¨utzenberger [LS78] proved that the generating function for tableaux weighted by this statistic is precisely K λµ ( t ). Consequently, the spectrum of topics surrounding Kostka-Foulkes polynomials is accessible from a purelycombinatorial study of cocharge. For example,˜ H µ ( x ; t ) = X t cocharge( T ) s shape( T ) ( x ) , (1)summing over all Young tableaux T with µ ones, µ twos, etc.In the 1980’s, Macdonald introduced a basis for Λ over the field Q ( q , t ) to unify the Hall-Littlewood andthe Jack polynomials (wave equations of the Calogero-Sutherland-Moser model [For92]). Ensuant studies ofthe basis have impacted an impressive range of areas including the representation theory of quantum groups[EAK94], double a ffi ne Hecke algebras [Che95], the shu ffl e algebra and diagonal harmonics [HHL + ffi ne Schubert calculus [LLM03], the elliptic Hall algebra of Shi ff mann-Vasserot [SV11], and extensions of HOMFLY polynomials for knot invariants [ORS12].Early characterizations of Macdonald polynomials were oblique, revealing little more than that they are el-ements of Q ( q , t )[ x , x , . . . ]. Nevertheless, using brute force to compute examples, Macdonald conjectured thatthe entries K λµ ( q , t ) of certain of their transition matrices lie in N [ q , t ]. Garsia modified Macdonald’s polynomialsso that K λµ ( q , t ) appeared as Schur expansion coe ffi cients of the resulting polynomials ˜ H µ ( x ; q , t ), thus piquingthe interest of representation theorists for whom Schur functions are synonymous with irreducible S n -modules.The q , t-Kostka coe ffi cients in ˜ H µ ( x ; q , t ) = X λ K λµ ( q , t ) s λ ( x ) , (2)have since been a matter of great interest.Rich theories were born from the compelling feature that the q , t -Kostka coe ffi cients reduce to the Kostka-Foulkes polynomials at q =
0. In [GH93], Garsia and Haiman introduced S n -modules R µ , for µ a partitionof n , given by the space of polynomials in variables x , . . . x n ; y , . . . y n spanned by all derivatives of a certainsimple determinant ∆ µ . They conjectured that the dimension of R µ equals n !, and that the modules provide arepresentation theoretic framework for (2). Their interpretation was designed to imply the Macdonald positivityconjecture. Haiman spent years putting together algebraic geometric tools which ultimately led him to prove theconjectures in [Hai01].Formula (1) set the gold standard for defining Macdonald polynomials, but cocharge was abandoned aftere ff orts to give a manifestly positive formula for generic ˜ H µ ( x ; q , t ) led no further than the most basic examples. In2004, an explicit formula for Macdonald polynomials was established by Haglund-Haiman-Loehr. Rather thanusing Young tableaux and cocharge, the formula involves the major index and an intricate inversion-like statistic:˜ H λ ( x ; q , t ) = X F q inv( F ) t maj( F ) Y i x F ( i ) , (3)over all Z + -valued functions (fillings) F on the partition λ . The Schur expansion was expected to come shortlybehind this breakthrough, but it took another decade even to recover the Hall-Littlewood case. In [Rob17], Austin2oberts converted the q = H λ ( x ; 0 , t ) = X F ∈U inv( F ) = t maj( F ) s weight( F ) ( x ) , (4)over a mysterious subset U of fillings (see § tabloids , words with an increasing condition used by Young to define (Specht) modules. We prove thatMacdonald polynomials are colored tabloid generating functions, weighted by cocharge and a betrayal statisticwhich measures the variation of cocharge on colored words from its value on usual words. Theorem.
For any partition µ , ˜ H µ ( x ; q , t ) = X T q betrayal( T ) t cocharge( T ) x shape( T ) , (5) over colored tabloids with µ ones, µ twos, and so forth. Further applications of colored words are geometrically inspired. The classical example in Schubert cal-culus addresses the cohomology of the Grasmann variety where the structure constants c νλµ count Yamanouchitableaux. Schubert calculus vastly expanded with e ff orts to characterize the structure of K -theory and (quantum)cohomology of other varieties; the problems are a combinatorial search for alternative, or more refined notions,of Yamanouchi. Thus, the combinatorial ideas surrounding the plactic monoid are often revisited in Schubertcalculus. In fact, c νλµ can be viewed as the number of skew tableaux with zero cocharge and the broader scope ofcolored words fits in well.We extend Van Leeuwen’s approach [vL01] to the Yamanouchi condition using Young tableaux companions .We show that colored tabloids serve as companions for the generic Z + -valued functions used in the Macdonaldpolynomials (3). From this point of view, a super-Yamanouchi condition arises and is applicable to K -theoreticSchubert calculus problems as well as Kostka-Foulkes polynomials. The companion map c simultaneously givesrelations between • the formulas (1) and (4) for q = • genomic tableaux of Pechenik-Yong [PY17] and set-valued tableaux of [Buc02], introduced to study K -theoretic problems in Schubert calculus, and • cocharge and the Lenart-Schilling statistic [LS13] for computing the (negative of the) energy function ona ffi ne crystals.Colored words also support equivariant K -theory of Grassmannians and Lagrangians, but details are sequesteredin a forthcoming paper.We investigate representation theoretic lines with the theory of crystal bases, introduced by Kashiwara [Kas90,Kas91] in an investigation of quantized enveloping algebras U q ( g ) associated to a symmetrizable Kac–Moody Liealgebra g . Integrable modules for quantum groups play a central role in two-dimensional solvable lattice models.When the absolute temperature is zero ( q = crystal basis with many striking features.The most remarkable is that the internal structure of an integrable representation can be combinatorially realizedby associating the basis to a colored oriented graph whose arrows are imposed by the Kashiwara (modified root)3perators. From the crystal graph, characters can be computed by enumerating elements with a given weight,and the tensor product decomposition into irreducible submodules is encoded by the disjoint union of connectedcomponents. Hence, progress in the field comes from having a natural realization of crystal graphs.A double crystal structure on colored tabloids using only the type- A crystal operators and jeu-de-taquin pro-vides a lens giving clarity to problems in Macdonald theory and in Schubert calculus. Several crystal graphsarise simultaneously through di ff erent colored tabloid manifestations of tabloids. From these, we deduce Schurexpansion formulas for dual Grothendieck polynomials and the q = , q = q =
0. In particular,
Theorem.
