Bernard Leclerc
We give a combinatorial description of the composition factors of the induction product of two evaluation modules of the affine Iwahori-Hecke algebra of type GL(m). Using quantum affine Schur-Weyl duality, this yields a combinatorial description of the composition factors of the tensor product of two evaluation modules of the quantum affine algebra of type sl(n).
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We present a Robinson-Schensted-Knuth type one-to-one correspondence between the set of pictures and the set of pairs of Littlewood-Richardson crystals.
We construct a natural bijection between the set of admissible pictures and the set of $U_q(gl(m,n))$-Littlewood-Richardson tableaux.
We show that the Littlewood-Richardson coefficients are values at 1 of certain parabolic Kazhdan-Lusztig polynomials for affine symmetric groups. These q-analogues of Littlewood-Richardson multiplicities coincide with those previously introduced in terms of ribbon tableaux.
We give a survey on the Littlewood-Richardson rule. Using Gelfand-Tsetlin patterns as the main machinery of our analysis, we study the interrelationship of various combinatorial descriptions of the Littlewood-Richardson rule.
We shall describe the one-to-one correspondence between the set of pictures and the set of Littlewood-Richardson crystals.
In previous work with Mikhail Khovanov and Aaron Lauda we introduced two odd analogues of the Schur functions: one via the combinatorics of Young tableaux (odd Kostka numbers) and one via the odd symmetrization operator. In this paper we introduce a third analogue, the plactic Schur functions. We show they coincide with both previously defined types of Schur function, confirming a conjecture. Using the plactic definition, we establish an odd Littlewood-Richardson rule. We also re-cast this rule in the language of polytopes, via the Knutson-Tao hive model.
We give new bounds and asymptotic estimates for Kronecker and Littlewood--Richardson coefficients. Notably, we resolve Stanley's questions on the shape of partitions attaining the largest Kronecker and Littlewood--Richardson coefficients. We apply the results to asymptotics of the number of standard Young tableaux of skew shapes.
The classical Littlewood-Richardson coefficients C(lambda,mu,nu) carry a natural $S_3$ symmetry via permutation of the indices. Our "carton rule" for computing these numbers transparently and uniformly explains these six symmetries; previously formulated Littlewood-Richardson rules manifest at most three of the six.
We find and prove a factorization formula for certain Macdonald Littlewood-Richardson coefficients $c_{λμ}^ν(q,t)$. Namely, we consider the case that the Kostka number $K_{μ, ν-λ}$ is $1$. This settles a particular case of a more general conjecture of Richard Stanely. This conjecture proposes that a factorization formula exists whenever the corresponding regular Littlewood-Richardson coefficient $c^ν_{λμ}$ is $1$.
We give a closed formula of the Littlewood-Richardson coefficients.
We present several direct bijections between different combinatorial interpretations of the Littlewood-Richardson coefficients. The bijections are defined by explicit linear maps which have other applications.
We introduce a Littlewood-Richardson rule based on an algorithmic deformation of skew Young diagrams and present a bijection with the classical rule. The result is a direct combinatorial interpretation and proof of the geometric rule presented by Coskun. We also present a corollary regarding the Specht modules of the intermediate diagrams.
The main aim of the paper is to present a~combinatorial algorithm that, applying Littlewood-Richardson tableaux with entries equal to $1$, computes generic extensions of semisimple invariant subspaces of nilpotent linear operators. Moreover, we discuss geometric properties of generic extensions and their connections with combinatorics.
