aa r X i v : . [ m a t h . QA ] N ov THE ODD LITTLEWOOD-RICHARDSON RULE
ALEXANDER P. ELLISA
BSTRACT . In previous work with Mikhail Khovanov and Aaron Lauda we introduced twoodd analogues of the Schur functions: one via the combinatorics of Young tableaux (oddKostka numbers) and one via the odd symmetrization operator. In this paper we introduce athird analogue, the plactic Schur functions. We show they coincide with both previously de-fined types of Schur function, confirming a conjecture. Using the plactic definition, we estab-lish an odd Littlewood-Richardson rule. We also re-cast this rule in the language of polytopes,via the Knutson-Tao hive model. C ONTENTS
1. Introduction 12. Review of odd symmetric functions 33. Odd plactic Schur functions 84. Littlewood-Richardson rules 12References 191. I
NTRODUCTION
The classical story.
Schur functions play a central role in the beautiful circle of ideasaround symmetric functions, symmetric and general linear group representations, and linearenumerative geometry. On the one hand, the Schur functions s λ corresponding to partitions λ of k give an integral orthonormal basis of the algebra of symmetric functions Λ in degree k . Under the Frobenius correspondence (see, for instance, [Sta99, Section 7.18]) M n ≥ K ( C [ S n ]) ∼ = Λ , the Schur function s λ corresponds to the irreducible representation L λ associated to the par-tition λ . Recall that the product on the symmetric group side is given as follows. If V is arepresentation of S k and W is a representation of S ℓ , then(1.1) [ V ] · [ W ] = [Ind S k + ℓ S k × S ℓ ( V ⊗ W )] . On the other hand, the ring Λ n obtained by setting s (1 n +1 ) = s (1 n +2 ) = . . . = 0 can be viewedas the character ring for polynomial representations of GL n ( C ) . Under this correspondence,the image of s λ in Λ n equals the character of the irreducible representation V λ associated to λ . This time the product is the usual product of characters, that is, s µ s ν is the character of Date : November 15, 2011. ALEXANDERP.ELLIS V µ ⊗ V ν . Either of these two descriptions immediately implies that the structure coefficientsof a product(1.2) s µ s ν = X λ c λµν s λ are all non-negative integers; they are called Littlewood-Richardson coefficients . The deter-mination of these coefficients is a classical and important problem. By the above, c λµν hasinterpretations as(1) the multiplicity of L λ in Ind S k × ℓ S k × S ℓ ( L µ ⊗ L ν ) ,(2) the multiplicity of V λ in V µ ⊗ V ν , and(3) the dimension of the space of GL n ( C ) -invariant vectors in V µ ⊗ V ν ⊗ V ∗ λ .The combinatorics of Young tableaux become an important tool at this point: partitions cor-respond to Young diagrams, and the coefficients c λµν can be computed by counting certainskew tableaux. We review this Littlewood-Richardson rule in Section 4.1; more completeexpositions can be found in [Ful97, Chapter 5] and [Sta99, Appendix A1].Schur functions also arise in linear enumerative geometry. The cohomology ring of theGrassmannian Gr( k, n ) of complex k -planes in C n is isomorphic to the quotient of Λ k bythe ideal generated by all complete symmetric functions h m for m > n − k . Geometrically,elementary functions e m correspond to the Chern classes of the tautological k -plane bundleand complete functions h m correspond (up to sign) to the Chern classes of a complemen-tary ( n − k ) -plane bundle. Under this identification, the image of the Schur function s λ in H ∗ (Gr( k, n )) equals zero unless λ ≤ n − k and ℓ ( λ ) ≤ k . The nonzero Schur functions give abasis of the cohomology ring, and s λ is Poincar´e dual to the class of a corresponding Schubertvariety . Relative to a fixed full flag V ⊂ V · · · ⊂ V n = C n , a point W ∈ Gr( k, n ) is in theSchubert variety C λ for the partition λ if and only if dim( W ∩ V i ) ≥ i + λ i − ( n − k ) for ≤ i ≤ n . By (1.2), then, c λµν is the coefficient of [ C λ ] in the cup product [ C µ ] · [ C ν ] .Therefore we have a geometric interpretation of c λµν as(4) the multiplicity of [ C λ ] in the cup product [ C µ ] · [ C ν ] .When | µ | + | ν | = dim(Gr( k, n )) = k ( n − k ) , the coefficient c ( n − k ) k µν is the (finite!) number of k -planes satisfying the dimension criteria of both C µ and C ν . For example, one can use thisanalysis to compute the number of lines which intersect five general 3-planes in P = P ( C ) .All of the above is to say that Schur functions provide important insight into areas ofgeometry, algebra, and representation theory. The goal of this paper is to provide a smallfirst step in this direction in the odd setting, in a sense we will now describe.1.2. Outline of this paper.
An ongoing project initiated in joint work with Khovanov andLauda is an attempt to give a construction of the Ozsv´ath-Rasmussen-Szab ´o odd Khovanovhomology [ORS07] via a 2-representation-theoretic approach analogous to Webster’s con-struction of (even) Khovanov homology [Web10a], [Web10b]. As a byproduct of this inves-tigation, an “odd” analogue OΛ of the algebra Λ was defined and found to admit signedanalogues of many of the combinatorial properties of Λ [EK11], [EKL11]. We gave two dif-ferent candidates for elements playing the role analogous to that of Schur functions. Thepresent work gives a third candidate, proves all three definitions are equivalent, and provesan odd analogue of the Littlewood-Richardson rule. HE ODD LITTLEWOOD-RICHARDSON RULE 3
Section 2.1 reviews the definitions and basic properties of odd symmetric functions andodd Schur functions; the short Section 2.2 gives a convenient notation for the many signswhich arise in the odd setting. Section 3.1 contains the new definition of odd Schur functions,and Section 3.2 proves the following.
Theorem.
The three definitions of odd Schur functions all coincide.In Section 4.1 we review the even Littlewood-Richardson rule, and in Section 4.2 we formu-late and prove an odd analogue.