For any partition µ , ˜ H µ ( x ; 1 , t ) = X t cocharge( T ) s shape( T ) , over colored tabloids with column increasing entries. The algebra Λ of symmetric functions in infinitely many indeterminants x , x , . . . , over the field Q ( q , t ) hasbases are indexed by partitions , λ = ( λ ≥ . . . ≥ λ ℓ > ∈ Z ℓ . The monomial basis is defined by elements m λ = P α x α taken over all distinct rearrangements α of ( λ , . . . , λ ℓ , , . . . ). The homogeneous basis has elements h λ = h λ · · · h λ ℓ , where h r ( x ) = P i ≤ ... ≤ i r x i · · · x i r , and the power basis has elements p λ = p λ · · · p λ ℓ , with p r ( x ) = P i x ri . A basis element indexed by partition λ of degree d = P i λ i (denoted by λ ⊢ d ) is a sum ofmonomials of degree d .The Hall-inner product , h , i , is defined on Λ by h p λ , p µ i = δ λµ z λ where z λ = Y i α i ! i α i for λ = ( · · · , α , α ) , and δ λµ evaluates to 0 when λ , µ and is otherwise 1. In fact, the basis of Schur functions , s λ , can be defined asthe unique orthonormal basis which is unitriangularly related to the monomial basis; for each λ ⊢ d , s λ = m λ + X µ ⊢ d µ ✁ λ a λµ m µ , where dominance order µ ✁ λ is defined by λ + · · · + λ k ≤ µ + · · · + µ k for all k .Macdonald [Mac95] proved the existence of another basis of polynomials, P λ ( x ; q , t ), also unitriangularlyrelated to the monomials, but orthogonal with respect to the q , t -deformation of h , i , h p λ , p µ i q , t = δ λµ z λ Y j − q λ j − t λ j . Of interest to combinatorialists, but not apparent from the definition, he conjectured that P λ ( x ; q , t ) have cer-tain transition coe ffi cients lying in Z ≥ [ q , t ]. Garsia modified P λ ( x ; q , t ) into polynomials ˜ H λ ( x ; q , t ) to rephraseMacdonald’s conjecture as one about Schur positivity: the q , t-Kostka coe ffi cients in˜ H µ ( x ; q , t ) = X λ K λµ ( q , t ) s λ ( x ) (6)4ie in Z ≥ [ q , t ]. See (3) for a precise definition of ˜ H µ ( x ; q , t ).Garsia’s approach appealed to a broader audience. Namely, results of Frobenius dictate that a positive sum ofSchur functions models the decomposition of an S n -representation into its irreducible submodules. Namely, for σ ∈ S n and λ ⊢ n , the value of the irreducible character χ λ of S n at σ arises in s λ = n ! X σ ∈ S n χ λ ( σ ) p τ ( σ ) , (7)where τ ( σ ) is the cycle-type of σ . Define the linear Frobenius map from class functions on S n to symmetricfunctions of degree n by F χ = n ! X σ ∈ S n χ λ p τ ( σ ) , and consider the Frobenius image of a doubly-graded S n -module M = L r , s M r , s , F char( M ) ( x ; q , t ) = X r , s t r q s F char( M r , s ) . The function F char( M ) ( x ; q , t ) is thus a positive sum of Schur functions with coe ffi cients in Z ≥ [ q , t ] by (7).So launched the search for a bi-graded module M for which ˜ H λ ( x ; q , t ) is the Frobenius image. Garsia andProcesi settled the q = K λµ (0 , t ) describes the multiplicities of S n characters χ λ in the graded characterof the cohomology ring of a Springer fiber, B µ . The cohomology ring H ∗ ( B µ ) can be defined by a particularquotient, R µ ( y ) = C [ y , . . . , y n ] / I µ , of the coinvariant ring R n ( y ) = C [ y , . . . , y n ] / h e , . . . , e n i . R µ ( y ) is the Garsia-Procesi module under the natural S n -action permuting variables; they proved the ideal I µ is generated by Tanisaki generators , defined to be theelementary symmetric functions e k ( S ) in the variables S = { y i , . . . , y i r } ⊂ { y , . . . , y n } when r > k > | S | − µ weakly east of column r .The simplicity of Garsia and Procesi’s definition led them to an algebraic proof that K λµ (0 , t ) ∈ N [ t ] ando ff ered an attack on the q , t -Kostka polynomials. Given that the Frobenius image of R µ ( y ) is ˜ H µ ( x ; 0 , t ), the taskwas to define an S n -module R µ ( x ; y ) = C [ x , . . . , x n ; y , . . . , y n ] / J µ , under the the diagonal S n -action, simultaneously permuting the x and y variables, so that˜ H µ ( x ; q , t ) = F char( R µ ( x ; y )) ( x ; q , t ) . (8)Garsia and Haiman found just the candidate; it is the ideal J µ = ( f : f ∂∂ x , · · · , ∂∂ x n , ∂∂ y , · · · , ∂∂ y n ! ∆ µ = ) , where ∆ µ is a generalization of the Vandermonde defined using a graphical depiction of µ . A lattice square ( i , j )lies in the i th row and j th column of N × N . The (Ferrers) shape of a composition α = ( α , . . . , α ℓ ) ∈ Z ℓ ≥ is thesubset of N × N made up of α i lattice squares left-justified in the i th row, for 1 ≤ i ≤ ℓ . A lattice square inside ashape α is called a cell . Given µ ⊢ n , the cells { ( r , c ) , . . . , ( r n , c n ) } in µ define ∆ µ = det x c y r x c y r · · · x c n y r n ... ... x c n y r n x c n y r n · · · x c n n y r n n . R µ ( x ; y ) is quite simple, the proof of (8) required sophisticated geomet-ric techniques developed by Haiman [Hai01]. How R µ ( x ; y ) decomposes into irreducible submodules remains an open problem. It is particularly intriguing inlight of the perfect description for decomposing R µ ( y ) in terms of the following statistic on words. Given a word w in the alphabet A , w B is the subword of w restricted to letters of B ⊂ A . When B = { i } , we use simply w i = w B .The weight of a word w is the composition α , where α i is the number of times i appears in w . A word with weight(1 , . . . ,
1) is called standard . The cocharge of a standard word w ∈ S n is defined by writing w counter-clockwiseon a circle with a ⋆ between w and w n , attaching a label to each letter, and summing these labels. The labels aredetermined iteratively starting by labeling 1 with a zero. Letter i is then given the same label as i − ⋆ lies between i − i (reading clockwise) and it is otherwise incremented by 1.The cocharge of a word w with weight µ ⊢ n is defined by writing w counter-clockwise on a circle andcomputing the cocharge of µ standard subwords of w . Letters of the i th standard subword are adorned with asubscript i and this subword is determined iteratively from i = ⋆ , choose the firstoccurrence of letter 1 and proceed on to the first occurrence of letter 2. Continue in this manner until µ ′ has begiven the index 1. Start again at ⋆ with i +
1, repeating the process on letters without a subscript. The cochargeof w is the sum of the cocharge of each standard subword. The charge of a word w of weight µ ischarge( w ) = n ( µ ) − cocharge( w ) , where n ( µ ) = P i ( i − µ i . Example 1.
The words w = and w = written counter-clockwise on circles ⋆ = ⇒ cocharge is 6 . ⋆ = ⇒ cocharge is 15 . Kostka-Foulkes polynomials require only words coming from Young tableaux. Use α | = n to denote that α isa composition of degree n = | α | = α + α + · · · . For compositions α and β where α i ≤ β i for all i , we say α ⊆ β .The skew shape of α ⊆ β is β/α , defined by the set theoretic di ff erence of their cells and of degree | β | − | α | . A (semi-standard) tableau is the filling of a skew shape with positive integers which increase up columns and arenot decreasing along rows (from west to east). Definition 2.
The reading order of any collection S ⊂ N × N is the total ordering on elements in S defined bysaying that lattice squares decrease from left to right, starting in the highest row and moving downward. Given a tableau T , the reading word w = word( T ) is defined by taking w i to be the letter in the i th cell of T ,where cells are read in decreasing reading order. The weight of tableau T is the weight of its reading word and T is called standard when word( T ) is standard. For skew shape λ/µ of degree n and γ | = n , the set of tableaux ofshape λ/µ and weight γ is denoted by SSYT ( λ/µ, γ ). Lascoux and Sch¨utzenberger [LS78] proved, for partitions λ and µ of the same degree, K λµ (0 , t ) = X T ∈SSYT ( λ,µ ) t cocharge( T ) , (9)6here the cocharge of a tableau T is defined by cocharge(word( T )).A similarly beautiful formula for the q , t -Kostka polynomials has been actively pursued for decades. Because K λµ (1 , = |SSYT ( λ, n ) | , the endgame is to establish a formula for K λµ ( q , t ) by attaching a q and a t weight toeach standard tableau. Although the Schur expansion of Macdonald polynomial still eludes us, Jim Haglund made a breakthroughin 2004 by proposing a combinatorial formula for ˜ H µ ( x ; q , t ). Rather than using semi-standard tableaux andcocharge, di ff erent statistics are associated to arbitrary fillings.A filling F of shape β/α and weight γ | = | β | − | α | is any placement of letters from a word with weight γ intoshape β/α . The entry in row r and column c of F is denoted by F ( r , c ) , and the set of fillings of shape β/α andweight γ is F ( β/α, γ ). Immediate from the definition is |F ( α/β, γ ) | = | γ | γ , γ , . . . , γ ℓ ( γ ) ! . (10)For a filling F of partition shape λ , an inversion triple is a triple of entries ( r , t , s ) which are arranged in acollection of cells in F of the form r . . . ts , and meeting the criteria that r = s , t or some cycle of ( r , t , s ) is decreasing, i.e. r > s > t , s > t > r , or t > r > s .If the cells containing r and t are in the first row we envision that s =
0. The inversion statistic is the numberinv( F ) of inversion triples in F . The major index of F ismaj( F ) = X F ( r , c ) > F ( r − , c ) ( λ ′ c − r + λ ′ c is the number of cells in column c of λ . Every partition λ has a conjugate λ ′ given by reflecting shape λ about y = x . Alternatively, a filling F has a descent at cell ( r , c ) when F ( r , c ) > F ( r − , c ) and maj( F ) is thenumber of descents of F , each weighted by the number of cells appearing weakly above it in F . Example 3.
