Francois Bergeron, Riccardo Biagioli, Mercedes H. Rosas
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.
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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 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 $w$.
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…
Littlewood Richardson coefficients are structure constants appearing in the representation theory of the general linear groups ($GL_n$). The main results of this paper are: 1. A strongly polynomial randomized approximation scheme for certain Littlewood-Richardson coefficients. 2. A proof of approximate log-concavity of certain Littlewood-Richardson coefficients.
Using a generalization of the Schensted insertion algorithm to rc-graphs, we provide a Littlewood-Richardson rule for multiplying certain Schubert polynomials by Schur polynomials.
An introduction is given to the Littlewood-Richardson rule, and various combinatorial constructions related to it. We present a proof based on tableau switching, dual equivalence, and coplactic operations. We conclude with a section relating these fairly modern techniques to earlier work on the Littlewood-Richardson rule.
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…
In this note, we study two generalizations of the Catalan numbers, namely the $s$-Catalan numbers and the spin $s$-Catalan numbers. These numbers first appeared in relation to quantum physics problems about spin multiplicities. We give a combinatorial description for these numbers in terms of Littlewood-Richardson coefficients, and explain some of the properties they exhibit in terms of Littlewood-Richardson polynomials.
The classical Littlewood-Richardson rule is a rule for computing coefficients in many areas, and comes in many guises. In this paper we prove two Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions that are analogous to the famed version of the classical Littlewood-Richardson rule involving Yamanouchi words. Furthermore, both our rules contain this classical Littlewood-Richardson rule as a special case. We then apply our rules to combinatorially classify symmetric skew quasisymmetric Schur functions. This answers affirmatively a conjecture of Bessenrodt, Luoto and van…
We compute with SageMath the group of all linear symmetries for the Littlewood-Richardson associated to the representations of $SL_3$. We find that there are 144 symmetries, more than the 12 symmetries known for the Littlewood-Richardson coefficients in general.
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…
This note is an extended abstract of my talk at the workshop on Representation Theory and Symmetric Functions, MSRI, April 14, 1997. We discuss the problem of finding an explicit description of the semigroup $LR_r$ of triples of partitions of length $\leq r$ such that the corresponding Littlewood-Richardson coefficient is non-zero. After discussing the history of the problem and previously known results, we suggest a new approach based on the ``polyhedral'' combinatorial expressions for the Littlewood-Richardson coefficients.
Suppose μand μ' are two partitions. We will let μ\oplus μ' denote the "direct sum" of the partitions, defined as the sorted partition made of the parts of $μ$ and $μ'$. In this paper, we define a summation operation on two Littlewood-Richardson fillings of type (μ, ν;λ) and (μ', ν';λ'), which results in a Littlewood-Richardson filling of type (μ\oplus μ', ν\oplus ν' ;λ\oplus λ'). We give an algorithm to produce the sum, and show that it terminates in a Littlewood-Richardson filling by defining a bijection between a Littlewood-Richardson filling and a flow on a honeycomb, and then showing that …
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.
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 present a new characterization of the shape equivalent class and the Littlewood-Richardson correspondence of Young tableaux in terms of tropical (ultradiscrete) integrable systems. As an application, an alternative proof of the "shape change theorem" is given.
A contratableau model for K-theoretic Littlewood--Richardson rule
本文针对K-理论Littlewood-Richardson规则，给出基于集值可收缩表格的显式组合公式，推广了经典规则的可收缩表格模型，为计算对称Grothendieck多项式乘积系数提供新方法。
For a given skew shape, we build a crystal graph on the set of all reverse plane partitions that have this shape. As a consequence, we get a simple extension of the Littlewood-Richardson rule for the expansion of the corresponding dual stable Grothendieck polynomial in terms of Schur polynomials.
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.
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…
We give a Littlewood-Richardson type rule for expanding the product of a row-strict quasisymmetric Schur function and a symmetric Schur function in terms of row-strict quasisymmetric Schur functions. This expansion follows from several new properties of an insertion algorithm defined by Mason and Remmel (2010) which inserts a positive integer into a row-strict composition tableau.
Shuffle Tableaux, Littlewood--Richardson Coefficients, and Schur Log-Concavity
本文提出了基于peelable tableaux和shuffle tableaux的Littlewood--Richardson系数新公式，可用于高效计算广义系数并恢复其对称性，同时证明了Schur对数凹性猜想的一个特例。
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.
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 present a one-to-one correspondence between the set of admissible pictures and the Littlewood-Richardson crystals. As a simple consequence, we shall show that the set of pictures does not depend on the choice of admissible orders.
We give a combinatorial rule for computing intersection numbers on a flag manifold which come from products of Schubert classes pulled back from Grassmannian projections. This rule generalizes the known rule for Grassmannians.
Jack polynomials generalize several classical families of symmetric polynomials, including Schur polynomials, and are further generalized by Macdonald polynomials. In 1989, Richard Stanley conjectured that if the Littlewood-Richardson coefficient for a triple of Schur polynomials is 1, then the corresponding coefficient for Jack polynomials can be expressed as a product of weighted hooks of the Young diagrams associated to the partitions indexing the coefficient. We prove a special case of this conjecture in which the partitions indexing the Littlewood-Richardson coefficient have at most 3 par…
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 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 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 construct a natural bijection between the set of admissible pictures and the set of $U_q(gl(m,n))$-Littlewood-Richardson tableaux.