Theorem.
The odd Littlewood-Richardson coefficient c λµν is a signed count of semistandardskew tableaux S of shape λ/µ and content ν such that the row word of S is Yamanouchi.Finally, in Section 4.3, we re-cast the odd Littlewood-Richardson rule in the language ofKnutson-Tao hives.1.3. Acknowledgments.
The author thanks Mikhail Khovanov for many helpful conversa-tions and suggestions. For the duration of this work, the author was supported by an NSFGraduate Research Fellowship.2. R
EVIEW OF ODD SYMMETRIC FUNCTIONS
Odd symmetric polynomials.
Now we will briefly review the constructions of [EK11],[EKL11]. Let n ≥ and let OPol n = Z h x , . . . , x n i / ( x i x j + x j x i if i = j ) be the ring of skew polynomials in n variables. The ascend-sorting of a monomial x i · · · x i r in OPol n is defined to be the monomial obtained by sorting the subscripts i , . . . , i r into non-decreasing order without introducing any sign. For example, the ascend-sorting of x x x is x x x . We will sometimes write : X : for the ascend-sorting of a monomial X .Although the generators x , . . . , x n pairwise supercommute, the algebra OPol n is not su-percommutative: ( x + x ) x = x ( x − x ) . The algebra
OPol n is finite dimensional in each degree and has no zero divisors, but it doesnot have unique factorization: ( x − x ) = x + x = ( x + x ) . Have the symmetric group S n act on the free algebra Z h x , . . . , x n i as(2.1) s i ( x j ) = − x i +1 if j = i, − x i if j = i + 1 , − x j if j = i, i + 1 , where s i is the transposition of i and i + 1 and the action of s i is a ring endomorphism. For i = 1 , . . . , n − , the i -th odd divided difference operator is the Z -linear map ∂ i : Z h x , . . . , x n i → Z h x , . . . , x n i defined by ∂ i ( x j ) = ( if j = i, i + 1 , otherwise, ∂ i ( f g ) = ∂ i ( f ) g + s i ( f ) ∂ i ( g ) . (2.2) ALEXANDERP.ELLIS
The second line is called the Leibniz rule. It is easy to check that ∂ i ( x j x k + x k x j ) = 0 for all j, k , so we can consider ∂ i as an operator on OPol n . On OPol n , the kernel and imageof ∂ i coincide; by contrast with the even case, however, these do not equal the space ofinvariants or anti-invariants of the action of s i . Definition 2.1.
The ring of odd symmetric polynomials is the subring(2.3) OΛ n = n − \ i =1 ker( ∂ i ) ⊂ OPol n . We endow OΛ n with a Z - and a Z / -grading, whose degree functions we denote by deg Z and deg s respectively. Write deg( f ) = (deg Z ( f ) , deg s ( f )) for short. The degree of each x i isdefined to be deg( x i ) = (2 , . We will consider OΛ n as a superalgebra via the Z / -grading. In particular, the productstructure on OΛ n ⊗ OΛ n is, on homogeneous elements, ( f ⊗ g )( f ′ ⊗ g ′ ) = ( − deg s ( g ) deg s ( f ′ ) ( f f ′ ) ⊗ ( gg ′ ) . For i = 1 , . . . , n , let e x i = ( − i − x i . We define elements of OPol n for each k ≥ , e k = X ≤ i <... n , then e k = 0 . Both these families of skewpolynomials are odd symmetric, and they satisfy the relation ℓ X k =0 ( − k ( k +1) e k h ℓ − k = 0 if ℓ ≥ . They also satisfy the odd defining relations , e a e b = e b e a if a + b is even ,e a e b + ( − a e b e a = ( − a e a +1 e b − + e b − e a +1 if a + b if odd . (2.4)The relations (2.4) also hold if all e ’s are replaced by h ’s.This next proposition follows from Section 2.1 of [EKL11]. Proposition 2.2.
The algebra OΛ has a presentation by generators e = 1 , e , e , . . . and re-lations (2.4). A basis for OΛ in Z -degree k is given by all products e λ · · · e λ r with λ ≥ . . . λ r ≥ , | λ | + . . . + | λ r | = k . The same is true if all e ’s are replaced by h ’s.The products e λ = e λ · · · e λ r and h λ = h λ · · · h λ r with λ ≥ . . . ≥ λ r are called oddelementary symmetric functions and odd complete symmetric functions , respectively. The sets ofelementary and complete functions in Z -degree k are naturally indexed by all partitions of k . HE ODD LITTLEWOOD-RICHARDSON RULE 5
There are maps
OPol n +1 → OPol n x i x i if ≤ i ≤ n,x n +1 . These induce maps OΛ n +1 → OΛ n which send h k h k and e k e k for all k . The inverselimit in the category of graded rings of the resulting system is called the ring of odd symmetricfunctions , OΛ = lim ←− OΛ n . The ring OΛ can be given the structure of a Hopf superalgebra [EK11]. The coproduct is ∆( h k ) = X i + j = k h i ⊗ h j , ∆( e k ) = X i + j = k e i ⊗ e j . As a Hopf superalgebra, OΛ is neither commutative nor cocommutative.As a result of Proposition 2.2, we can define certain symmetries of OΛ , ψ ( h k ) = e k bialgebra automorphism (not an involution), ψ ( h k ) = ( − k ( k +1) h k algebra involution (not a coalgebra homomorphism), ψ ( h k ) = h k algebra anti-involution (not a coalgebra homomorphism).The map ψ ψ is an involution, and both ψ , ψ commute with ψ . The antipode of the Hopfsuperalgebra structure on OΛ is S = ψ ψ ψ . For a partition λ = ( λ , . . . , λ r ) , define ǫ λ = ( − P j λ Tj ( λ Tj − , where λ T is the transpose (or dual, or conjugate) partition to λ . That is, λ Tj is the height ofthe j -th column of λ . The involution ψ ψ exchanges the elementary and complete bases upto sign,(2.5) ψ ψ ( e λ ) = ( − | λ | ǫ λ T h λ , ψ ψ ( h λ ) = ( − | λ | ǫ λ T e λ . In [EK11], an odd analogue of the Schur functions was given as follows. From here on, wewill identify partitions and Young diagrams without comment. Recall that a