The following filling F ∈ F ((333) , (342)) has maj( F ) = + + = and inv( F ) = . Descents: 3 2 32 2 12 1 1 Inversion triples: 3 2 32 2 12 1 1 3 2 32 2 12 1 1 3 2 32 2 12 1 1 3 2 32 2 12 1 1
Theorem 4. [HHL05a] For any partition λ , ˜ H λ ( x ; q , t ) = X F ∈F ( λ, · ) q inv( F ) t maj( F ) x weight( F ) . (11) We introduce a new combinatorial structure and prove that the Macdonald polynomials are generating functionsattached to cocharge and a second statistic called betrayal.7 .1 Colored words and circloids A colored letter x i is a letter x in an alphabet A adorned with a subscript (its color ) i from A . A colored wordw is a string of distinct colored letters. The weight of w is a skew composition recording the colors which adorneach letter; weight( w ) = α/β where { β x + , . . . , α x } are the colors attached to letter x in w . When β | =
0, wesimply say the weight of w is α .Colored words also come equipped with shapes which are assigned using the prismatic order on coloredletters: u v > r c when r < u or ( r = u and c > v ). Equivalently, u v > r c ⇐⇒ cell ( u , v ) precedes ( r , c ) in reading order.A strict composition (one without zero entries) γ | = n is a shape admitted by a colored word w = w n · · · w if w > · · · > w γ , w γ + > · · · > w γ + γ , and so forth. A weak composition γ ′ | = n is a shape admitted by w whenthe strict composition obtained by removing the zeroes from γ ′ is a shape admitted by w .The set of all shapes admitted by a colored word w = w · · · w n corresponds multisets that containDes( w ) = { p : w p < w p + } , under the bijection sending compositions of degree n to sub-multisets of { , . . . , n − } defined by( γ ℓ , . . . , γ ) set( γ ) = { γ , γ + γ , . . . , γ + · · · + γ ℓ ( γ ) − } . Ashapeisastrictcompositioni f andonlyi f itcorrespondstoatrueset . Example 5.
The colored word w = has weight (3 , , and Des( w ) = { , , } . It thus admitsshape λ = (3 , , , and, for example, β = (3 , , , , . A circular representation of colored words is convenient when attaching statistics. We write a colored word w counter-clockwise on a circle and separate its letters into sectors to give a concept of shape. Definition 6.
A circloid C of shape γ | = n is a placement of n distinct colored letters on the perimeter of asubdivided circle such that, reading clockwise from a distinguished point ⋆ , γ x colored letters lie in decreasingprismatic order in sector x, for x = , . . . , ℓ ( γ ) . Each circloid C is uniquely associated to a colored word w of the same shape by reading the letters of C incounter-clockwise order. The weight of C is defined to be weight( w ). The set of circloids of weight α/β andshape γ is denoted by C ( γ, α/β ). Note that letter b appears in a circloid C ∈ C ( · , α/β ) exactly α b − β b times sincethere are α b − β b colors needed to adorn the set of b ’s. Example 7.
Circloids C ∈ C ((3 , , , , (3 , , and C ∈ C ((3 , , , , , (3 , , with underlying w = are C = ⋆ C = ⋆ If unspecified, entries and positions of a circloid are always taken clockwise. For example, 3 and 3 liebetween 1 to 2 in C since 3 and 3 are passed when reading clockwise from 1 to 2 . We consider thefollowing two restrictions of a circloid C , C i = { x y ∈ C : y = i } and C ≥ j = { x y ∈ C : x ≥ j } . γ and α/β , a coloredtabloid T ∈ CT ( γ, α/β ) is a filling of shape γ with colored letters so that row entries are increasing from westto east under the prismatic order and the colors adorning letter x in T are { β x + , . . . , α x } . As expected, α/β iscalled the weight of T . It is straightforward to see that a bijection is given by the map ι : C ( γ, α/β ) −→ CT ( γ, α/β ) , defined by putting the colored letters of sector r from circloid C into row r of shape γ so that the row is coloredincreasing from west to east, for each r = , . . . , ℓ ( γ ). The cocharge statistic on words naturally extends to circloids. Macdonald polynomials turn out to be generatingfunctions of circloids, weighted by cocharge and a second statistic which measures the variation of cocharge fromthe Lascoux-Sch¨utzenberger statistic.For a circloid C ∈ C ( · , µ ) of partition weight µ , the cocharge of C is defined bycocharge( C ) = ℓ ( µ ) X i = cocharge( C i ) . Remark . Any word w with partition weight µ can be uniquely identified with a circloid C ∈ C (1 n , µ ) of the samecocharge. Place the letters of w counter-clockwise on a circle and color according to the labeling for standardsubwords. That is, moving clockwise from ⋆ , label the first 1 with i =
1. By iteration, the first x + x is colored 1. Once µ ′ letters have been labeled by 1, repeat with 2 on uncoloredletters, and so on.The second statistic measures how di ff erent a coloring is from standard subwords. When choosing whichletter x to color j , each candidate passed over in clockwise order increases the statistic by 1. Precisely,betrayal( C ) = X i ≥ X j s i , j , where s i , j is the number of i ˆ with ˆ > j lying between ( i − j and i j in C , with the understanding that 0 j = ⋆ forall j = , . . . , µ . Example 9.
The circloids in the previous example have a betrayal of 2 and cocharge of 4.
It is through the lens of circloids that we can prove cocharge is as fundamental to the q , t -Macdonald settingas it is to Kostka-Foulkes and Hall-Littlewoods polynomials. We show that a Macdonald polynomial is noneother than the shape generating function of circloids weighted by cocharge and betrayal. Moreover, the resultfollows straightforwardly from a correspondence between circloids and skew fillings. Definition 10.
The map f acts on a circloid C of shape γ by placing entry x in cell ( r , c ) , for each colored letterr c in sector x, moving through sectors x = , . . . , ℓ ( γ ) . Example 11.
The action of f on two circloids: ⋆ ⋆ . heorem 12. For any partition λ , ˜ H λ ( x ; q , t ) = X C ∈C ( · ,λ ) q betrayal( C ) t cocharge( C ) x shape( C ) . (12) Proof.
Consider any compositions γ and β ⊆ α where | γ | = | α | − | β | . We first establish that f is a bijection where f : C ( γ, α/β ) −→ F ( α/β, γ ) . (13)Given a circloid C , let f = f ( C ). Each letter r in sector x of C corresponds to an entry x in row r of f implyingthat C ∈ C ( γ, α/β ) if and only if f ∈ F ( α/β, γ ).Note similarly that any other circloid D where f ( C ) = f ( D ) lies in C ( γ, α/β ). Consider the set of colored letters { r (1) c (1) , . . . , r ( γ x ) c ( γ x ) } in sector x of C . By definition of f , f ( r ( i ) , c ( i ) ) = x for i = , . . . , γ x . Therefore, { r (1) c (1) , . . . , r ( γ x ) c ( γ x ) } is alsothe set of colored letters in sector x of D . Since colored letters lie in unique decreasing prismatic order withinsectors, every sector of C and D is the same and we see that C = D . That f is bijective then follows by noting thatthe number of fillings given in (10) matches the number of circloids C ∈ C ( γ, · ). That is, again viewing lettersfrom sector x of C as a subset { r (1) c (1) , . . . , r ( γ x ) c ( γ x ) } of the distinct | γ | colored letters in C , we see that |C ( γ, α/β ) | = | γ | γ , γ , . . . , γ ℓ ( γ ) ! . (14)We next restrict our attention to circloid C ∈ C ( γ, λ ) for λ ⊢ n and claim thatcocharge( C ) = maj( f ) and betrayal( C ) = inv( f ) , (15)for f = f ( C ). Since maj is computed on columns, we need only verify that cocharge( C i ) equals the maj ofcolumn i in f to prove that maj( f ) = cocharge( C ). A slight reinterpretation of the cocharge definition givescocharge( C i ) = P r L r where L r = λ ′ i − r + r i occurs (clockwise) between ( r − i and ⋆ in C and otherwise L r =
0. In fact, since r i > ( r − i and sectors are prismatic order decreasing, L r , r i lies in asector y strictly larger than the sector x containing ( r − i . On the other hand, the action of f dictates that r i is insector y and ( r − i is in sector x of C precisely when y lies in cell ( r , i ) above x in cell ( r − , i ) of f .We next claim that betrayal( C ) = inv( f ) using the observation that betrayal( C ) = P i ≥ (cid:18) | I i | + · · · + | I i λ ′ i | (cid:19) ,where I ij = { ˆ > j : i ˆ lies between ( i − j and i j } , with the convention that 0 j = ⋆ for all j . For any j , 1 j is in sector x of C if and only if entry x lies in (1 , j ) of f . Note that ˆ ∈ I j implies i ˆ lies in sector y < x since i j < i ˆ and therefore ˆ ∈ I j corresponds uniquely to aninversion of x with the entry y in ( i , ˆ ) of f . When i >
1, for each pair of ( i − j in sector x and i j in sector y of C , one of the following relations concerning the sector z with ˆ ∈ I ij must be true: x = y and z , x , x < z < y , y < x < z , or z < y < x . Correspondingly, entries x , y , and z in cells ( i − , j ) , ( i , j ) , and ( i , ˆ ) of f respectively,form a triple inversion. (cid:3) We can extend the definitions of cocharge and betrayal to colored tabloid,cocharge( T ) = cocharge( ι − T ) and betrayal( T ) = betrayal( ι − T ) . Immediately following from Theorem 12 is an expression using charge and one using fixed weight coloredtabloid.
