Pierre-Emmanuel Chaput, Nicolas Perrin
We prove a general combinatorial formula yielding the intersection number $c_{u,v}^w$ of three particular $Λ$-minuscule Schubert classes in any Kac-Moody homogeneous space, generalising the Littlewood-Richardson rule. The combinatorics are based on jeu de taquin rectification in a poset defined by the heap of $w$.
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We give a positive equivariant Littlewood-Richardson rule also discovered independently by Molev. Our proof generalizes a proof by Stembridge of the ordinary Littlewood-Richardson rule. We describe a weight-preserving bijection between our indexing tableaux and the Knutson-Tao puzzles.
We give elementary and short proofs of the Littlewood-Richardson rules for type A Grassmannians and maximal isotropic Grassmannians, based on the corresponding Pieri rules.
We give a closed formula of the Littlewood-Richardson coefficients.
A Littlewood-Richardson rule for Grassmannian Schubert varieties
本研究针对完全旗流形的舒伯特结构常数，提出了一种适用于其中一个因子为格拉斯曼流形情形的组合模型，完善了相关组合表示理论研究。
A consequence of work of Klyachko and of Knutson-Tao is the Horn recursion to determine when a Littlewood-Richardson coefficient is non-zero. Briefly, a Littlewood-Richardson coefficient is non-zero if and only if it satisfies a collection of Horn inequalities which are indexed by smaller non-zero Littlewood-Richardson coefficients. There are similar Littlewood-Richardson numbers for Schur P- and Q- functions. Using a mixture of combinatorics of root systems, combinatorial linear algebra in Lie algebras, and the geometry of certain cominuscule flag varieties, we give Horn recursions to determi…
We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture holds for a finite number of pairs, for any given height. Moreover, we propose a natural generalization of the conjecture to the case of skew shapes.
We describe an explicit geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties so that they break into Schubert varieties. There are no restrictions on the base field, and all multiplicities arising are 1; this is important for applications. This rule should be seen as a generalization of Pieri's rule to arbitrary Schubert classes, by way of explicit homotopies. It has a straightforward bijection to other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao's puzzles. This gives the first geometric proof and interpretation of the…
Upper cluster structure on Kac--Moody Richardson varieties
本文证明了对任意可对称化 Kac--Moody 类型，开 Richardson 簇及其在旗簇扭曲积上的广义版本的坐标环均为上簇代数，推广了有限类型中的相关结果。
We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are polynomials in the parameters which we call the Littlewood-Richardson polynomials. We give a combinatorial rule for their calculation by modifying an earlier result of B. Sagan and the author. The new rule provides a formula for these polynomials which is manifestly positive in the sense of W. Graham. We apply this formula for the calculation of the product of eq…
Constructed tableaux and a new shifted Littlewood-Richardson rule
针对各向同性格拉斯曼流形的Schubert结构系数，提出一种新的Littlewood-Richardson规则，首次给出Serrano公式的代数证明，并基于Yamanouchi表格的类比得到易于实现且时间复杂度更优的算法。
We study the Littlewood-Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant $K$-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang-Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood-Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by Knutson-T…
Littlewood--Richardson rules from quivers for two-step flag varieties
该研究通过分拆对索引的箭图构造对称函数代数张量积的向量空间基，给出两步旗簇中Schubert多项式与特定Schur多项式乘积的Littlewood-Richardson系数正表公式，覆盖Pieri规则情形。
We prove an explicit combinatorial formula for the structure constants of the Grothendieck ring of a Grassmann variety with respect to its basis of Schubert structure sheaves. We furthermore relate K-theory of Grassmannians to a bialgebra of stable Grothendieck polynomials, which is a K-theory parallel of the ring of symmetric functions.
We develop the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. One key ingredient of this algorithm is our new optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. Our implementation can solve problem instances with tens of thousands of solutions.
This is a companion paper to arXiv:0909.2280. It is mostly expository and focuses on the representation-theoretic and combinatorial aspects of the main problems considered in the other article.
Solving Schubert Problems with Littlewood-Richardson Homotopies
提出一种基于Vakil的Littlewood-Richardson规则几何证明的数值同伦延拓算法，用于求解Grassmannian上的Schubert问题的所有解。
We study the T-equivariant quantum cohomology of the Grassmannian. We prove the vanishing of a certain class of equivariant quantum Littlewood-Richardson coefficients, which implies an equivariant quantum Pieri rule. As in the equivariant case, this implies an algorithm to compute the equivariant quantum Littlewood-Richardson coefficients.