Young tableau T on a partition λ with entries in an ordered alphabet A is an assignment of an element of A toeach box of λ . When considering OΛ we will take A = Z > and when considering OΛ n wewill take A = { , . . . , n } . A tableau T is called semistandard if its entries are non-decreasingin rows (left to right) and strictly increasing in columns (top to bottom). The row word w r ( T ) of a tableau T is the word in the alphabet A obtained by reading the entries of T from left toright, bottom to top. For example, T = 1 1 22 3 is a semistandard Young tableau of shape (3 , , and w r ( T ) = 23112 . ALEXANDERP.ELLIS
The content of a tableau T is the tuple ( a , . . . , a r ) , where a i is the number of entries of T equal to i . For each partition λ , there is a unique Young tableau T λ of shape and content bothequal to λ . For example, T (21) = 1 12 , T (311) = 1 1 123 . If T is a Young tableau, write sh( T ) for its shape and ct( T ) for its content. Let SSY T ( λ ) be theset of semistandard Young tableaux of shape λ and let SSY T ( λ, µ ) be the set of semistandardYoung tableaux of shape λ and content µ .In order to use tableaux in the odd setting, define the sign of a tableau T , denoted sign ( T ) ,to be the sign of the minimal length permutation which sorts w r ( T ) into non-decreasingorder. For instance, with T as above,sign ( T ) = ( − = − , sign ( T (21) ) = ( − = 1 , sign ( T (311) ) = ( − = − . For partitions λ, µ , the odd Kostka number K λµ is defined to be a signed count of semistandardtableaux of shape λ and content µ ,(2.6) K λµ = sign ( T λ ) X T ∈ SSY T ( λ,µ ) sign ( T ) . It is readily verified that as a matrix, ( K λµ ) | λ | = | µ | = k is lower triangular and unimodular whenthe partitions basis is ordered lexicographically, so we can define the family of combinatorial(or Kostka) odd Schur functions by the change of basis relation(2.7) h µ = X λ ⊢ k K λµ s Kλ , where µ ⊢ k .The following proposition combines several of the statements from Section 3.3 of [EK11]. Proposition 2.3.
The family { s Kλ } λ ⊢ k is an integral unimodular basis for OΛ in Z -degree k .This family is signed-orthonormal,(2.8) ( s Kλ , s Kµ ) = ǫ λ δ λµ , with respect to the bilinear form of [EK11]. The involution ψ ψ and the anti-involution ψ act on s λ as(2.9) ψ ψ ( s Kλ ) = ( − ℓ ( w λ )+ | λ | s Kλ T , ψ ( s Kλ ) = ǫ λ sign ( T λ ) s Kλ . Here, w λ is the element of S k which is combinatorially defined in Proposition 2.14 of [EK11](it is the minimal representative of the unique double coset in S λ T \ S k /S λ which gives anonzero homomorphism from Ind S k S λ ( V triv ) to Ind S k S λT ( V sign ) ) and ℓ is the Coxeter length func-tion.In [EKL11], another family of Schur functions was introduced. Let ∂ w = ∂ ( ∂ ∂ ) · · · ( ∂ n − · · · ∂ ) , a particular choice of longest odd divided difference operator (odd divided difference oper-ators corresponding to other choices of reduced expression could differ by a factor of − ). HE ODD LITTLEWOOD-RICHARDSON RULE 7
For a skew polynomial f , let f w be the result of acting by w on f via the action (2.1). The odd-symmetrized Schur functions are defined by(2.10) s sλ = ( − n ) (cid:16) ∂ w ( x λ · · · x λ n n x n − x n − · · · x n − ) (cid:17) w . For the motivation behind this name, see Section 2.2 of [EKL11]. The goal of Section 3.2 is toprove that for all λ , s sλ = s Kλ .2.2. Boxterpretations.
It will be convenient to have a systematic way to handle the signswhich arise in the odd setting. Most combinatorial quantities arising in these signs canbe described by counting certain boxes in some Young diagram or tableau; we call thesedescriptions “boxterpretations.” Here are some which have already arisen: • For a Young diagram λ , the sum P j λ Tj ( λ Tj − can be described as: for each box B , add the number of boxes directly above B . This is the exponent of − used indefining the sign ǫ λ . • For a Young diagram λ , the quantity ℓ ( w λ ) can be described as: for each box B , addthe number of boxes above and to the right of B . • For a Young tableau T : for each box B , add the number of boxes above and withsmaller entry. Then sign ( T ) equals − raised to this count. • As a special case of the previous, to obtain sign ( T λ ) , for each box B , count the numberof boxes above B .We will use notations adapted from [Ful97] to help condense these descriptions. If a box B of a Young diagram is in a row above that of a box B ′ , we say B is North of B ′ ; if B is Northof or in the same row as B ′ , we say B is north of B ′ . Likewise for East/east, West/west,and South/south. Write N ( B ) for the number of boxes North of B , sW ( B ) for the numberof boxes southWest (both south and West) of B , and so forth. If we are considering a Youngtableau and decorate a direction with one of { <, ≤ , >, ≥} , this means to only count boxeswith entry lower than (lower than or equal to, greater than, greater than or equal to) B .So E > ( B ) means the number of boxes East of B with strictly greater entry than B . If weevaluate one of these counting functions on a diagram or a tableau, we mean to sum over allboxes, evaluating the function at each. For instance, SW ≥ ( T ) = X B ∈ T { boxes of T SouthWest of B with entry ≥ that of B } . One last decoration: dN means directly North, that is, North and neither East nor West (andlikewise for the other counts). Example 2.4.