Corollary 13.
For any partition µ , ˜ H µ ( X ; q , / t ) t n ( µ ) = X C ∈C ( · ,µ ) q betrayal( C ) t charge( C ) x shape( C ) = X T ∈CT ( · ,µ ) q betrayal( T ) t charge( T ) x shape( T ) . Colorful companions
The subset of Young tableaux with an additional
Yamanouchi condition is of particular importance; its cardinalitygives tensor product multiplicities of GL n , the Schur expansion coe ffi cients in a product of Schur functions, andthe Schubert structure constants in the cohomology of the Grassmannian Gr( k , n ) of k -dimensional subspaces of C n . For partition λ , T λ denotes the unique tableau of shape and weight λ . A word w is λ -Yamanouchi when w · word( T λ ) = b n · · · b b has the property that the weight of each su ffi x b j · · · b b is a partition. A filling is λ -Yamanouchi when its reading word is λ -Yamanouchi and a circloid is λ -Yamanouchi when the counter-clockwisereading of its letters is λ -Yamanouchi. A ∅ -Yamanouchi object is simply called Yamanouchi . Remark . Since a word of weight µ has zero charge only when every standard subword is the maximal lengthpermutation, zero charge matches the Yamanouchi condition.Because many open problems in representation theory, geometry, and symmetric function theory involvea search for contemporary notions of Yamanouchi and tableaux to characterize mysterious invariants, the Ya-manouchi condition has been revisited often from di ff erent viewpoints. The combinatorics of circloids naturallycaptures several of these simultaneously. Van Leeuwen addresses the classical Littlewood-Richardson rule by rephrasing the Yamanouchi condition onskew tableaux P in terms of companion tableaux . A companion of P is any skew tableau Q such that the entriesin row x match the row positions of letters x ∈ P and are aligned to meet the condition that entries increase upcolumns. He proves that a Yamanouchi tableau P always has a companion tableau of (straight) partition shape µ .We forsake the column increasing condition and instead view a companion as the tabloid where rows areuniquely aligned into a straight shape. Such a companion of semi-standard tableau P is precisely the tabloidobtained by ignoring colors of ι ◦ f − ( P ). This approach opens the door to a more inclusive study allowing forcompanions of arbitrary fillings. Definition 15.
The companion map is the bijection, c = ι ◦ f − : F ( ν/λ, µ ) −→ CT ( µ, ν/λ ) . The companion of a filling F ∈ F ( ν/λ, µ ) is the unique colored tabloid c ( F ) . Following directly from the definition of f and ι , the action of c on a filling F takes entry e in cell ( r , c ) to thecolored letter r c placed in row e of T , arranged so that each row of T is colored increasing. Companions give avaluable mechanism to study Yamanouchi related problems. Definition 16.
A filling F is super-Yamanouchi when the non-decreasing rearrangement of entries within eachrow is a Yamanouchi tabloid.
Proposition 17.
Given partitions µ and ν/λ , consider a filling F ∈ F ( ν/λ, µ ) and its companion T = c ( F ) .1. F is Yamanouchi if and only if entries of T are prismatic increasing in columns,2. F is a super-Yamanouchi filling if and only if letters of T increase in columns,3. letters of F increase in columns if and only if T is λ -Yamanouchi. roof. (1) F is Yamanouchi if and only if the letter x in a cell ( r , c ) of F can be paired uniquely with an x − r , ˆ c ) occurring after ( r , c ) in reading order. Equivalently, each entry r c in row x of T pairs uniquelywith an entry ˆ r ˆ c ≤ r c in row x − r c is smaller in prismaticorder.(2) Consider a colored tabloid T where letters do not strictly increase up some column. If columns of T arenot prismatic increasing, F is not Yamanouchi by (1). Otherwise, we can choose b to be the rightmost column of T with an r c in row x and an r ˆ c in row x − c < ˆ c . Correspondingly, F has an x in cell ( r , c ) and an x − r , ˆ c ). Since entries in rows of T are prismatic non-decreasing and r c and r ˆ c lie in column b , the subset of cellsin F weakly smaller than ( r , ˆ c ) in reading order contain b x − b x ’s. However, the filling ˆ F obtained byrearranging letters in row r of F into weakly increasing order is not Yamanouchi since the x in column c < ˆ c of F moves to the east of all x − T with letters increasing up columns has prismatic increasing columnsand therefore F is Yamanouchi by (1). Suppose that F has an x and an x − r , c ) and ( r , ˆ c ), respectively,such that when letters in row r are put into weakly increasing order, the resulting filling is not Yamanouchi. Since F is Yamanouchi, this can only happen if ˆ c > c and there is an equal number of x − x ’s in the subset ofcells of F occurring weakly after ( r , ˆ c ) in the reading order of cells. However, under the f -correspondence, T hasan r c in row x and an r ˆ c in row x − F ∈ F ( ν/λ, µ ) is column increasing, construct the unique filling ˆ F ∈ F ( ν, (1 | λ | , µ )) by replacingeach i ∈ F with i + | λ | and putting 1 , , . . . , | λ | into cells of λ so the reading word taken from these cells is | λ | · · · C = f − ( ˆ F ) di ff ers from f − ( F ) by the deletion of the first | λ | sectors.A letter in an arbitrary cell ( r , c ) of ˆ F is larger than the letter in cell ( r − , c ) if and only if entry r c occursin a later sector then r − c of ˆ C = f − ( ˆ F ). This is equivalent to the Yamanouchi condition on ˆ C ; r can bepaired with a letter r − r in ˆ C . The claim follows by noting that thecounter-clockwise reading of letters from the first | λ | sectors of ˆ C is word( T λ ). (cid:3) The initial study of companions involved only the subset of fillings which are semi-standard tableaux. Proposi-tion 17 pinpoints that dropping the row condition and requiring only that letters increase in columns of a fillingimposes the λ -Yamanouchi condition on its companion circloid (or colored tabloid). On the other hand, it is alsonatural to examine the subset of fillings which are tabloids , that is, fillings which are non-decreasing in rows fromwest to east. Let T ( α, β ) be the set of tabloids of shape α and weight β .A distinguished coloring on circloids comes to light under these conditions. A circloid is reverse colored when the colors adorning letter x increase clockwise from ⋆ , for each fixed letter x . A tabloid T is reversecolored if ι − ( T ) is a reverse colored circloid. Remark . Since the reverse coloring uniquely assigns a color to each letter of a tabloid, reverse colored circloidsare a manifestation of tabloids.
Proposition 19.
Given compositions γ and β ⊆ α , the companion T of a filling F ∈ F ( α/β, γ ) is reverse coloredif and only if F is a tabloid.Proof. By definition of f , a filling F has the property that the letter in cell ( r , c ) is not smaller than the letter in( r , c −
1) if and only if r c does not occur before r c − in f − ( F ). (cid:3) xample 20. −→ ⋆ ≡ In particular, the companion Q of a semi-standard tableau filling P is a reverse colored λ -Yamanouchi circloid,or equivalently, is a manifestation of a λ -Yamanouchi tabloid by Remark 18. Furthermore, when F is bothYamanouchi and a semi-standard tableau, we recover the classical result that Yamanouchi tableaux of shape ν/λ and weight µ and λ -Yamanouchi tableaux of shape µ and weight ν − λ are equinumerous. Corollary 21.
For partitions ν/λ and µ , P ∈ SSYT ( ν/λ, µ ) if and only if its companion is a λ -Yamanouchitabloid, and P is Yamanouchi if and only if its companion is a λ -Yamanouchi tableau Q ∈ SSYT ( µ, ν − λ ) . In the combinatorial theory of K -theoretic Schubert calculus, tableaux are replaced by more intricate combi-natorial objects such as reverse plane partitions, set-valued tableaux, and genomic tableaux. The later were in-troduced recently by Pechunik and Yong [PY17] to solve a di ffi cult problem concerning equivariant K -theory ofthe Grassmannian. We have discovered that reverse-colored companions are closely related to genomic tableauxand carry out the details separately. A glimpse of this application is given in § K -homology classes of the Grassmannian. Another useful manifestation of tabloids arises from a second distinguished circloid coloring. A circloid C is faithfully colored when, for each i ≥
1, if entries of color j < i are ignored, the closest 1 to ⋆ (moving clockwise)has color i = x + x i has color i , for x ≥
1. A colored tabloid is defined to be faithfullycolored if it is the ι -image of a faithfully colored circloid. Example 22.
