We consider Buch's rule for K-theory of the Grassmannian, in the Schur
multiplicity-free cases classified by Stembridge. Using a result of Knutson,
one sees that Buch's coefficients are related to Moebius inversion. We give a
direct combinatorial proof of this by considering the product expansion for
Grassmannian Grothendieck polynomials. We end with an extension to the
multiplicity-free cases of Thomas and Yong

Snider, Michelle

English

arXiv

We consider Buch's rule for K-theory of the Grassmannian, in the Schur multiplicity-free cases classified by Stembridge. Using a result of Knutson, one sees that Buch's coefficients are related to Möbius inversion. We give a direct combinatorial proof of this by considering the product expansion for Grassmannian Grothendieck polynomials. We end with an extension to the multiplicity-free cases of Thomas and Yong

Michelle Snider

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FPSAC 2009, Hagenberg, Austria DMTCS proc. AK, 2009, 805–812A Combinatorial Approach to Multiplicity-FreeRichardson Subvarieties of theGrassmannianMichelle Snider11Cornell University, Ithaca, NY 14853, USA.Abstract. We consider Buch’s rule for K-theory of the Grassmannian, in the Schur multiplicity-free cases classifiedby Stembridge. Using a result of Knutson, one sees that Buch’s coefficients are related to Möbius inversion. We give adirect combinatorial proof of this by considering the product expansion for Grassmannian Grothendieck polynomials.We end with an extension to the multiplicity-free cases of Thomas and Yong.Résumé. On examine la règle de Buch pour la K-théorie de la variété grassmannienne dans les cas sans multiplicité deSchur, qui ont étés classifiés par Stembridge. En utilisant un résultat de Knutson, on démontre que les coefficients deBuch sont liés à l’inversion de Möbius. On en fait une preuve directe et combinatoire qui passe par le developpementde produits de polynômes de Grothendieck. Pour conclure, on donne une application de cette théorie aux cas sansmultiplicité de Thomas et Yong.Keywords: Grassmannian, Richardson varieties, Grothendieck polynomials, Schur multiplicity free1 Motivation1.1 Schubert and Richardson varietiesWe consider the Grassmannian GrkCn := {V ≤ Cn | dim(V ) = k}. For a partition λ contained in ak× (n− k) box, consider the path from the northeast corner to its southwest corner of the box that tracesthe partition. For the standard flag (Ci = (∗1, . . . , ∗i, 0, . . . 0)), we define the Schubert variety asXλ = {V ∈ GrkCn | dim(V ∩ Ci) ≥ #( south steps in the first i steps of the path )}.We denote the Schubert class in cohomology as Sλ := [Xλ]H ∈ H?(GrkCn). The set{Sλ | λ ⊂ k × (n− k) box}forms a Z−basis for H?(GrkCn), whereSλSµ =∑cνλµSν1365–8050 c© 2009 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France806 Michelle Sniderfor |ν| = |λ| + |µ|, ν ⊂ k × (n − k) box, and cνλµ the Littlewood-Richardson coefficients. This followsfrom the surjective homomorphism{ring of symmetric polynomials} {H?(GrkCn)}sλ 7−→{Sλ, if λ fits in k × (n− k) box;0, otherwise.for Schur functions sλ.The Möbius function µP(ν) is defined recursively on a poset P as the unique function satisfying∑α≥PνµP(α) = 1The connection of this definition to K-classes is shown in [Kn08]. Since we are primarily interested inworking in K-theory, we will use [A] to denote the K-class of a subscheme of A, and [A]H to denoteits homology class. Any subvariety X of a flag manifold is rationally equivalent to a linear combina-tion of Schubert cycles with uniquely determined non-negative integer coefficients [Br03]. We say X ismultiplicity-free if these coefficients are 0 or 1.Theorem 1 [Kn08] Let X be a multiplicity-free irreducible subvariety of G/P , in the sense of [Br03],with [X]H =∑d∈D[Xd]H ,D a subset of the Bruhat order, and P a parabolic subgroup. LetP ⊆W/WPbe the set of Schubert varieties contained in ∪d∈DXd (an order ideal in the Bruhat order on W/WP ).Then as an element of K(G/P ),[X] =∑Xe⊆Sd∈D XdµP(Xe) [Xe].We will give an independent combinatorial proof of this fact in the case that X is a multiplicity-freeRichardson variety in a Grassmannian, the intersection of a Schubert variety Xλ with an opposite Schu-bert variety w0 · Xµ, for w0 the longest word. For any Xλ ⊂ GrkCn, let Gλ := [Xλ]. We have that{Gλ | λ ⊂ k × (n− k) box} form a basis for K(GrkCn). For certain symmetric polynomials gλ whichwe will define in the next section, we have a surjective homomorphism [Bu02]:{ring of symmetric functions} {K(GrkCn)}gλ 