Following the methods used by Derksen-Weyman in \cite{DW11} and Chindris in \cite{Chi08}, we use quiver theory to represent the generalized Littlewood-Richardson coefficients for the branching rule for the diagonal embedding of $\gl(n)$ as the dimension of a weight space of semi-invariants. Using this, we prove their saturation and investigate when they are nonzero. We also show that for certain partitions the associated stretched polynomials satisfy the same conjectures as single Littlewood-Richardson coefficients. We then provide a polytopal description of this multiplicity and show that its…
Starting with Knutson and Tao's hive model (in J. Amer. Math. Soc., 1999) we characterize the Littlewood-Richardson coefficient $c_{λ,μ}^ν$ of given partitions $λ,μ,ν\in N^n$ as the number of capacity achieving hive flows on the honeycomb graph. Based on this, we design a polynomial time algorithm for deciding $c_{λ,μ}^ν>0$. This algorithm is easy to state and takes $O(n^3 \log ν_1)$ arithmetic operations and comparisons. We further show that the capacity achieving hive flows can be seen as the vertices of a connected graph, which leads to new structural insights into Littlewood-Richardson coe…
In this paper we explicitly compute all Littlewood-Richardson coefficients for semisimple or Kac-Moody groups G, that is, the structure coefficients of the cohomology algebra H^*(G/P), where P is a parabolic subgroup of G. These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the Littlewood-Richardson coefficients is given in terms of the Cartan matrix and the Weyl group of G. However, if some off-diagonal entries of the Cartan matrix are 0 or -1, the formula may contain negative summands. On the other hand, if the Cart…
A super Littlewood--Richardson type rule
本文提出了带符号字母表上超 Schur 函数的超 Littlewood--Richardson 规则，并利用超 Young 表的性质给出了相应系数的组合解释，为表示论等领域提供了应用工具。
We prove a special case of a conjecture of Naito-Sagaki about a branching rule for the restriction of irreducible representations of $\mathfrak{sl}(2n,\mathbb{C})$ to $\mathfrak{sp}(2n,\mathbb{C})$. The conjecture is in terms of certain Littelmann paths, with the embedding given by the folding of the type $A_{2n-1}$ Dynkin diagram. We propose and motivate an approach to the conjecture in general, in terms of Littlewood-Richardson Sundaram tableaux.
We present a bijection between vacillating tableaux and pairs consisting of a standard Young tableau and an orthogonal Littlewood-Richardson tableau for the special orthogonal group $\mathrm{SO}(2k+1)$. This bijection is motivated by the direct-sum-decomposition of the $r$th tensor power of the defining representation of $\mathrm{SO}(2k+1)$. To formulate it, we use Kwon's orthogonal Littlewood-Richardson tableaux and introduce new alternative tableaux they are in bijection with. Moreover we use a suitably defined descent set for vacillating tableaux to determine the quasi-symmetric expansion o…
We use Hopf algebras to prove a version of the Littlewood-Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood-Richardson rules for Schur P- and Q-functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group.
We present a polynomiality property of the Littlewood-Richardson coefficients c_{λμ}^ν. The coefficients are shown to be given by polynomials in λ, μand νon the cones of the chamber complex of a vector partition function. We give bounds on the degree of the polynomials depending on the maximum allowed number of parts of the partitions λ, μand ν. We first express the Littlewood-Richardson coefficients as a vector partition function. We then define a hyperplane arrangement from Steinberg's formula, over whose regions the Littlewood-Richardson coefficients are given by polynomials, and relate thi…
Saturation for Flagged Skew Littlewood-Richardson Coefficients
本文定义了带旗的斜 Littlewood-Richardson 系数，并证明其饱和性，推广了已有结果。
Flagged Littlewood-Richardson tableaux and branching rule for classical groups
本文推广Littlewood限制公式，提出了从${\ m GL}_n$到${\ m O}_n$分支规则的新公式，该公式基于带标志条件的Littlewood-Richardson表，并据此给出了$B_n$和$D_n$型Lusztig $t$-权重重数的组合公式。
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…
Macdonald polynomials are orthogonal polynomials associated to root systems, and in the type A case, the symmetric kind is a common generalization of Schur functions, Macdonald spherical functions, and Jack polynomials. We use the combinatorics of alcove walks to calculate products of monomials and intertwining operators of the double affine Hecke algebra. From this, we obtain a product formula for Macdonald polynomials of general type.