A faithfully colored circloid and its corresponding (faithfully colored) tabloid:C = ⋆ ι ( C ) = . Proposition 23.
For composition α and partition λ where | α | = | λ | , the image of the companion map c on F ∗ ( λ, α ) = { F ∈ F ( λ, α ) : inv( F ) = } is the subset of faithfully colored tabloid in CT ( α, λ ) .Proof. The number of inversion triples in a filling F matches the betrayal of f ( F ) by (15). It thus su ffi ces to note,by definition, that restricting the set of circloids to those with zero betrayal gives the subset of faithfully coloredelements. (cid:3) Theorem 24.
For any partition µ , ˜ H µ ( x ; 0 , t ) = X F ∈F ∗ ( µ, · ) F super − Yamanouchi t maj( F ) s weight( F ) . roof. Consider an inversionless filling F ∈ F ∗ ( µ, · ) which is super-Yamanouchi. Note that the weight of F mustbe a partition λ ⊢ | µ | since F is Yamanouchi. Propositions 17 and 23 give that the c -image of F is a faithfullycolored tabloid with letters which increase up columns. Since each tabloid has a unique faithful coloring, ignoringcolors gives the bijection between { F ∈ F ( µ, λ ) : inv( F ) = F super − Yamanouchi } ↔ SSYT ( λ, µ ) . (16)The maj( F ) = cocharge( f ( F )) by (15), and any faithfully colored circloid C ∈ C ( λ, µ ) has the same cocharge asthat of its manifest tabloid T by Remark 8. The result then follows from (9). (cid:3) The comparison of Theorem 24 to (4) suggests that the set U defined by Roberts is related to the super-Yamanouchi condition. Roberts’ formula requires inversionless, Yamanouchi fillings with an additional propertyimposed upon entries in a pistol configuration, one that is made up of cells in row r lying in columns 1 , . . . , c andcells of row r + c , . . . , µ r + , for any fixed r , c . In our language, a filling is jammed if its reversecoloring results in a pistol containing both x y and ( x + y + for some letter x with color y . When a filling is notjammed, we say it is jamless . Lemma 25.
The set of inversionless, super-Yamanouchi fillings is the same as the set of inversionless, jamless,Yamanouchi fillings.Proof.
Suppose an inversionless, super-Yamanouchi filling F is jammed and for convenience, consider its reversecoloring. Since F is jammed, it has rows r and r + x y and ( x + y + . If x i does not liein a lower row than ( x + j of a super-Yamanouchi filling, then j < i . Therefore, x y must lie in cell ( r , c ) of F and ( x + y + in cell ( r + , ˆ c ), for some ˆ c ≥ c . Moreover, x y + lies in row r of F . Consider the minimal color i adorning x in the set of rows higher than row r . Since x i − , x i − , . . . , x y + lie west of column ˆ c in row r , and F isinversionless, an x + x ’s. Therefore, ( x + i lies in row r + F issuper-Yamanouchi.On the other hand, suppose that a filling F is jamless, inversionless, and Yamanouchi but not super-Yamanouchi.Then there is some row r and letter x + F where the number y of x + r is greater than thenumber of x ’s below row r . In particular, y is the color adorning the leftmost x + r in the reverse coloringof F . Further, the x colored y must lie after this ( x + y in reading order since F is Yamanouchi. However, it isnot super-Yamanouchi and therefore x y lies in row r . Since F is inversionless and has ( x + y west of x y in row r , x must lie below ( x + y . When reverse colored, this x has color z < y . In turn, z < y implies ( x + z + must lieweakly after ( x + y (in reading order). However, the Yamanouchi condition requires that ( x + z + lies before x z . Therefore, the pistol based at x z contains ( x + z + contradicting that F is not jammed. (cid:3) Corollary 26.
Roberts’ formula (4) and the Lascoux-Sch¨utzenberger formula (9) for q = Macdonald polyno-mials are related by the companion map c . The quantum enveloping algebra U q ( sl n + ) is the Q ( q )-algebra generated by elements e i , f i , t i , t − i , for 1 ≤ i ≤ n ,subject to certain relations. For a U q ( sl n + )-module M and λ ∈ Z n + , the weight vectors (of weight λ ) are elementsof the set M λ = { u ∈ M : t i u = q λ i − λ i + u } . A weight vector is said to be primitive if it is annihilated by the e i ’s. A highest weight U q ( sl n + )-module is a module M containing a primitive vector v such that M = U q ( sl n + ) v . Theirreducible highest weight module with highest weight λ is denoted V λ .Kashiwara [Kas90, Kas91] introduced a powerful theory whereby combinatorial graphs are used to under-stand finite-dimensional integrable U q ( s l n )-modules M . The crystal of M is a set B equipped with a weightfunction wt : B → { x α : α ∈ Z ∞≥ } and operators ˜ e i , ˜ f i : B → B ∪ { } satisfying the properties, for a , b ∈ B ,14 ˜ e i a = b ⇐⇒ ˜ f i b = a • ˜ e i a = b = ⇒ wt( b ) = x i x i + wt( a ).The crystal graph associated to B is a directed, colored graph with vertices from B and edges E = {{ a , b } : b = ˜ e i ( a ) for some i } labeled by color i ∈ I = { , . . . , n − } . Irreducible submodules are in correspondence with highestweight vectors b ∈ B where ˜ e i ( b ) = i . These are the vertices of the crystal graph with no incoming edges;each connected component of B represents an irreducible and contains a single highest weight vector. The subsetof highest weight vectors b ∈ B where wt( b ) = x γ is denoted by Y ( B , γ ).The tensor product crystal graph B ⊗ · · · ⊗ B k has vertices in the Cartesian product ( b , . . . , b k ) ∈ B × · · · × B k which are denoted b = b ⊗ · · · ⊗ b k . Its weight function is defined bywt( b ) = n Y j = wt B j ( b j ) . A morphism Φ : B → B ′ is a map on crystal graphs where Φ (0) = b ∈ B , • Φ ( ˜ f i ( b )) = ˜ f i ( Φ ( b )) • Φ (˜ e i ( b )) = ˜ e i ( Φ ( b )) • wt( Φ ( b )) = wt( b ).Lascoux and Sch¨utzenberger anticipated the necessary ingredients for the Kashiwara type- A crystal in theirdevelopment of the plactic monoid on words [LS81, LLT02]. It is given by the set B (1) n = B (1) ⊗ · · · ⊗ B (1)of words in the alphabet B (1) = [ n ]; the crystal action ˜ e i , ˜ f i is defined on b ∈ B (1) n by changing a single i (or i +
1) to an i + i ) in the restriction of b to the subword w { i , i + } . Regarding each letter as a parenthesis, i + i as a right, adjacent pairs of parentheses “()” are matched and declared to be invisible until nomore matching can be done. It is a letter in the remaining subword, z = i p ( i + q for some p , q ∈ Z ≥ , which ischanged. Precisely, ˜ e i ( b ) = q =
0, ˜ f i ( b ) = p =
0, and otherwise ˜ e i ( b ) is the word formed from w by replacing the subword z with i p + ( i + q − and ˜ f i ( b ) is formed by replacing z with i p − ( i + q + . Remark . Parentheses pairing of any b ∈ B (1) n has the property that every adjacent ( i + i is paired, and thefirst i in any adjacent pair ii is never the rightmost unpaired entry. Therefore, descents of such pairs are preservedby the action of ˜ e i , ˜ f i and Des(˜ e i ( b )) = Des( ˜ f i ( b )) = Des( b ) when b is not anhilated.For µ ⊢ n , since ˜ e i annihilates only the Yamanouchi words, the set of highest weights of B = B (1) n withwt( b ) = x µ is Y ( B , µ ) = { b ∈ B : b is Yamanouchi of weight µ } . (17)As dictated by Kashiwara’s theory, the crystal graph B ( µ ) of the irreducible submodule V µ is isomorphic to aconnected subgraph of B (1) n which contains a Yamanouchi word of weight µ , and B (cid:27) M µ ⊢ n B ( µ ) × Y ( B , µ ) . (18)The crystal graph B ( m ) is isomorphic to the subgraph of elements b ∈ B (1) m with no descents since b = (1 , . . . ,
1) is the only element in Y ( B (1) m , ( m )). Therefore, for any γ | = n of length ℓ , the tensor product crystal B = B ( γ ) ⊗ B ( γ ) ⊗ B ( γ ℓ )has highest weight elements given by Yamanouchi words which are non-decreasing in the first γ positions, inthe next γ positions, and so forth. 15 .1 Singly graded Garsia-Haiman modules A crystal structure on circloids leads us to a characterization for the singly graded decomposition of Garsia-Haiman modules which preserves the spirit of the Garsia-Procesi module decomposition given by (9).We first refine the decomposition of B = B (1) ⊗ · · · ⊗ B (1). For any D ⊂ { , . . . , n − } , define the inducedsubposet B ( D ) of B by restriction to vertex set { b ∈ B : Des( b ) = D } . Theorem 28.