7−→{Gλ, if λ fits in k × (n− k) box;0, otherwise.Our main theorem will show that, for a poset P that we will define,Gλ ·Gµ =∑νµP(Gν) Gνwhere the sum is over ν such that ν ⊆ (k × n− k) box and |ν| ≥ |λ|+ |µ|. Our proof will proceed withsign-reversing involutions on this poset, and many reductions in the sizes of the partitions in the product.A Combinatorial Approach to Multiplicity-Free Richardson Subvarieties of the Grassmannian 8071.2 Grothendieck PolynomialsFor finite non-empty sets in Z+, a and b, we say a < b if max(a) < min(b), and a ≤ b if max(a) ≤min(b). For a partition λ, Buch defined a set-valued (English) tableau (SVT) as a filling of a Young dia-gram with nonempty sets in Z+ [Bu02]. If each box has a single entry, it is a Young tableau. A tableau isa semistandard tableau (SS) if it is weakly increasing across rows and strictly increasing down columns.The superstandard filling of a tableau is the one in which each box (i, j) has a single entry, i (its row).In all of our examples, we will use numbers smaller than 10, so we can avoid the use of set notation: weuse 45 to denote the set {4, 5}.Recall the combinatorial definition for the Schur polynomials,sλ =∑T∈SSY T (λ)xT .We consider the Grothendieck polynomials gλ of Lascoux and Schützenberger [LS82]. For λ a partition,Buch [Bu02] gives the formulagλ =∑T∈SS−SV T (λ)(−1)|T |−|λ|xTwhere|T | =∑i,j|T (i, j)|.He proves that this is a special case of the Lascoux-Schützenberger formula (which we will not need) forgπ in the case when π is a Grassmannian permutation. These are the {gλ} representing the Gλ in section1.1. As with the Schur polynomials, it is not obvious from the combinatorial definition that these poly-nomials are in fact symmetric and a basis for the symmetric polynomials [Bu02]. Linear independencefollows from the fact that the lowest homogeneous component of gλ is sλ.We define the word of a tableau w(T ) to be the entries read right to left, top to bottom. Note thatentries in a set are listed in increasing order, so that they occur in decreasing order in the word. A word iscalled a reverse lattice word (RLW) if for any initial string (e.g. 001 in 001110101),multiplicity(i) ≥ multiplicity(i+ 1) ∀ i ≥ 1A word that satisfies this condition is sometimes called an election word (or ballot sequence). Fortableaux of shape λ and µ, we define the shape λ × µ as the skew tableau formed by placing µ directlysouthwest of λ. When we refer to a filling of the shape λ×µ, we will call λ the “northeast” partition, andµ the “southwest” partition. Buch [Bu02] gives a combinatorial rule for the product of two Grothendieckpolynomials:gλgµ =∑c′νλµgνwhere the coefficients are given byc′νλµ = (−1)|ν|−|λ|−|µ|#(T )808 Michelle Sniderfor SS-SVT T of shape λ × µ, content ν, with w(T ) a RLW. We call these the K-theoretic Littlewood-Richardson numbers, since if |ν| = |λ|+ |µ|, then c′νλµ = cνλµ, the usual Littlewood-Richardson number.Figure 1 shows the calculation of the expansion of g1 · g1.1 1112 12Fig. 1: g1g1 = g2 + g11 − g21First, we note that the reverse lattice word condition requires that the filling of the northeast tableauλ always be superstandard. We will construct a poset out of all of the allowed fillings of the southwesttableau µ, where each vertex is labeled with all tableaux of a given content, and for vertices ν, ν′, ν ≤P ν′if content(ν) ⊃ content(ν′). We are interested in a poset, since its Möbius function will allow us tocompute structure constants. Note that for each tableau, the row in the poset corresponds to the numberof “extra” elements in the filling (e.g. the top row has only semistandard Young tableaux, correspondingto the product sλ · sµ).Example 1 Consider the product G2,1 ·G2,2 and its poset in Figure 2. From the second line of this poset,we can see that the coefficient of G431 is 2 since there are two tableaux on the corresponding vertex,while G422 has coefficient 1. Note that this product is H-multiplicity-free, but not K-multiplicity-free. TheK-multiplicity-free cases are extremely rare, occurring only when both partitions λ and µ are rectanglesor one of them is a single box or empty ([Bu02, Proposition 7.2]).32121312132221 1 3323 4 432112 421312 313 412 31434133122 413 4123 4123 3141122312 132323 23 1212312 123 2312323Fig. 2: The poset corresponding to G2,1 ·G2,2: the product satisfies Stembridge cases (3) and (4) from Theorem 3.We will consider our products as being inside an ambient box of size k× (n− k). That is, we limit theterms in the expansion to those indexed by