Let T = 1 1 2 22 3 3 43 4 45 6 . Then dN ( T ) = 16 , E > ( T ) = 28 , and sW ( T ) = 47 .In this notation, we have the following boxterpretations: ǫ λ = ( − dN ( λ ) , ( − ℓ ( w λ ) = ( − NE ( λ ) , sign ( T ) = ( − N < ( T ) , sign ( T λ ) = ( − N ( λ ) . ALEXANDERP.ELLIS
3. O
DD PLACTIC S CHUR FUNCTIONS
Definition and basic properties.
Let A = { a , a , . . . } be an ordered alphabet. In prac-tice, we will take A = Z > when working with OΛ and A = { , , . . . , n } when working with OΛ n . In order to add, multiply, and assign signs to tableaux, we will use the odd plactic ring Z P l , which is the unital ring defined bygenerators: A relations: yzx = − yxz if x < y ≤ z ( K ′ ) ,xzy = − zxy if x ≤ y < z ( K ′′ ) . (3.1)When we want to emphasize that the alphabet in question is { , , . . . , n } , we will sometimeswrite Z P l n instead of Z P l . The relations ( K ′ ) , ( K ′′ ) are called elementary Knuth transformations.We define a map from the set of semistandard Young tableaux with entries in the alphabet A to the odd plactic ring by { SSYTs } → Z P l,T w r ( T ) . (3.2)Since both the relations ( K ′ ) , ( K ′′ ) are transpositions of letters with a minus sign, Z P l n sitsas an intermediate quotient between a free algebra and the skew polynomial ring, Z h x , . . . , x n i ։ Z P l n ։ OPol n , where the first map sends e x i to i and the second map sends i to e x i . If w is a word in Z P l n ,we will write e x w for the image of w in OPol n . In particular, a semistandard Young tableau T is sent to e x w r ( T ) .The utility of the plactic ring is in large part due to the following remarkable theorem. Theorem 3.1 ([Ful97], Section 2.1) . Every word is equivalent, via relations ( K ′ ) and ( K ′′ ) , tothe row word w r ( T ) of a unique tableau T .Thus the set of all Young tableaux with entries in A forms a basis of Z P l . We will informallyrefer to the multiplication of tableaux in the following; what we mean is the multiplication oftheir row words in Z P l . In terms of tableaux, the relations ( K ′ ) and ( K ′′ ) can be interpretedas “bumping transformations”: ( K ′ ) y z · x = − x zy if x < y ≤ z, ( K ′′ ) x z · y = − x yz if x ≤ y < z. For a detailed exposition of bumping, see Section 1.1 of [Ful97].If a word w is known to be the row word of some tableau, then it is easy to reconstructthe tableau from the word. Since the row entries of a tableau never decrease from left toright and the column entries must always increase from top to bottom, reading the word w from left to right until the first adjacent decreasing pair simply gives the bottom row of thetableau. Then continuing to read until the next adjacent decreasing pair gives the second tobottom row, and so forth. HE ODD LITTLEWOOD-RICHARDSON RULE 9Example 3.2.
Using Z > as the ordered alphabet, w = 53422331112 corresponds to . Definition 3.3.
Let λ be a partition. Define an element of Z P l n by(3.3) b s λ = ( − dN ( λ )+ N ( λ ) X T ∈ SSY T ( λ ) T. Its image in
OPol n ,(3.4) s pλ = ( − dN ( λ )+ N ( λ ) X T ∈ SSY T ( λ ) e x w r ( T ) , is called the plactic odd Schur function corresponding to λ .The label p in the notation for the plactic odd Schur function is to distinguish it from thecombinatorial odd Schur functions s Kλ of [EK11] and the odd-symmetrized Schur functions s sλ of [EKL11]; once we prove these three objects are equal later in this section, we will dropthe extra labels. Remark 3.4.
For the rest of the paper, we will not always specify whether we are workingin OΛ or in OΛ n . Generally speaking, results will hold in OΛ . The only change required inpassing to OΛ n is the understanding that certain elements become zero. It is easy to see that OΛ n = OΛ / ( e m : m > n ) , and it will follow from Theorem 3.8 that s λ = 0 in OΛ n if and onlyif λ has height greater than n . With this understood, the proofs and results in the rest of thispaper work in either context. Lemma 3.5. If λ = (1 k ) , then up to sign, all three Schur functions coincide with the corre-sponding elementary polynomial:(3.5) s K (1 k ) = s p (1 k ) = s s (1 k ) = ( − k ( k − e k . If λ = ( k ) , the combinatorial and plactic Schur functions coincide with the correspondingcomplete polynomial:(3.6) s K ( k ) = s p ( k ) = h k . The last equation equals s s ( k ) too, but we will prove this later. Proof.
The equality s K (1 k ) = ( − k ( k − e k follows from Proposition 3.10 of [EK11] and theequality s s (1 k ) = ( − k ( k − e k is Lemma 2.25 of [EKL11]. Since the row word of a semis-tandard Young tableau on the shape (1 k ) is i k · · · i for positive integers i < . . . < i k and ittakes k ( k − transpositions to ascend-sort the monomial e x i k · · · e x i , the equality with s p (1 k ) holds. It is similar but easier to show s p ( k ) = h k , and s K ( k ) = h k is obvious because the onlysemistandard Young tableau of content ( k ) is a row of 1’s. (cid:3) Comparison with previous definitions.