Jack Littlewood-Richardson Coefficients and the Nazarov-Sklyanin Lax Operator
本文研究 Nazarov-Sklyanin 量子 Lax 算子多项式本征函数性质，通过考虑本征函数乘积推广 Kerov 公式，推导出基于分区乘法运算的 Jack Littlewood-Richardson 系数约束系统。
Based on the direct-sum-decomposition of the rth tensor power of the defining representation of the special orthogonal group SO(2k+1) one is interested in a bijective approach for determining the Frobenius characters of the isotypic components. In particular this leads us to a bijection between vacillating tableaux and pairs of standard Young tableaux and orthogonal Littlewood-Richardson tableaux, which we present for SO(3). Moreover we introduce the descent set of a vacillating tableau. As our bijection preserves this descent set, we also obtain the quasi-symmetric expansion of the Frobenius …
J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood-Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas-A. Yong. Our proof then combines a saturation theorem of D. Anderson-E. Richmond-A. Yong, a reading order independence property, and E. Tardos' algo…
On the Shifted Littlewood-Richardson Coefficients and Littlewood-Richardson Coefficients
本文通过给出移位 Littlewood-Richardson 系数的新解释，将 Schur Q-函数分解中的系数与经典 Littlewood-Richardson 系数联系起来，证明了对称性与不等式关系，并提出相关猜想。
We develop structural insights into the Littlewood-Richardson graph, whose number of vertices equals the Littlewood-Richardson coefficient c(λ,μ,ν) for given partitions λ, μ, and ν. This graph was first introduced by Bürgisser and Ikenmeyer in arXiv:1204.2484, where its connectedness was proved. Our insights are useful for the design of algorithms for computing the Littlewood-Richardson coefficient: We design an algorithm for the exact computation of c(λ,μ,ν) with running time O(c(λ,μ,ν)^2 poly(n)), where λ, μ, and ν are partitions of length at most n. Moreover, we introduce an algorithm for d…
Littlewood-Richardson (LR) coefficients and Kostka Numbers appear in representation theory and combinatorics related to $GL_n$. It is known that Kostka numbers can be represented as special Littlewood-Rischardson coefficient. In this paper, we show how one can represent LR coefficient as a signed sum of Kostka numbers, and use the formulation to give a polynomial time algorithm for the same, hence showing that they belong to the same class of decision problems. As a corollary, we will prove Steinberg's formula using Kostant's partition function.
The Littlewood-Richardson (LR) coefficient counts among many other things the LR tableaux of a given shape and a given content. We prove, that the number of LR tableaux weakly increases if one adds to the shape and the content the shape and the content of another LR tableau. We also investigate the behaviour of the number of LR tableaux, if one repeatedly adds to the shape another shape with either fixed or arbitrary content. This is a generalisation of the stretched LR coefficients, where one repeatedly adds the same shape and content to itself.
This is an expanded version of the notes of my three lectures at a NATO Advanced Study Institute ``Symmetric functions 2001: surveys of developments and perspectives" (Isaac Newton Institute for Mathematical Sciences, Cambridge, UK; June 25-July 6, 2001). Lecture I presents a unified expression due to A. Berenstein and the author for generalized Littlewood-Richardson coefficients (= tensor product multiplicities) for any complex semisimple Lie algebra. Lecture II outlines a proof of this result; the main idea of the proof is to relate the LR-coefficients with canonical bases and total positivi…
We consider the example from invariant theory concerning the conjugation action of the general linear group on several copies of the $n \times n$ matrices, and examine a symmetric function which stably describes the Hilbert series for the invariant ring with respect to the multigradation by degree. The terms of this Hilbert series may be described as a sum of squares of Littlewood-Richardson coefficients. A "principal specialization" of the gradation is then related to the Hilbert series of the $\K$-invariant subring in the $\GL_n$-harmonic polynomials, where $\K$ denotes a block diagonal embe…
We prove a Littlewood-Richardson type formula for $(s_{λ/μ},s_{ν/κ})_{t^k,t}$, the pairing of two skew Schur functions in the MacDonald inner product at $q = t^k$ for positive integers $k$. This pairing counts graded decomposition numbers in the representation theory of wreath products of the algebra $C[x]/x^k$ and symmetric groups.
We are interested in identities between Littlewood-Richardson coefficients, and hence in comparing different tensor product decompositions of the irreducible modules of the linear group GL n (C). A family of partitions-called near-rectangular-is defined, and we prove a stability result which basically asserts that the decomposition of the tensor product of two representations associated to near-rectangular partitions does not depend on n. Given a partition $λ$, of length at most n, denote by V n ($λ$) the associated simple GL n (C)-module. We conjecture that, if $λ$ is near-rectangular and $μ$…
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.