partitions that fit inside this box. We note that this restrictionA Combinatorial Approach to Multiplicity-Free Richardson Subvarieties of the Grassmannian 809gives us a sub-poset of the full poset. The Möbius function on the remaining terms is unaffected by theremoval of vertices with content exceeding the box size, since all terms above a vertex ν have contentcontained in the content of ν. That is, for a given vertex ν with content in the ambient box, no vertex inits upwards order ideal will have content exceeding the ambient box. We are interested in cases in whichthe terms in the Grothendieck expansion which correspond to the Schur expansion are multiplicity-free,i.e. that their coefficients are 0 or 1.Then Theorem 1 implies the following:Theorem 2 Consider partitions λ = (λβ11 , . . . , λβll ) and µ = (µα11 , . . . , µαmm ) such that Gλ · Gµ in ak× (n− k) box is a Schur-multiplicity-free product. In the corresponding poset, for each vertex ν′, µ(ν′)gives the coefficient of Gν in the Buch expansion of the product, where ν = ν′⋃(1λ1 , . . . , lλl).These Schur-multiplicity-free cases have been classified by Stembridge [St01] as follows, and our proofexplicitly uses his analysis. We begin by recalling Stembridge’s definitions and classification of Schur-multiplicity-free cases.Definition 1 [St01] A partition µ with at most one part size (i.e., empty, or of the form (cr) for suitablec, r > 0) is said to be a rectangle. If it has k rows or k columns (i.e., k = r or k = c), then we say ν isa k-line rectangle. A partition µ with exactly two part sizes (i.e., µ = (brcs) for suitable b > c > 0 andr, s > 0) is said to be a fat hook. If it is possible to obtain a rectangle by deleting a single row or columnfrom the fat hook µ, then we say that µ is a near rectangle.We will call these top, bottom, left, or right near rectangles, to denote the location of the extra row orcolumn. We say that a product of Schur functions is multiplicity-free if all of the Littlewood-Richardsoncoefficients of the expansion are 0 or 1.Theorem 3 [St01] The product of Schur functions sλ · sµ is multiplicity-free if and only if1. λ and µ are rectangles, or2. (Pieri rule) λ is arbitrary, and µ is a 1-line rectangle3. λ is a rectangle and µ is a near-rectangle4. µ is a fat hook and λ is a 2-line rectangleWe now mention some speculative geometry that motivated our combinatorial proof of Theorem 2.Buch shows that the expansion of Xλ ∩ (w0 · Xµ) into Schubert classes has signs that alternate withdimension ([Bu02]). This suggests that there exists an exact sequence on sheaves0→ OSν∈P Xν→⊕T,|T |=|ν|+1OXcontent(T ) → · · · →⊕T,|T |=|ν|+k−1OXcontent(T ) → · · · (1)where the kth nonzero term sums over Buch Littlewood-Richardson tableaux with k − 1 extra entries.This leads to a sequence for the point in the Grassmannian corresponding to each λ,0→ C1 → · · · →⊕T,|T |=|ν|+k−1,content(T )⊆λC1 → · · ·One can hope that this sequence is in fact exact. Our main result is810 Michelle SniderTheorem 4 There exists such an exact sequence of vector spaces, and it can be explicitly constructed asa direct sum of exact sequences with exactly two non-zero terms.The proof requires an involution which pairs terms differing in size by one. In some cases, we provide asingle rule that matches all terms required. In other cases however, we must resort to a multistage divideand conquer approach, where the involution is defined differently on several disjoint subsets. AssumingTheorem 4, we can prove Theorem 2 as a corollary.Proof of Theorem 2: The exactness of the sequence (1) gives us that the alternating sum of dimensionsis 0. Thus the sum of the coefficients of the pairs of Buch tableaux, with signs alternating with numberof extra numbers (i.e. how many more than a single entry), is also 0. Together with the extra 1 fromthe single fixed point tableau, this is equivalent to the statement that the coefficient of ν′ is given by theMöbius function. 