For λ ⊢ n, ˜ H λ ( x ; 1 , t ) = X D ⊂ [ n − t maj λ ′ ( D ) X highest weight b ∈ B ( D ) s wt( b ) ( x ) , (19) where maj ν ( D ) = ℓ ( ν ) X i = X d ∈ D | ν< i − | < d < | ν< i | ν i − d . Proof.
For any D ⊂ [ n − B ( D ) is a crystal graph made up of the disjoint union ofconnected components in B (1) n . Therefore, the Frobenius image of the module associated to B ( D ) is X highest weight b ∈ B ( D ) s wt( b ) ( x ) . The highest weights are Y ( B ( D ) , µ ) = { b ∈ B (1) n : Des( b ) = D and b is Yamanouchi of weight µ } . (20)More generally, the right hand side of (19) reflects the graph decomposition of the crystal B (1) n into B ( D ), gradedby maj λ ′ ( D ).On the other hand, Macdonald polynomials at q = λ -shaped fillings graded by maj. Each filling f ∈ F ( λ, · ) can be uniquely identified with a vertex b ∈ B (1) n byreading the columns of f from top to bottom (choosing any fixed column order). Consider the filling f b identifiedby vertex b . Since the computation of maj( f b ) relies only on descents in columns of f b , precisely the subset ofdescents involved in the computation of maj λ ′ (Des( b )), we have that maj λ ′ (Des( b )) = maj( f b ). By definition,maj λ ′ (Des( b )) is constant over all elements b ∈ B ( D ) and therefore maj( f ) is constant on all fillings f associatedto b ∈ B ( D ). (cid:3) Remark . Although each vertex b ∈ B (1) n could be uniquely identified with the filling f of shape λ ⊢ n whose reading word is b , maj is not constant on all fillings in the same connected component under this correspondence.The interaction of crystals with the cocharge statistic comes out of a directed, colored graph B ( γ ) whosevertices are circloids of weight γ . An i -colored edge between circloids is imposed by operators, ˜ e i and ˜ f i , whichmove an entry from sector i + i or vice versa using a method of pairing colored letters.Pairing is a process which iterates over each entry in a given sector. Entries are considered from smallest tolargest with respect to the co-prismatic order , defined on colored letters by x y > ′ u v ⇐⇒ y > v or y = v and x > u . Pairing is done by writing the entries from sectors i and i + i + i a right parenthesis. Entries are then pairedas per the Lascoux and Sch¨utzenberger rule for parenthesis.16 efinition 30. For a composition α and i ∈ { , . . . , ℓ ( α ) − } , the operator ˜ e i acts on C ∈ C ( · , α ) by moving thelargest unpaired entry in sector i + to the unique position of sector i which preserves the prismatic decreasingcondition on circloids. In contrast, ˜ f i acts on C by moving the smallest unpaired entry in sector i to sector i + .Remark . For α, γ | = n and a circloid C ∈ C ( γ, α ), if ˜ e i does not anhilate C then its action preserves weight sinceit involves only moving an entry; the entry is moved from sector i + i , implying that ˜ e i ( C ) ∈ C ( β, α )where β = ( γ , . . . , γ i − , γ i + , γ i + − , . . . ). Theorem 32.
For any composition γ | = n, B ( γ ) is a crystal graph and its highest weights (of weight µ ⊢ n) are inbijection with SSCT ( · , µ ) = { T ∈ CT ( · , µ ) : colored letters increase up columns of T , with respect to < ′ } . When γ is a partition, the connected components of B ( γ ) are constant on cocharge .Proof. Given γ | = n , let φ i act on F ( γ, · ) by the induced action φ i = f ◦ ˜ e i ◦ f − . When an entry x y in circloid C ispaired with u v < ′ x y by the action of ˜ e i , an i + x , y ) of φ i ( f − ( C )) is paired with an i in cell ( u , v ) whereeither y > v or ( y = v and x > u ). Consider the graph on F ( γ ′ , · ) where an i -colored edge connects f and ˆ f whenˆ f ′ = φ i ( f ′ ); when each filling is replaced by its reading word b , this is the crystal graph B (1) n . In particular, acrystal morphism Φ : B ( γ ) → B (1) n is given by Φ ( C ) = b , where b is the reading word obtained by reading down the columns of the filling f ( C ) from right to left . That is,cell ( x , y ) of f ( C ) is read before cell ( u , v ) if y > v or ( y = v and x > u ). This is equivalent to x y > ′ u v . The weightfunction on B ( γ ) maps circloid C to its shape by (13).A highest weight C ∈ B ( γ ) satisfies ˜ e i ( C ) = i if and only if each entry x y in row i + T = ι ( C ) ∈CT ( µ, · ) pairs with an entry u v < ′ x y in row i . By rearranging the rows of T so that they are co-prismaticallynon-decreasing this will result in co-prismatic increasing columns. Note that µ must be a partition, for if a sector i + C has more entries than sector i , then C has an unpaired entry and is not anhilated. (cid:3) The circloid crystal captures a formula for the q = q = Corollary 33.
For any partition µ , ˜ H µ ( x ; 1 , t ) = X T ∈SSCT ( · ,µ ) t cocharge( T ) s shape( T ) ( x ) . Proof.
We have seen that each b ∈ B ( D ) corresponds to a filling f b with maj λ ′ ( D ) = maj( f b ). Theorem 28 givesthat ˜ H λ ( x ; 1 , t ) = X D ⊂ [ n − X µ ⊢ n X b ∈ Y ( B ( D ) ,µ ) t maj( f b ) s µ ( x ) = X µ ⊢ n X b ∈ Y ( B (1) n ,µ ) t maj( f b ) s µ ( x ) . The claim follows by recalling that maj( f b ) = cocharge( f − ( f b )) and that Φ is a morphism of crystals. (cid:3) From this, it is not di ffi cult to rederive Macdonald’s formula taken over standard tableaux. Corollary 34. [Mac95] For any partition µ ⊢ n, ˜ H µ ( x ; 1 , t ) = X T ∈SSYT ( · , n ) µ − Y i = t cocharge( T i ) s shape( T ) ( x ) where T i is the subtableau of T restricted to letters in [ µ ′ i + , µ ′ i + ] . roof. Give a prismatic column increasing circloid C ∈ C ( · , µ ), replace each entry i c of C with letter i + P j < c µ ′ j .The condition on C that x < u or x = u and y < v for any x y above u v implies that letters are strictly increasing incolumns of the tabloid T . Since the computation of cocharge on a circloid independently calculates cocharge onstandard subwords of a given color, µ − cocharge( T ) = cocharge( C ). (cid:3) Characterization of the doubly graded irreducible decomposition of Garsia-Haiman modules presents major ob-stacles. Although the identification of fillings with elements of B (1) n given by column reading yields connectedcomponents constant on the maj-statistic, it is incompatible with the inversion triples. Even the subset of ver-tices with zero inversion triples is not a connected component. The crystal cannot be applied to (3), even when q =
0, to gain insight on bi-graded decomposition of R µ ( x , y ) into its irreducible components. However, a doublecrystal structure using dual Knuth relations (jeu-de-taquin) and ˜ e i , ˜ f i operators on colored tabloids can be appliedto the Garsia-Procesi modules. Double crystal structures on B ( µ ) ⊗ · · · ⊗ B ( µ ℓ ) have been studied in variouscontexts [vL01, Shi, Las03], but without regard to graded modules.For any composition γ of length ℓ , we consider a crystal B † ( γ ) on vertices CT ( · , γ ) which is dual to B ( γ ). An i -colored edge is prescribed by a sliding operation defined on an inflation of rows i and i + i-inflation of a vertex b ∈ B † ( γ ) is defined by spacing out the colored letters in row i of b while preservingtheir relative order as follows: entries e are taken from west to east from row i of b and placed in the leftmostempty cell of row i without an entry e ′ < e directly above it. The operator e † i on b ∈ B † is then defined by ajeu-de-taquin sliding action whereby the largest entry of row i + i -inflation of b which lies immediatelyabove an empty cell is swapped with this empty cell, after which all empty cells are removed. When no emptycell lies in row i , e † i ( b ) = inflation a vertex b ∈ B † ( γ ) as the punctured colored tabloid obtained by inflatingrows of b in succession from top to bottom. Note that the inflation of b has entries (prismatic order) increasingup columns. Example 35.