The volume of the hive polytope (or polytope of honeycombs) associated with a Littlewood- Richardson coefficient of SU(n), or with a given admissible triple of highest weights, is expressed, in the generic case, in terms of the Fourier transform of a convolution product of orbital measures. Several properties of this function -- a function of three non-necessarily integral weights or of three multiplets of real eigenvalues for the associated Horn problem-- are already known. In the integral case it can be thought of as a semi-classical approximation of Littlewood-Richardson coefficients. We pr…
We give a new proof for the Littlewood-Richardson rule for the wreath product $F \wr S_{n}$ where $F$ is a finite group. Our proof does not use symmetric functions but more elementary representation theoretic tools. We also derive a branching rule for inducing the natural embedding of $F\wr S_{n}$ to $F\wr S_{n+1}$. We then apply the generalized Littlewood-Richardson rule for computing the ordinary quiver of the category $F \wr {\bf FI}_{n}$ where ${\bf FI}_{n}$ is the category of all injective functions between subsets of an $n$-element set.
We provide two shifted analogues of the tableau switching process due to Benkart, Sottile, and Stroomer, the shifted tableau switching process and the modified shifted tableau switching process. They are performed by applying a sequence of specially contrived elementary transformations called {\em switches} and turn out to have some spectacular properties. For instance, the maps induced from these algorithms are involutive and behave very nicely with respect to shifted Young tableaux whose reading words satisfy the lattice property. As an application, we give combinatorial interpretations of S…
For any ordinary irreducible character of a maximal reflection subgroup of type $D_aD_b$ of a type $D$ Weyl group we give an explicit decomposition of the induced character in terms of Littlewood-Richardson coefficients.
The Littlewood-Richardson coefficients describe the decomposition of tensor products of irreducible representations of a simple Lie algebra into irreducibles. Assuming the number of factors is large, one gets a measure on the space of weights. This limiting measure was extensively studied by many authors. In particular, Kerov computed the corresponding density in a special case in type A and Kuperberg gave a formula for the general case. The goal of this paper is to give a short, self-contained and pure Lie theoretic proof of the formula for the density of the limiting measure. Our approach is…
The Littlewood-Richardson coefficients $c^λ_{μν}$ give the multiplicity of an irreducible polynomial ${\rm GL}_n$-representation $F^λ_n$ in the tensor product of polynomial representations $F^μ_n\otimes F^ν_n$. In this paper, we generalize these coefficients to an $r$-fold tensor product of rational representations, and give a new method for computing them using an analogue of statistical contingency tables. We demonstrate special cases in which our method reduces to counting statistical contingency tables with prescribed margins. Finally, we extend our result from the general linear group to …
Proof of the Newell-Littlewood saturation conjecture
本文将蜂窝方法推广到莫比乌斯带上，从而证明了 Newell-Littlewood 数的饱和猜想。
Let K be a complex reductive algebraic group and V a representation of K. Let S denote the ring of polynomials on V. Assume that the action of K on S is multiplicity free. If V_λ is an irreducible representation of K, let S_λ denote the corresponding isotypic component of S. Write S_λ S_μ for the subspace of S spanned by products of S_λ and S_μ. If V_ν occurs as an irreducible constituent of the tensor product of V_λ and V_μ, is it true that S_ν is contained in S_λ S_μ? We investigate this question for representations arising in the context of Hermitian symmetric pairs. We show that the answer…
We point out that the remarkable Knutson and Tao Saturation Theorem and polynomial time algorithms for LP have together an important and immediate consequence in Geometric Complexity Theory. The problem of deciding positivity of Littlewood-Richardson coefficients for GLn(C) belongs to P. Furthermore, the algorithm is strongly polynomial. The main goal of this article is to explain the significance of this result in the context of Geometric Complexity Theory. Furthermore, it is also conjectured that an analogous result holds for arbitrary symmetrizable Kac-Moody algebras.