22 An Extension to the Thomas-Yong CasesConsider partitions λ and µ in a (k × (n − k)) box. We will review the notation introduced in [TY07].We call R = (λ, µ, k × (n− k)) a Richardson quadruple, and use the notation poset(R) to denote theassociated poset of allowed fillings of µ. Place λ in the upper left corner of the box, then rotate µ by 180◦(call this rotate(µ)) and place it in the lower right corner. This quadruple (λ, µ, k × (n − k)) is calledbasic if λ⋃rotate(µ) does not contain any full rows or columns. If it is not basic, we can remove all fullrows and columns to get a basic demolition R̃ = (λ̃, µ̃, k̃×(ñ− k̃)). We call each row (column) removala row (column) demolition. Notice that if λ⋂rotate(µ) 6= ∅, then Gλ · Gµ = 0, so the demolition isundefined. In order to determine whether a Richardson quadruple is multiplicity-free, we consider itsbasic demolition.Theorem 5 [TY07] A Richardson quadruple is multiplicity-free if and only if its basic Richardson quadru-ple is multiplicity-free. If a the basic demoltion of a Richardson quadruple (λ, µ, k×n−k) is multiplicity-free, then it must be in the cases classified by [St01].For example, consider the case ((4, 4, 2, 2, 1), (4, 3, 2, 1), 5 × 5). This product is not multiplicity-free,but has a basic demolition of (1, 1, 2× 2), which is multiplicity-free.We will show that our analysis of the [St01] multiplicity-free cases extends to this larger class of prod-ucts by showing that the posets of a Richardson quadruple and its basic demolition are isomorphic. Let usdefine the accessible word wA as the independent values of λ read in increasing order, or equivalentlywA(j) = 1 + #(rows of λ in column n− k − j).Lemma 1 (Column Demolition Lemma) For R = (λ, µ, k × (n − k)), if column c is full, let R̃ be thequadruple with column c removed. Then poset(R̃) is isomorphic to poset(R).Let λ = (λ1, . . . , λl) and µ = (µ1, . . . , µm). There is a full row in the diagram if and only if λr +µk−r+1 = n− k.Lemma 2 For R = (λ, µ, k × (n − k)), if λ1 = n − k, let R̃ be the quadruple with λ1 removed. Thenposet(R) is isomorphic to poset(R̃) for R̃ = ((λ2, . . . , λl), µ, (k − 1)× (n− k)).A Combinatorial Approach to Multiplicity-Free Richardson Subvarieties of the Grassmannian 811Lemma 3 (Row Demolition Lemma) For R = (λ, µ, k × (n− k)), if row r is full, then the poset(R̃) isisomorphic to poset(R).We note that basic demolition is well defined, i.e. independent of the order of full column/row removal,thus so are the corresponding isomorphisms between posets.Figure 3 is an example of a case with both a full row and column. (This product is Stembridgemultiplicity-free in any ambient box, but is a good example of the row and column demolition com-mutativity.)1 321 331 32312131231 2 2 212 21 212remove column 2remove row 2remove row 2remove column 2Fig. 3: Two demolition paths of ((2, 2), (2, 1), 3× 3) to ((1), (1), 2× 2).Proposition 1 Theorem 4 holds for any Richardson quadruple whose basic demolition is a Stembridgecase.AcknowledgementsI would like to extend special thanks to Allen Knutson for both the statement of the question and contin-uing guidance throughout the process.References[Br03] M. Brion, Multiplicity-free subvarieties of flag varieties, Contemporary Math. 331, 13-23, Amer.Math. Soc., Providence, 2003. math.AG/0211028[Bu02] A. Buch, A Littlewood-Richardson rule for the K-theory of Grassmannians, Acta Math. 189(2002), no. 1, 37–78. math.AG/0004137812 Michelle Snider[St01] J.R. Stembridge, Multiplicity-free products of Schur functions, Annals of Combinatorics (2001),no. 5, 113–121[TY07] H. Thomas, A. Yong, Multiplicity-free Schubert calculus, (2007). math/0511537[Kn08] A. Knutson, Frobenius splitting and Möbius inversion, In preparation (2008)[LS82] A. Lascoux and M.-P. Schützenberger, Structure de Hopf de l’anneau de cohomologie et del’anneau de Grothendieck d’une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982),no. 11, 629633. MR 84b:14030

A Combinatorial Approach to Multiplicity-Free Richardson Subvarieties of the Grassmannian

We consider Buch's rule for K-theory of the Grassmannian, in the Schur multiplicity-free cases classified by Stembridge. Using a result of Knutson, one sees that Buch's coefficients are related to Möbius inversion. We give a direct combinatorial proof of this by considering the product expansion for Grassmannian Grothendieck polynomials. We end with an extension to the multiplicity-free cases of Thomas and Yong.On examine la règle de Buch pour la K-théorie de la variété grassmannienne dans les cas sans multiplicité de Schur, qui ont étés classifiés par Stembridge. En utilisant un résultat de Knutson, on démontre que les coefficients de Buch sont liés à l'inversion de Möbius. On en fait une preuve directe et combinatoire qui passe par le developpement de produits de polynômes de Grothendieck. Pour conclure, on donne une application de cette théorie aux cas sans multiplicité de Thomas et Yong

A Combinatorial Approach to Multiplicity-Free Richardson Subvarieties of the Grassmannian

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