If $α\in S_n$ is a permutation of $\{1, 2, \ldots, n\}$, the inversion set of $α$ is $Φ(α) = \{(i, j) \, | \, 1 \leq i < j \leq n, α(i) > α(j)\}$. We describe all $r$-tuples $α_1, α_2, \ldots, α_r \in S_n$ such that $Δ_n^+ = \{(i, j) \, | \, 1 \leq i < j \leq n\}$ is the disjoint union of $Φ(α_1), Φ(α_2), \ldots, Φ(α_r)$. Using this description we prove that certain faces of the Littlewood-Richardson cone are simplicial and provide an algorithm for writing down their sets of generating rays. We also discuss analogous problems for the Weyl groups of root systems of types $B$, $C$ and $D$ provid…
We characterize the relationship between the singular values of a complex Hermitian (resp., real symmetric, complex symmetric) matrix and the singular values of its off-diagonal block. We also characterize the eigenvalues of an Hermitian (or real symmetric) matrix C=A+B in terms of the combined list of eigenvalues of A and B. The answers are given by Horn-type linear inequalities. The proofs depend on a new inequality among Littlewood-Richardson coefficients.
Let G be a semisimple algebraic group over an algebraically-closed field of characteristic zero. In this note we show that every regular face of the Littlewood-Richardson cone of G gives rise to a reduction rule: a rule which, given a problem "on that face" of computing the multiplicity of an irreducible component in a tensor product, reduces it to a similar problem on a group of smaller rank. In the type A case this result has already been proved by Derksen and Weyman using quivers, and by King, Tollu, and Tomazet using puzzles. The proof here is geometric and type-independent.
In his 1951 book "Infinite Abelian Groups", Kaplansky gives a combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group. In this paper we first use partial maps on Littlewood-Richardson tableaux to generalize this result to finite direct sums of such embeddings. We then focus on an application to invariant subspaces of nilpotent linear operators. We develop a criterion to decide if two irreducible components in the representation space are in the boundary partial order.
The Kronecker coefficients and the Littlewood-Richardson coefficients are nonnegative integers depending on three partitions. By definition, these coefficients are the multiplicities of the tensor product decomposition of two irreducible representations of symmetric groups (resp. linear groups). By a classical Littlewood-Murnaghan's result the Kronecker coefficients extend the Littlewood-Richardson ones.The nonvanishing of a Littlewood-Richardson coefficient implies linear inequalities on the triple of partitions, called Horn inequalities. In thispaper, we extend the essential Horn inequalitie…
I prove that the full Fock-Goncharov conjecture holds for $\mathrm{Conf}_3^\times(\widetilde{\mathcal{F}\hspace{-1.6pt}\ell})$-- the configuration space of triples of decorated flags in generic position. As a key ingredient of this proof, I exhibit a maximal green sequence for the quiver of the initial seed. I compute the Landau-Ginzburg potential $W$ on $\mathrm{Conf}_3^\times(\widetilde{\mathcal{F}\hspace{-1.6pt}\ell})^\vee$ associated to the partial minimal model $\mathrm{Conf}_3^\times(\widetilde{\mathcal{F}\hspace{-1.6pt}\ell}) \subset \mathrm{Conf}_3(\widetilde{\mathcal{F}\hspace{-1.6pt}…
Answering a question raised by S. Friedland, we show that the possible eigenvalues of Hermitian matrices (or compact operators) A, B, and C with C <= A + B are given by the same inequalities as in Klyachko's theorem for the case where C = A + B, except that the equality corresponding to tr(C) = tr(A) + tr(B) is replaced by the inequality corresponding to tr(C) <= tr(A) + tr(B). The possible types of finitely generated torsion modules A, B, and C over a discrete valuation ring such that there is an exact sequence B -> C -> A are characterized by the same inequalities.