The -inflation of T = is . The inflation of T is . Definition 36.
For any γ | = n, let B † ( γ ) be the graph on vertices CT ( · , γ ) with a directed, i-colored edge fromb → b ′ when e † i ( b ) = b ′ . We will establish that B † ( γ ) is a crystal graph doubly related to B (1) n = B (1) ⊗ · · · ⊗ B (1) by the companionmap. For each γ | = n , define the map c γ on B (1) n by c γ ( b ) = c ( f b ) , where f b is the unique filling of shape γ whose reading word is b . Theorem 37.
For each γ | = n, c γ is a crystal isomorphismB (1) ⊗ · · · ⊗ B (1) (cid:27) B † ( γ ) . The highest weights in B † ( γ ) of weight µ ⊢ n are Y ( B † ( γ ) , µ ) = { T ∈ CT ( µ, γ ) : entries of T prismatically increase up columns } . roof. Since the image of the companion bijection c on F ( γ, · ) is the set of vertices in B † ( γ ), the map c γ is abijection between B (1) ⊗ · · · ⊗ B (1) and the vertices of B † ( γ ). To check that edges match, we need to prove that˜ e i ( b ) = b ′ if and only if ˜ e † i ( c γ ( b )) = c γ ( b ′ ), for each b ∈ B (1) ⊗ · · · ⊗ B (1).We will show that an entry x y in row i + i -inflation of c γ ( b ) has an empty cell under it if and only if thecorresponding i + x , y ) of b is unpaired. The proof then follows because sliding the rightmost x y downto row i is the equivalent of changing the leftmost unpaired i + i in b .Suppose that x y lies above an empty cell in the i -inflation of c γ ( b ). Then for each u v < x y in row i the entry u ′ v ′ immediately above in row i + u v < u ′ v ′ < x y . The entry x y corresponds to an i + x , y ) of b , and for every i appearing afterward, there is a distinct i + x , y ). Therefore the i + x , y ) will be unpaired.Suppose that an i + x , y ) of b is unpaired. Then every i appearing afterward is paired with an i + x , y ) Therefore, for each u v in row i of c γ ( b ) with u v < x y , there is a unique u ′ v ′ inrow i + u v < u ′ v ′ < x y . Thus we are guaranteed that there is an empty cell under x y when we i -inflate c γ ( b ).The highest weights of B (1) n are defined by the Yamanouchi property and thus their companions are prismaticcolumn increasing by Proposition 17. Alternatively, e † i anhilates a colored tabloid T when there is no entry inrow i + T above an empty square in row i . In particular, T is its own inflation and thus hasprismatic increasing columns. (cid:3) As a first application, we show how the graded irreducible decomposition of a Garsia-Procesi module is readilyapparent in the crystal B † . For this, we need only the induced subposet B † ( γ ) on the restricted set of reverse-colored vertices in the crystal graph B † ( γ ). Proposition 38.
For a composition γ of length ℓ , the companion map is a crystal isomorphism B † ( γ ) (cid:27) B ( γ ) ⊗ · · · ⊗ B ( γ ℓ ) . The highest weights of B † ( γ ) are (reverse-colored) semi-standard tableaux of weight γ .Proof. Given b ∈ B † , consider b ′ = ˜ e † i ( b ) If b ′ ,
0, there is a (largest) unpaired x y in row i + b above an empty cell. The definition of inflation thus implies x y − cannot lie in row i . Therefore, the image of areverse colored tabloid under the crystal action remains as such since the action merely slides x y into row i . Thatis, each connected component in B † is a connected component of B † . We then note that the set of vertices B † is in bijection with B ( γ ) × · · · × B ( γ ℓ ) since reverse-colored tabloids of weight γ are the companion images oftabloids with shape γ by Proposition 19. In turn, B ( γ ) × · · · × B ( γ ℓ ) are defined as induced subposets of B (1) n allowing us to apply Theorem 37 to establish the isomorphism.The highest weights of B † ( γ ) are the colored tabloids with prismatically increasing columns by Theorem 37.Thus, a highest weight element b has entries x y above x y ′ in the same column only when y < y ′ . However, if b isalso reverse colored, then y > y ′ and therefore its letters increase up columns. (cid:3) Define the faithful recoloring of b ∈ B † to be the colored tabloid b obtained by stripping b of its colors andthen faithfully coloring its letters. Lemma 39.
For µ ⊢ n and b ∈ B † ( µ ) with the property that letters increase up columns of the inflation of b, cocharge( b ) = cocharge(( b ′ ) ) for b ′ = ˜ e † i ( b ) , . roof. Consider b ∈ B † such that letters increase up columns in its inflation b . If b ′ = ˜ e † i ( b ) ,
0, then there is alargest entry x y in row i + b which lies above an empty cell and it is moved into row i to obtain b ′ from b .Since letters must increase up columns of b and an x in row i + x − i than there are x ’s in row i of b by construction of b . For the number n ix of letters smaller than x in row i , n ix > n i − x − .Let d be the smallest color adorning an x in row i + b and note that there can be no ( x − d ′ with d ′ < d in row i of b by definition of faithful coloring. If ( x − d does not lie in row i of b , then b and ( b ′ ) are thesame with the exception of one entry; x d lies in row i + b and in row i of ( b ) ′ . Therefore, their cochargesequal since ( x − d is not in row i of either element.Otherwise, d ∈ D ∩ E for the set D of colors adorning letter x in row i + E of colors attached to x − i of b . In this case, b and ( b ) ′ again di ff er only by one entry; x e lies in row i of b and in row i − b ′ ) for the smallest element e ∈ D \ E . Therefore, their cocharges match since ( x − e is not in row i . (cid:3) Proposition 40.
For µ ⊢ n, the graded decomposition of the Garsia-Procesi module R µ ( y ) into its irreduciblecomponents is encoded in the crystal graph B † ( µ ) with cocharge( b ) attached to each vertex b.Proof. The Frobenius image of the module R µ ( y ) is obtained by setting q = ι to obtain ˜ H µ ( x ; 0 , t ) = X T t cocharge( T ) x shape( T ) , (21)over faithfully colored tabloids T of weight µ . The map b b = T is a bijection between B † ( µ ) and CT ( · , µ )since reverse colored and faithfully colored tabloids are both manifestations of uncolored tabloids. This allowsus to convert the previous identity to˜ H µ ( x ; 0 , t ) = X b ∈B † ( µ ) t cocharge( b ) x shape( b ) = X b ∈B † ( µ ) t cocharge( b ) x wt( b ) , (22)recalling that the weight function on the crystal B † ( µ ) sends b shape( b ). Now, letters increase up columns inthe inflation of a reverse colored tabloid b ; if x y lies in a higher row than x z , then x y < x z and thus these entriescannot lie in the same column of the inflation of b . This given, Lemma 39 implies that attaching each vertex b tococharge( b ) is a statistic which is constant on the connected components of B † ( µ ). Therefore,˜ H µ ( x ; 0 , t ) = X b ∈ Y ( B † ( µ ) ,λ ) t cocharge( b ) s λ ( x ) = X T ∈SSYT ( λ,µ ) t cocharge( T ) s λ ( x ) , (23)since Proposition 38 characterizes the highest weights by (reverse colored) semi-standard tableaux. (cid:3) ffi ne crystals Lenart and Schilling [LS13] connect the q = ffi ne crystals. The double crystal reveals that their statistic is precisely the companion of cocharge.For a partition λ of length ℓ and b ∈ B ( λ ) ⊗ · · · ⊗ B ( λ ℓ ), we definezmaj( b ) = X w ∈ Z ( b ) maj( w ) , Z ( b ) is a set of λ words extracted from the letters of b = ( b , . . . , b ℓ ). The first word w = w λ ′ · · · w isconstructed by selecting w to be the smallest entry in b . Iteratively, w i is selected to be the smallest entry largerthan w i − in b i (breaking ties by taking the easternmost). If there is no larger entry available, the smallest entry in b i is selected instead. The first word w is fully constructed after a letter has been selected from b ℓ . The remainingwords of Z ( b ) are constructed by the same process, ignoring previously selected letters of b . Example 41.