Necessary conditions for the positivity of Littlewood-Richardson and plethystic coefficients
本文给出Littlewood-Richardson系数和SXP系数正性的必要条件，并推导出plethystic系数正性的必要条件，主要结果刻画了$S^μ(S^ν(V))$分解中$S^λ(V)$出现时$ν$与$λ$图形间的包含关系。
Congruences of shifted Jack Littlewood-Richardson coefficients
本文证明了关于平移Jack Littlewood-Richardson系数的同余猜想，该猜想涉及两个三元组（其中一个分区通过单框移动变化）的系数模移动框的α-钩长同余。
Koornwinder polynomials are $q$-orthogonal polynomials equipped with extra five parameters and the $B C_n$-type Weyl group symmetry, which were introduced by Koornwinder (1992) as multivariate analogue of Askey-Wilson polynomials. They are now understood as the Macdonald polynomials associated to the affine root system of type $(C^\vee_n,C_n)$ via the Macdonald-Cherednik theory of double affine Hecke algebras. In this paper we give explicit formulas of Littlewood-Richardson coefficients for Koornwinder polynomials, i.e., the structure constants of the product as invariant polynomials. Our form…
Benkart, Sottile, and Stroomer have completely characterized by Knuth and dual Knuth equivalence a bijective proof of the conjugation symmetry of the Littlewood-Richardson coefficients. Tableau-switching provides an algorithm to produce such a bijective proof. Fulton has shown that the White and the Hanlon-Sundaram maps are versions of that bijection. In this paper one exhibits explicitly the Yamanouchi word produced by that conjugation symmetry map which on its turn leads to a new and very natural version of the same map already considered independently. A consequence of this latter construct…
We define the_hive ring_, which has a basis indexed by dominant weights for GL(n), and structure constants given by counting hives [KT1] (or equivalently honeycombs, or Berenstein-Zelevinsky patterns [BZ1]). We use the octahedron rule from [Robbins-Rumsey,Fomin-Zelevinsky,Propp,Speyer] to prove bijectively that this "ring" is indeed associative. This, and the Pieri rule, give a self-contained proof that the hive ring is isomorphic as a ring-with-basis to the representation ring of GL(n). In the honeycomb interpretation, the octahedron rule becomes "scattering" of the honeycombs. This recovers …
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.
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…
Clifford algebras and Littlewood-Richardson coefficients
本文提出利用Clifford代数技术描述等秩紧对称空间的de Rham上同调环，为计算Littlewood-Richardson系数提供了一种基于Hadamard积的新方法。
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.
The socle tableau as a dual version of the Littlewood-Richardson tableau
该研究提出基座表（socle tableau）作为LR-tableau的对偶版本，明确其行列递增规则与修正格置换性质，证明它可作为有限阿贝尔$p$-群子群嵌入的同构不变量且均可被实现，且与对偶嵌入的LR-tableau相互确定。
Let $α$, $β$, and $γ$ be partitions describing the isomorphism types of the finite abelian $p$-groups $A$, $B$, and $C$. From theorems by Green and Klein it is well-known that there is a short exact sequence $0\to A\to B\to C\to 0$ of abelian groups if and only if there is a Littlewood-Richardson sequence of type $(α,β,γ)$. Starting from the observation that a sequence of partitions has the LR property if and only if every subsequence of length 2 does, we demonstrate how LR-sequences of length two correspond to embeddings of a $p^2$-bounded subgroup in a finite abelian $p$-group. Using the kno…
Let G be a connected reductive algebraic group and H be a reductive closed and connected subgroup of G both defined on an algebraically closed field of characteristic zero. We consider the set C of the couple (x,y) of the dominant weights such that the irreducible H-module of highest weight x is a submodule of the G-module corresponding to y. This set identifies canonically with a semi-group finitely generated in a rational vector space. We consider the generalized Littlewood-Richardson cone C' generated by C. This cone is polyhedral. For any (x',y') in C', x' is dominant, and so, satisfies a …
A second reduction-type formula for the refined Littlewood--Richardson coefficients in type A
本文研究对称群\(S_n\)中置换及分划相关的\(A_n\)型精细化 Littlewood-Richardson 系数\(c^{\nu}_{\lambda,\mu}(w)\)，通过蜂巢模型建立其第二个约化型公式，扩展了经典系数的约化公式。
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…
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