The Littlewood-Richardson coefficients $c^ν_{λ,μ}$ are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group GL$(n, {\mathbb C})$. They are parametrized by the triples of partitions $(λ, μ, ν)$ of length at most $n$. By the so-called Fulton conjecture, if $c^ν_{λ,μ}=1$ then $c^{kν}_{kλ,kμ}= 1$, for any $k \geq 0$. Similarly, as proved by Ikenmeyer or Sherman, if $c^ν_{λ,μ}=2$ then $c^{kν}_{kλ,kμ} = k + 1$, for any $k\geq 0$. Here, given a partition $λ$, we set $λ(p, q) = p(qλ')'$ , where prime denotes the conjugate partition. We o…
Littlewood-Richardson Coefficient, Springer Fibers and the Annihilator Varieties of Induced Representations
本文利用几何Sakate等价证明了$GL(n,\mathbb{C})$中幂零轨道的包含关系与Littlewood-Richardson系数非零的等价性，并描述了用于研究Whittaker支撑和零化簇的空间$\mathcal{O}\cap\mathfrak{p}$的几何性质。
A Littlewood-Richardson Rule for Forest Polynomials via the Schubert Bialgebra
提出森林多项式乘积结构常数的组合规则，计数提升积落在特定森林RC图上的森林RC图对，并推广到对偶Schubert等基的LR规则。
A partial flag variety ${\mathcal {P}}_K$ of a Kac-Moody group $G$ has a natural stratification into projected Richardson varieties. When $G$ is a connected reductive group, a Bruhat atlas for ${\mathcal {P}}_K$ was constructed by He, Knutson and Lu: ${\mathcal {P}}_K$ is locally modeled with Schubert varieties in some Kac-Moody flag variety as stratified spaces. The existence of Bruaht atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties. A Bruhat atlas does not exist for partial flag varieties…
The set of possible spectra (λ,μ,ν) of zero-sum triples of Hermitian matrices forms a polyhedral cone. We give a complete determination of its facets, finishing a long story with recent highlights by [Helmke-Rosenthal, Klyachko, Belkale]. We introduce_puzzles_, which are new combinatorial gadgets to compute Grassmannian Schubert calculus, and will probably be the main point of interest for many readers. As the proofs indicate, the Hermitian sum problem is very naturally studied using puzzles directly, and their connection to Schubert calculus is quite incidental to our approach. In particular,…
On Seidel representation in quantum K-theory of Grassmannians
该研究通过具体分析投影格罗莫夫-维滕簇直接证明了格拉斯曼流形量子K理论中的赛德尔表示，给出了量子皮埃里法则的替代证明，简化了高次量子舒伯特结构常数，并为QK(Gr(3,n))提供了量子利特尔伍德-理查森法则。
It is well-known that the intersection multiplicities of Schubert classes in the Grassmanian are Littlewood-Richardson coefficients. We generalize this statement in the context of quiver representations. Here the intersection multiplicity of Schubert classes is replaced by the number of subrepresentations of a general quiver reprsentation, and the Littlewood-Richardson coefficients are replaced by the the dimension of a certain space of semiinvariants.
We prove Pieri formulas for the multiplication with special Schubert classes in the K-theory of all cominuscule Grassmannians. For Grassmannians of type A this gives a new proof of a formula of Lenart. Our formula is new for Lagrangian Grassmannians, and for orthogonal Grassmannians it proves a special case of a conjectural Littlewood-Richardson rule of Thomas and Yong. Recent work of Thomas and Yong and of E. Clifford has shown that the full Littlewood-Richardson rule for orthogonal Grassmannians follows from the Pieri case proved here. We describe the K-theoretic Pieri coefficients both as i…
The jeu-de-taquin-based Littlewood-Richardson rule of H. Thomas and A. Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of "unique rectification targets" in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to $K$-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's "$d$-complete posets" to extend the Thomas-Yong rule from minu…
The Newell-Littlewood numbers are tensor product multiplicities of Weyl modules for the classical groups in the stable range. Littlewood-Richardson coefficients form a special case. Klyachko connected eigenvalues of sums of Hermitian matrices to the saturated LR-cone and established defining linear inequalities. We prove analogues for the saturated NL-cone: an eigenvalue interpretation; a minimal list of defining linear inequalities; a description by Extended Horn inequalities, as conjectured in part II of this series; and a factorization of NL-numbers, on the boundary.
We prove that structure constants related to Hecke algebras at roots of unity are special cases of k-Littlewood-Richardson coefficients associated to a product of k-Schur functions. As a consequence, both the 3-point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to \hat{su}(\ell) are shown to be k-Littlewood Richardson coefficients. From this, Mark Shimozono conjectured that the k-Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas k-Schur copro…
In their 2004 paper, Stretched Littlewood-Richardson and Kostka Coefficients, King, Tollu, and Toumazet conjectured that if a Littlewood-Richardson coefficient of value 2 is stretched by a factor of N, the resulting coefficient has value N+1. A generalization of this is proven using geometric methods and Schubert calculus.
We give counterexamples to Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients.
Richardson varieties play an important role in intersection theory and in the geometric interpretation of the Littlewood-Richardson Rule for flag varieties. We discuss three natural generalizations of Richardson varieties which we call projection varieties, intersection varieties, and rank varieties. In many ways, these varieties are more fundamental than Richardson varieties and are more easily amenable to inductive geometric constructions. In this paper, we study the singularities of each type of generalization. Like Richardson varieties, projection varieties are normal with rational singula…
We establish a relation between the generating functions appearing in the S-duality conjecture of Vafa and Witten and geometric Eisenstein series for Kac-Moody groups. For a pair consisting of a surface and a curve on it, we consider a refined geometric function E (involving G-bundles with parabolic structures along the curve) which depends both on elliptic and modular variables. We prove a functional equation for E with respect to the affine Weyl group, thus establishing the elliptic behavior. When the curve is P^1, we calculate the Eisenstein-Kac-Moody series explicitly and it turns out to b…
Stretching the parameters of a Littlewood-Richardson coefficient of value 2 by a factor of n results in a coefficient of value n+1. We give a geometric proof of a generalization for representations of quivers.
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