For b = (223 , , ∈ B (3) ⊗ B (3) ⊗ B (2) , the words in Z ( b ) are Z ( b ) = { , , } implyingthat zmaj( b ) = + + . Proposition 42.
For partition λ of length ℓ and b ∈ B ( λ ) ⊗ · · · ⊗ B ( λ ℓ ) , zmaj( b ) = cocharge( c ( f b ) ) , where f b is the unique λ -shaped tabloid with reading word b.Proof. For b ∈ B ( λ ) ⊗ · · · ⊗ B ( λ ℓ ), consider first the case that the tabloid f b contains an x in row i and letter z > x in row i +
1. For the colored tabloid T = c ( f b ), there is an i in row x of T and and z is the lowest row above x withan i +
1. If f b does not have any z > x in row i +
1, instead take z ≤ x to be the minimal entry in row i + z is the lowest row in T containing an i +
1. More generally, for the i th word w = w λ ′ i · · · w in Z ( b ), w j records the row containing j i in T . Since each w j + > w j contributes λ ′ i − j to the maj and each j + j contributes the same to cocharge( T ) the claim follows. (cid:3) Lemmas 39 and Proposition 42 imply that connected components of the crystal graph B ( λ ) ⊗ · · · ⊗ B ( λ ℓ ) areconstant on zmaj. Corollary 43.
For a partition λ of length ℓ and b ∈ B ( λ ) ⊗ · · · ⊗ B ( λ ℓ ) , zmaj( b ) = zmaj(˜ e i ( b )) and zmaj( b ) = zmaj( ˜ f i ( b )) , for any i ∈ { , . . . , n − } . Theorem 44.
For any partition λ , ˜ H λ ( x ; 0 , t ) = X b ∈ B ( λ ⊗···⊗ B ( λℓ ) b Yamanouchi t zmaj( b ) s weight( b ) ( x ) . Proof.
The expansion (21) of ˜ H µ ( x ; 0 , t ) over elements of B † can be converted to one involving B ( λ ) ⊗ · · ·⊗ B ( λ ℓ )by Proposition 38. As reviewed in (17), the highest weights of B = B ( λ ) ⊗ · · · ⊗ B ( λ ℓ ) are characterized by theYamanouchi condition. Corollary 43 implies that zmaj is constant on connected components of B , which are incorrespondence with Schur functions indexed by the the highest weights. (cid:3) The faithful recoloring of vertices in B † not only gives a formula for the energy function, it exposes an identifi-cation between inversionless fillings and tabloids used in [HHL05a]. Define s : { F ∈ F ( λ, α ) : inv ( F ) = } → T ( λ, α )on a filling F simply by rearranging entries in each row into non-decreasing order from west to east. The inverseof s is defined in [HHL05a](Proof of Proposition 7.1) by uniquely constructing an inversionless filling from a21ollection of multisets, m = { m , m , . . . , m k } . The unique placement of entries from m i into row i of f so thatinv( f ) = m are put into the bottom row in non-decreasing order, from west to east.Proceeding to the next row r =
2, letters of m r are placed in columns c from west to east as follows: an emptycell ( r , c ) is filled with the smallest value that is larger than the entry in ( r − , c ). If there are no values remainingthat are larger than that in ( r − , c ), the smallest available value is chosen. The filling f arises from iteration onrows and by construction, f has no inversion triples.In fact, s − ( f ) is none other than the companion preimage of the faithful recoloring of f ’s companion. Thatis, the companions of f and s ( f ) are both manifestations of the same tabloid, one is reverse colored and the otheris faithful. Proposition 45.
For any tabloid f ∈ T ( γ, · ) , c ( s − ( f )) = c ( f ) . Proof.
For any tabloid f , s − ( f ) and f di ff er only by the rearrangement of entries within rows. Thus, by definitionof companions, T ′ = c ( s − ( f )) and T = c ( f ) di ff er only by their colorings. Since f is a tabloid, T is reversecolored by Proposition 19 and since s − ( f ) is inversionless, T ′ is faithfully colored by Proposition 23. Each ofthese colorings is uniquely defined on the manifest tabloid and the claim follows. (cid:3) K -theoretic implications To give a flavor of how circloid crystals fit into K -theoretic Schubert calculus, consider tabloids with the propertythat their conjugate is also a tabloid. Such a filling is called a reverse plane partition . If the weight of a reverseplane partition is defined to be the vector α where α i records the number of columns containing an i , the weightgenerating functions are representatives for K -homology classes of the Grassmannian: for skew partition ν/λ , g ν/λ ( x ) = X r ∈ RPP( ν/λ, · ) x weight( r ) . From this respect, repeated entries in a column of the reverse plane partition r are superfluous motivating usto instead identify r with the tabloid f obtained by deleting any letter that is not the topmost in its column andthen left-justifying letters in each row. The inflated shape of tabloid f is defined to be the shape of its inflation.This recovers the shape of the reverse plane partition r from whence f came. Proposition 46.
For skew partition ν/λ ,g ν/λ ( x ) = X Yamanouchi tabloid Tinflatedshape ( T ) = ν/λ s weight( f ) ( x ) . (24) Proof.
For any composition γ , since B = B ( γ ) ⊗ · · · ⊗ B ( γ ℓ ) is a crystal graph under ˜ e i , it su ffi ces to show thatthe inflated shape of f ∈ T ( γ, · ) is the same as the inflated shape of ˜ e i ( f ). From this, the induced subposet of B on vertices with fixed inflated shape is also a crystal and the Schur expansion comes from the highest weights.Consider a filling f and f ′ = ˜ e i ( f ) di ff ering by only one letter i + i in some row r . The shapeof the inflation of f can di ff er from that of f ′ only if there is an i + r + i + f lies in row r , the process of pairing ensures that every i + r + i in row r . Therefore, there are more i ’s in row r of f than there are i + r + f must have an entry smaller than i + i + r + (cid:3) An expression for the Schur expansion of g ν/λ over a sum of semi-standard tableaux arises as a corollary ofProposition 46 by applying the companion map to (24) and using Corollary 21. Such an expression opens upthe study of problems in K -theoretic Schubert calculus to the classical theory of tableaux. For example, a simplebijective proof of the K -theoretic Littlewood-Richardson was given in [LMS17] using this approach.22 Quasi-symmetric expansion
It is not di ffi cult to use dual equivalence graphs instead of crystals to deduce our previous results. For this, we for-mulate Macdonald polynomials using colored words, without mention of shape, in terms of Gessel’s fundamentalquasisymmetric function . Defined for any S ⊆ [ n ], let Q S ( x ) = X i ≤···≤ i n i j < i j + i ff j ∈ S x β i · · · x β n i n . Betrayal and cocharge are defined on a colored word w by computing these statistics on the circloid C obtainedby writing entries of w counter-clockwise on a circle. Since the shape of a circloid has no bearing on the statistics,it su ffi ces to write each entry in its own sector so that C has shape (1 ℓ ( w ) ). Theorem 47.
For partition µ ⊢ n, ˜ H µ ( X ; q , t ) = X colored word w weight( w ) = µ q betrayal( w ) t cocharge( w ) Q Des n ( w ) ( x ) , where Des n ( w ) = { n − d : d ∈ Des( w ) } . Proof.
Given a fixed colored word w of weight µ , consider the set of weak compositions β for which shape( w ) = β . For each such β , Des( w ) ⊆ set( β ). Therefore, a unique circloid C ∈ C ( β, µ ) is obtained by counter-clockwise inscribing the entries of w on a circle, separated into sectors of sizes β ℓ , . . . , β . Since the compu-tation of cocharge and betrayal of circloid C does not involve its shape, every C arising in this way satisfiescocharge( C ) = cocharge( w ) and betrayal( C ) = betrayal( w ). Theorem 12 can thus be rewritten as˜ H µ ( X ; q , t ) = X C ∈C ( · ,µ ) q betrayal( C ) t cocharge( C ) x sh( C ) = X colored word w weight( w ) = µ X β :sh( w ) = β q betrayal( w ) t cocharge( w ) x β . The claim follows by noting that X β :sh( w ) = β x β = X γ | = ℓ ( w ) n − Des( w ) ⊆ set( γ ) X i < ··· < i ℓ ( γ ) x γ i · · · x γ ℓ ( γ ) i ℓ ( γ ) . (cid:3) References [Bry89] R. Brylinski. Limits of weight spaces, lusztig’s q-analogs, and fiberings of adjoint orbits.
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