For a partition λ , let iλ be the Young diagramobtained by removing rows 1 through i from the diagram corresponding to λ . Similarly,let λi , i | λ , and λ | i be obtained by removing rows i through the bottom, columns through i , and columns i through the rightmost respectively. We say that a skew shape is a verticalstrip (respectively horizontal strip ) if no two of its boxes are in the same row (respectivelycolumn). We say that a diagram µ is obtained from λ by adding a vertical strip if λ ⊂ µ and µ/λ is a vertical strip; likewise for horizontal strips. The following proposition was provedin [EKL11]. Proposition 3.6 (Odd Pieri rule, e -right odd-symmetrized version) . Let λ be a partition. Then(3.7) s sλ s s (1 k ) = X µ ( − (cid:12)(cid:12)(cid:12) i λ (cid:12)(cid:12)(cid:12) + ... + (cid:12)(cid:12)(cid:12) ikλ (cid:12)(cid:12)(cid:12) s sµ . The sum is over all µ obtained from λ by adding a vertical strip of size k , and i , . . . , i k arethe rows of λ to which a box was added.The plactic odd Schur functions satisfy the same relation. We prove the horizontal stripvariant instead, for simplicity. Proposition 3.7 (Odd Pieri rule, h -right plactic version) . Let λ be a partition. Then(3.8) ( − NE ( λ ) s pλ s p ( k ) = X µ ( − NE ( µ ) ( − | i | λ | + ... + | i k | λ | s pµ . The sum is over all µ obtained from λ by adding a horizontal strip of size k , and i , . . . , i k are the columns of λ to which a box was added. Proof.
Expanding all Schur functions as odd plactic sums, we want to prove ( − dN ( λ )+ N ( λ )+ NE ( λ ) e x w r ( T ) e x w r ( V ) = ( − dN ( µ )+ N ( µ )+ NE ( µ ) ( − | i | λ | + ... + | i k | λ | e x w r ( U ) whenever T is a Young tableau of shape λ , V is a Young tableau of shape ( k ) , T V = U in the even plactic ring with sh( U ) = µ , and the boxes of µ/λ are in columns i , . . . , i k .This is because mod 2 the even and odd plactic rings are isomorphic (by the obvious map, w r ( T ) w r ( T ) ), so the set of products e x w r ( T ) e x w r ( V ) and the set of terms e x w r ( U ) which occuron the right-hand side of (3.8) are in bijection, with ( T, V ) corresponding to U if and only if T V = U in the even plactic ring. Suppose the leftmost box of V has entry j . If that box endsup in column i , we claim ( − dN ( λ )+ N ( λ )+ NE ( λ ) e x w r ( T ) e x j = ( − dN ( µ )+ N ( µ )+ NE ( µ )+ | i | λ | e x w r ( U ) . For any partition ν , ( − dN ( ν )+ N ( ν )+ NE ( ν ) = ( − NW ( ν ) . The new box of µ is not NorthWestof any other box (since it must be a southeast corner), so the sign discrepancy only countsthose boxes NorthWest of the new box. The sign ( − | i | λ | counts boxes NorthEast of the newbox, so the overall sign is ( − P j ( λ j − . And this is precisely the sign between e x w r ( T ) e x j and e x w r ( U ) , since bumping a box past a row of length r incurs a sign of ( − r − . Finally, note thatas the boxes of V are added one at a time, the signs cancel telescopically so as to yield thesign of equation (3.8). (cid:3) We now have the tools necessary to prove the main result of this section.
HE ODD LITTLEWOOD-RICHARDSON RULE 11Theorem 3.8.
The three notions of odd Schur function all coincide: for any partition λ ,(3.9) s Kλ = s pλ = s sλ . Proof.
The proof that s Kλ = s pλ is similar in spirit to the proof of the odd Pieri rule, h -rightplactic version. By the same sort of analysis as in that proof, one shows that for a partition µ = ( µ , . . . , µ r ) , h µ = h µ · · · h µ r = X ct( T )= µ ( − NE ( T )+ NE < ( T )+ dN ( T )+ N ( T ) e x w r ( T ) = X λ X T ∈ SSY T ( λ,µ ) ( − NE ( λ )+ NE < ( T ) s pλ . Since h µ = P λ K λµ s Kλ and both { h µ } , { s Kλ } are integral bases of the ring of odd symmetricfunctions, it suffices to check ( − NE ( λ )+ NE < ( T ) = ( − N ( λ )+ N < ( T ) whenever T is a Young tableau of shape λ (the right-hand side is the sign with which T iscounted in defining K λµ ; see Section 2.1). The sum computing the sign from a particularbox B of T involves two types of other box: those Northwest and those NorthEast of B . Forthose NorthEast, the sign is identical. Those Northwest are ignored by the left-hand sign,but the entry of such a box is necessarily less than that of B (since T is semistandard), so theright-hand sign is +1 . Hence s Kλ = s pλ .We now use the Pieri rules (Propositions 3.6, 3.7) to prove s pλ = s sλ . For λ = (1 k ) , this istrue by Lemma 3.5. Using this as a base case, we now induct on the width of λ , and withineach particular width we induct with respect to the lexicographic order of λ T . Since we haveshown s Kλ = s pλ , we can apply the involution ψ ψ to equation (3.8) to show the plactic Schurfunctions obey an e -right Pieri rule of the same form and signs as the one obeyed by theSchur functions s Kλ ,(3.10) s pλ s p (1 k ) = X µ ( − (cid:12)(cid:12)(cid:12) i λ (cid:12)(cid:12)(cid:12) + ... + (cid:12)(cid:12)(cid:12) ikλ (cid:12)(cid:12)(cid:12) s pµ . Let λ be a partition of width λ = r . Using the e -right Pieri rule, both types of Schur functionsatisfy s ( λ T ) · · · s ( λ Tr ) = X µ ± s µ , where each µ has width at most r and is lexicographically greater than or equal to λ . Thecoefficient of s λ on the right-hand side is ± . All the signs ± are the same for the two typesof Schur function, so this allows us to solve for both s pλ , s sλ in terms of elementary functionsby the same expressions; hence s pλ = s sλ . (cid:3) Corollary 3.9.
The span of the b s λ in Z P l n is a subalgebra isomorphic to OΛ n , and this subal-gebra is taken isomorphically onto OΛ n ⊂ OPol n by the map w e x w .For the rest of this paper, we will drop the superscript labels on odd Schur functions.
4. L
ITTLEWOOD -R ICHARDSON RULES
The even Littlewood-Richardson rule.