We give an inductive procedure for finding the extremal rays of the equivariant Littlewood-Richardson cone, which is closely related to the solution space to S. Friedland's majorized Hermitian eigenvalue problem. In so doing, we solve the "rational version" of a problem posed by C. Robichaux, H. Yadav, and A. Yong. Our procedure is a natural extension of P. Belkale's algorithm for the classical Littlewood-Richardson cone. The main tools for accommodating the equivariant setting are certain foundational results of D. Anderson, E. Richmond, and A. Yong. We also study two families of special rays…
We extend the theory of Littlewood-Richardson fillings (defined over the non-negative integers) to include diagrams with rows and boxes of real-valued length. We realize such fillings as invariants of matrix pairs over rings with a real-valued valuation. We then prove a bijection between pairs of fillings determined by a matrix pair, and connect this to the classical row-switching bijection for Littlewoo-Richardson fillings. We utilize matrix calculations to interpret continuous deformations of generalized Littlewood-Richardson fillings.
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 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.
In this work we continue our study of Fully Packed Loop (FPL) configurations in a triangle. These are certain subgraphs on a triangular subset of the square lattice, which first arose in the study of the usual FPL configurations on a square grid. We show that, in a special case, the enumeration of these FPLs in a triangle is given by Littlewood-Richardson coefficients. The proof consists of a bijection with Knutson-Tao puzzles.
Kostka numbers and Littlewood-Richardson coefficients play an essential role in the representation theory of the symmetric groups and the special linear groups. There has been a significant amount of interest in their computation. The issue of their computational complexity has been a question of folklore, but was asked explicitly by E. Rassart. We prove that the computation of either quantity is #P-complete. The reduction to computing Kostka numbers, is from the #P-complete problem of counting the number of 2 x k contingency tables having given row and column sums. The main ingredient in this…
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.
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。
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…
Young's lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing Young's lattice, we introduce a new partial order on weak compositions that we call the key poset. Saturated chains in this poset correspond to standard key tableaux, the combinatorial objects that generate the key polynomials, a nonsymmetric polynomial generalization of the Schur basis. Generalizing skew Schur functions, we define skew key polynomials in terms of this new pos…
We use Young's raising operators to derive a Pieri rule for the ring generated by the indeterminates $h_{r,s}$ given in Macdonald's 9th Variation of the Schur functions. Under an appropriate specialisation of $h_{r,s}$, we derive the Pieri rule for the ring $\La(a)$ of double symmetric functions, which has a basis consisting of the double Schur functions. Together with a suitable interpretation of the Jacobi--Trudi identity, our Pieri rule allows us to obtain a new proof of a rule to calculate the Littlewood--Richardson polynomials, which gives a multiplication rule for the double Schur functi…
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 prove an identity for Littlewood--Richardson coefficients conjectured by Pelletier and Ressayre (arXiv:2005.09877). The proof relies on a novel birational involution defined over any semifield.
Lower and Upper Bounds for Nonzero Littlewood-Richardson Coefficients
本文针对非零Littlewood-Richardson系数$c^γ_{λ,μ}>0$的条件，提出了一种基于Littlewood-Richardson表数量刻画和（广义）支配序的算法，以确定划分$μ$必须满足的下界和上界。
A bijection is defined from Littlewood-Richardson tableaux to rigged configurations. It is shown that this map preserves the appropriate statistics, thereby proving a quasi-particle expression for the generalized Kostka polynomials, which are q-analogues of multiplicities in tensor products of irreducible general linear group modules indexed by rectangular partitions.
A finite cover for coefficient positivity of stretched Littlewood-Richardson polynomials in the seven-row, size-thirty box
本文通过有限覆盖论证，结合精确枚举和Ehrhart计算，证明了七行、尺寸三十内所有拉伸Littlewood-Richardson多项式系数的非负性。
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…
On the largest Littlewood--Richardson coefficient
本研究聚焦最大Littlewood-Richardson系数对应的分拆特征，证明其在满足μ包含于ν且ν/μ为不交方格并的分拆处取得，提出n≥16时均满足该性质的猜想，并在16≤n≤45范围内数值验证了猜想。
Let M and N be two r x r matrices over a discrete valuation ring of characteristic zero. The orders (with respect to a uniformizing parameter) of the invariant factors of M form a partition of non-negative integers, called the invariant partition of M. Let the invariant partition of M be mu, of N be nu, and of the product MN be lambda. In this paper we construct a Littlewood-Richardson filling of the skew shape lambda/mu with content nu, and show that this filling is an invariant of the orbit of the pair (M,N) with respect to a natural group action on the pair. We relate the algebraic combinat…
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 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。
The slack data of the recording tableaux in the quantum Littlewood-Richardson map determines its inverse: some applications
本文引入量子Littlewood-Richardson映射中记录表形的松弛概念，证明其继承LR-Sundaram表形所需数据以定义量子LR映射的逆，并将松弛数据应用于$\mathfrak{k}$-最高权辛表形。
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.
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