For this section only, we work in the even setting.
Let µ, ν, λ be partitions. The
Littlewood-Richardson coefficient c λµν is the coefficient of s λ in s µ s ν , s µ s ν = X λ c λµν s λ . If | µ | + | ν | 6 = | λ | , then c λµν = 0 . Schur functions are generating functions for semistandardYoung tableaux of a given shape, s λ = X T ∈ SSY T ( λ ) x w r ( T ) . It follows that, for any fixed semistandard Young tableau T of shape λ ,(4.1) c λµν = { U ∈ SSY T ( µ ) , V ∈ SSY T ( ν ) : U V = T } . Here, the product of tableaux is taken in the (even) plactic ring. If T = T λ , then it is not hardto see that U V = T λ forces V to be T ν ; equation (4.1) becomes(4.2) c λµν = { U ∈ SSY T ( µ ) : U T ν = T λ } . Since computing the set on the right-hand side of (4.2) can be tricky in practice, we wouldlike to have a simpler combinatorial description of the c λµν . The Littlewood-Richardson ruleprovides one of many such simpler descriptions. Good accounts of the rule and its proof aregiven in [Ful97, Chapter 5] and [Sta99, Appendix A1]. We will review the terminology andstatement here.Recall that a (Young) skew shape λ/µ is the complement of a subdiagram µ ⊆ λ . A semi-standard skew tableau is a skew shape which has been filled with entries from some orderedalphabet, subject to the same rules as for a semistandard tableau: entries must strictly in-crease in columns (top to bottom) and must not decrease in rows (left to right). We write SSY T ( λ/µ ) for the set of semistandard skew tableaux of shape λ/µ and SSY T ( λ/µ, ν ) forthe set of semistandard skew tableaux of shape λ/µ and content ν .A word w = w · · · w r in some ordered alphabet is called Yamanouchi (or a reverse latticeword ) if, when read backwards, each initial subword has at least as many a ’s as b ’s whenever a < b . For example, is Yamanouchi but and are not. A skew tableau S is called a Littlewood-Richardson tableau if w r ( S ) is a Yamanouchi word. The following isProposition 3, Chapter 5 of [Ful97] and Theorem A1.3.3 of [Sta99]. Theorem 4.1 ((Even) Littlewood-Richardson Rule) . The coefficient c λµν equals the number ofLittlewood-Richardson tableaux S of shape λ/µ and content ν .One specific bijection between the set described in the theorem and the set in equation(4.2) is described in the following section, where we use it to deduce an odd analogue ofTheorem 4.1. Example 4.2.
Let µ ⊆ λ be a subdiagram and let k = | λ | − | µ | ≥ . If S is a Littlewood-Richardson tableau of shape λ/µ and content ( k ) , then no two boxes of S can be in the same HE ODD LITTLEWOOD-RICHARDSON RULE 13 column (since all entries equal 1). And on any such skew shape λ/µ , there is exactly onetableau of content ( k ) . Thus c λµ ( k ) = 1 if λ/µ is a horizontal strip and equals 0 otherwise. Wehave deduced the (even) Pieri rule,(4.3) s µ s ( k ) = X λλ/µ is ahorizontal strip s λ . Using the standard involution ω on Λ , or just arguing in analogy with the above, the same istrue if ( k ) is replaced by (1 k ) and “horizontal” is replaced by “vertical.” Example 4.3.
The lowest degree product which is not described by the Pieri rule is s s = s + s + s + 2 s + s + s + s . The odd Littlewood-Richardson rule.
The sign of a Young tableau T is the sign be-tween its row word monomial e x w r ( T ) and the ascend-sorting of that monomial. As explainedin Section 2.1, this sign equals ( − N < ( T ) . If S is a skew tableau of shape λ/µ , let j be an el-ement of the alphabet less than every entry of S and let b S be the Young tableau of shape λ formed by placing j in each box of µ and filling the rest so as to match S . We then define thesign of S to be sign ( S ) = sign ( b S ) = N < ( b S ) . When µ = (0) , this reduces to the sign of a tableau as defined earlier. More generally, when-ever we write a count E, nW ≥ , dS, . . . evaluated on S , read b S for S . Example 4.4.
To either alphabet Z > or { , , . . . , n } , we can always adjoin 0 and take j = 0 .If λ = (3 , , , , µ = (2 , , , and S = 11 223 , then b S = 0 0 10 1 20 23 and sign ( S ) = ( − = 1 . Remark 4.5.
We consider the partition µ to be part of the data of S . For example, both (1 , / (1) and (2) / (1) consist of a single box, but if we fill these two boxes with equal entries,the resulting skew tableaux have opposite signs. Definition 4.6.
Let µ, ν, λ be partitions. The odd Littlewood-Richardson coefficient c λµν is thecoefficient of s λ when s µ s ν is expanded in the basis of odd Schur functions, s µ s ν = X λ c λµν s λ . If | µ | + | ν | 6 = | λ | , then c λµν = 0 .Note that the odd Pieri rules compute certain odd Littlewood-Richardson coefficients. Us-ing the involution ψ ψ and the anti-involution ψ , we know the odd Littlewood-Richardsoncoefficient c λµν whenever µ or ν has either height or width 1.If Y, Z are two nonzero monomials in
OPol n such that Y = ± Z , then let sign ( Y, Z ) denotethe sign between them; for example sign ( x x x , x x x ) = − . The odd Littlewood-Richardson coefficient c λµν is(4.4) c λµν = ( − dN ( µ )+ dN ( ν )+ dN ( λ )+ N ( µ )+ P i ( λ/ν ) i | νi | X U ∈ SSY T ( µ ) UT ν = T λ ( − N < ( U ) . In the summation condition, the product
U T ν is taken in the even plactic ring (so it is anequality up to sign in Z P l n ). Proof.
First note that c λµν = ( − dN ( µ )+ dN ( ν )+ dN ( λ )+ N ( µ )+ N ( ν )+ N ( λ ) X U ∈ SSY T ( µ ) V ∈ SSY T ( ν ) UV = T λ sign ( e x w r ( U ) e x w r ( V ) , e x w r ( T λ ) )= ( − dN ( µ )+ dN ( ν )+ dN ( λ )+ N ( µ )+ N ( ν )+ N ( λ ) X U ∈ SSY T ( µ ) UT ν = T λ sign ( e x w r ( U ) e x w r ( T ν ) , e x w r ( T λ ) ) . (4.5)The first equality is immediate from the definition of c λµν and equation (3.4). The secondfollows from the following fact: if U, V are semistandard tableaux of shapes µ, ν and
U V = T λ in the even plactic ring, then V = T ν [Ful97, Section 5.2]. To turn e x w r ( U ) e x w r ( T ν ) into e x w r ( T λ ) , we proceed in two steps:(1) Ascend-sort each of e x w r ( U ) , e x w r ( T ν ) , and e x w r ( T λ ) separately. This incurs the sign ( − N < ( U )+ N ( ν )+ N ( λ ) . The monomials are now : e x w r ( T λ/ν ) :: e x w r ( T ν ) : and : e x w r ( T λ ) : ,where the colons denote ascend-sorting.(2) To sort these monomials together requires P i ( λ/ν ) i (cid:12)(cid:12) νi (cid:12)(cid:12) transpositions.The lemma follows. (cid:3) Theorem 4.8 (Odd Littlewood-Richardson rule) . The odd Littlewood-Richardson coefficient c λµν is(4.6) c λµν = ( − N ( µ )+ N ( λ ) X S ∈ SSY T ( λ/µ,ν ) w r ( S ) is Yamanouchi ( − N < ( S ) . Proof.
The Littlewood-Richardson bijection of [Ful97, Section 5.2],(4.7) { S ∈ SSY T ( λ/µ, ν ) : w r ( S ) is Yamanouchi } → { U ∈ SSY T ( µ, λ/ν ) : U T ν = T λ } , is described as follows. Let U be a semistandard Young tableau of shape µ , all of whoseentries are less than all of those of S . Then form T S ∈ SSY T ( λ ) by giving it the entries of U on µ ⊆ λ and the entries of S on λ/µ . Under the RSK correspondence, ( T λ , T S ) ↔ (cid:0) t ux v (cid:1) , where x, t have length | µ | and u, v have length | ν | . We can describe what these sub-wordscorrespond to under the RSK correspondence as well: ( U, U ) ↔ (cid:0) tx (cid:1) , ( T ν , T ν ) ↔ ( uv ) for some U ∈ SSY T ( µ, λ/ν ) . The Littlewood-Richardson bijection (4.7) assigns U to S . HE ODD LITTLEWOOD-RICHARDSON RULE 15
In order to prove the theorem, we have to relate the quantities N < ( U ) and N < ( S ) . Theodd RSK correspondence of [EK11] allows us to do this; it impliessign ( x v ) = ( − dN ( λ )+ N ( λ )+ N < ( T S ) = ( − dN ( λ )+ N ( λ )+ N < ( U )+ N < ( S ) , sign ( x ) = ( − dN ( µ )+ N < ( U )+ N < ( U ) , sign ( v ) = ( − dN ( ν ) . (4.8)Since we know the contents of x , v , and x v , the sign of the latest can be expressed in termsof the former two,(4.9) sign ( x v ) = sign ( x ) sign ( v )( − P i ( λ/ν ) i | νi | . Comparing the signs in equations (4.8) and (4.9), ( − dN ( µ )+ dN ( ν )+ dN ( λ )+ N ( λ )+ N < ( U )+ N < ( S )+ P i ( λ/ν ) i | νi | = 1 . Applying this to Lemma 4.7, the theorem follows. (cid:3)
Example 4.9.
The first interesting cancellation is c (3 , , , , = 0 (in the even case, it equals 2).The Littlewood-Richardson skew tableaux of shape (3 , , / (2 , and content (2 , are , . They have N < ( S ) equal to and , respectively. Remark 4.10.
It follows from equation (2.9) that c λµν = ( − dN ( µ )+ dN ( ν )+ dN ( λ )+ N ( µ )+ N ( ν )+ N ( λ ) c λνµ ,c λµν = ( − NE ( µ )+ NE ( ν )+ NE ( λ ) c λ T µ T ν T . (4.10)These symmetries constrain some signs associated to Young diagrams. For instance, if c λµµ =0 , then ( − dN ( λ )+ N ( λ ) = 1 .4.3. Translation to Knutson-Tao hives.
There are many combinatorial expressions for theeven Littlewood-Richardson coefficients. Among them, the several which are expressible interms of integer points of rational convex polytopes are especially interesting; one reason isthat as result of Knutson and Tao’s proof of the Saturation Conjecture [KT99], Klyachko’ssystem of inequalities [Kly98] gives a necessary and sufficient criterion for a Littlewood-Richardson coefficient to be nonzero.These expressions in terms of polytopes include Gelfand-Zeitlin (GZ) patterns [GZ86],Berenstein-Zelevinsky triangles [BZ92], the Littlewood-Richardson triangles of Pak and Vallejo[PV05], and the honeycombs and hives of Knutson and Tao [KT99]. As explained in the ex-position of [Buc00], the hive model is particularly convenient and flexible. In this section,we will write down the bijection of [PV05] from Littlewood-Richardson skew tableaux tohives, by way of Littlewood-Richardson triangles. In both the triangle and hive settings, thesign associated to a diagram will come from a quadratic form on the ambient space of thepolytope in question.
The following exposition follows [PV05] rather closely; our only contribution is to trackthe signs which arise in the odd case. For fixed n ≥ , we will work in V Z = Z ( n +22 ) and V = V Z ⊗ Z R , with coordinates { a i,j : 0 ≤ i ≤ j ≤ n } . We call elements of V triangles andwrite them pictorially as (for n = 3 ) a , a , a , a , a , a , a , a , a , a , . Definition 4.11. A Littlewood-Richardson triangle is an element A = ( a i,j ) ∈ V such that(1) a i,j ≥ for ≤ i < j ≤ n ,(2) i − X p =0 a p,j ≥ i X p =0 a p,j +1 for ≤ i ≤ j < n, and(3) j X q = i a i,q ≥ j +1 X q = i +1 a i +1 ,q for ≤ i < j < n. Note that a ,j < and a j,j < are both possible.Let △ LR denote the set of all Littlewood-Richardson triangles, a cone in V . Given aLittlewood-Richardson triangle A = ( a i,j ) ∈ △ LR , define partitions λ, µ, ν by λ j = j X p =0 a p,j , µ j = a ,j , ν i = k X q = i a i,q . Let △ LR ( λ, µ, ν ) be the set of Littlewood-Richardson triangles with fixed λ, µ, ν . Each set △ LR ( λ, µ, ν ) is a convex polytope in V .To a Littlewood-Richardson skew tableau S ∈ SSY T ( λ/µ, ν ) , associate an element A S of △ LR ( λ, µ, ν ) by setting a , = 0 , a ,j = µ j ,a i,j = { entries equal to i in row j of S } (0 < i ≤ j ) . (4.11) Lemma 4.12 ([PV05], Lemma 3.1) . Suppose | µ | + | ν | = | λ | . Then the assignment S A S is abijection between the set of Littlewood-Richardson tableaux in SSY T ( λ/µ, ν ) and the set ofinteger points of △ LR ( λ, µ, ν ) .It is easy to read off N < ( S ) from the triangle A S = ( a i,j ) . For each entry a i,j , write Y i,j for the sum of all the a p,q in the shaded region below, where the ( i, j ) place is the dot drawnwith a hollow center: HE ODD LITTLEWOOD-RICHARDSON RULE 17
More formally,(4.12) Y i,j = i − X p =0 j − X q = p a p,q . Consider the quadratic form(4.13) Q △ ( A ) = X i,j a i,j Y i,j = n X i =0 n X j = i a i,j i − X p =0 j − X q = p a p,q . Then it is immediate from the description of the bijection above that N < ( S ) = Q △ ( A S ) , so(4.14) c λµν = ( − N ( µ )+ N ( λ ) X A ∈△ LR ( λ,µ,ν ) ∩ V Z ( − Q △ ( A ) , as long as λ, µ, ν all have at most n parts. Example 4.13. If S = 121 , then A S = 02 11 0 10 1 0 0 and Q △ ( A S ) = 6 .We now translate this result into the language of hives. A hive is an element H = ( h i,j ) ∈ V with h , = 0 which satisfies theinequalities ( R ) h i,j − h i,j − ≥ h i − ,j − h i − ,j − for ≤ i < j ≤ n, ( V ) h i − ,j − h i − ,j − ≥ h i,j +1 − h i,j for ≤ i ≤ j < n, ( L ) h i,j − h i − ,j ≥ h i +1 ,j +1 − h i,j +1 for ≤ i ≤ j < n. (4.15)Let H be the set of all hives; this is a cone in V . The inequalities (4.15) have a geometricinterpretation when we express H as a triangle. Inside a triangle, there are three types ofrhombi which can be made out of two adjacent smallest-size triangles: RVL
We call these right-slanted (R), vertical (V), and left-slanted (L) rhombi. The inequalities(4.15) say that the sum of the entries at the obtuse angles of any such rhombus is greaterthan or equal to the sum at the acute angles; the three inequalities are the right-slanted,vertical, and left-slanted cases, respectively.As with Littlewood-Richardson triangles, we associate three partitions to a hive, λ j = h j,j − h j − ,j − , µ j = h ,j − h ,j − , ν i = h i,n − h i − ,n . Let H ( λ, µ, ν ) be the set of all hives with corresponding partitions λ, µ, ν . Each H ( λ, µ, ν ) is aconvex polytope in V . Theorem 4.15 ([PV05], Theorem 4.1) . Let
Φ : V → V be the linear map which takes A = ( a i,j ) to H = ( h i,j ) , where(4.16) h i,j = i X p =0 j X q = p a p,q . Then Φ is a volume-preserving isomorphism and induces bijections △ LR ( λ, µ, ν ) → H ( λ, µ, ν ) for all λ, µ, ν .As a matter of convention, let h i,j = 0 if either i > j , i < , or j > n . It follows fromequation (4.16) and an inclusion-exclusion argument that h i,j = a i,j + h i − ,j + h i,j − − h i − ,j − if ≤ i < j ≤ n,h i,i = a i,i + h i − ,i if ≤ i = j ≤ n. (4.17) HE ODD LITTLEWOOD-RICHARDSON RULE 19
Let Q H = Q △ ◦ Φ − . Then for a hive H = ( h i,j ) , equations (4.17) imply(4.18) Q H ( H ) = n X i =1 n X j = i h i − ,j − ( h i,j − h i − ,j − h i,j − + h i − ,j − ) − n − X i =1 h i,i . Note that the parenthesized term is non-negative by the right-slanted rhombus inequality ( R ) . It follows that if µ, ν, λ are partitions with at most n parts, then(4.19) N < ( S ) = Q △ ( A S ) = Q H (Φ( A S )) , so(4.20) c λµν = ( − N ( µ )+ N ( λ ) X H ∈ H ( λ,µ,ν ) ∩ V Z ( − Q H ( H ) . The quantity N ( µ ) + N ( λ ) can also be expressed as a quadratic form in either the variables ( a i,j ) or the variables ( h i,j ) . Example 4.16.
With S and A S as in Example 4.13,(4.21) Φ( A S ) = 02 33 4 53 5 6 6 and Q H (Φ( A S )) = 6 . R EFERENCES [Buc00] A. Buch. The saturation conjecture (after A. Knutson and T. Tao).
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