Littlewood-Richardson rules for Grassmannians
Anders Skovsted Buch, Andrew Kresch, Harry Tamvakis
基于相应的Pieri规则，给出了A型Grassmannian和极大迷向Grassmannian的Littlewood-Richardson规则的初等和简短的证明.
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.
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Equivariant Littlewood-Richardson Skew Tableaux
我们给出了一个正等变的Littlewood-Richardson规则，该规则也是由Molev独立发现的。我们的证明推广了普通Littlewood-Richardson规则的Stembridge证明。我们描述了我们的索引表和克努森陶难题之间的重量保持双射。
Towards a Littlewood-Richardson rule for Kac-Moody homogeneous spaces
本文证明了在任意Kac-Moody齐性空间中三个特殊的$Λ$ -极小Schubert类的相交数$c_{u,v}^w$的一般组合公式，推广了Littlewood-Richardson规则.组合数学是基于jeu de taquin整流在一个偏序集定义的堆$w$。
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。
The Horn recursion for Schur P- and Q- functions: Extended Abstract
Klyachko和Knutson-Tao的工作的一个结果是Horn递归来确定Littlewood-Richardson系数何时为非零。简而言之，Littlewood-Richardson系数是非零的，当且仅当它满足由较小的非零Littlewood-Richardson系数索引的Horn不等式的集合。Schur P-函数和Schur Q-函数也有类似的Littlewood-Richardson数。使用组合学的根系，组合线性代数在李代数，和几何的某些cominuscule旗品种的混合物，我们给霍恩递归，以确定当这些其他Littlewood-Richardson数是非零的。我们的不等式来自通常的Littlewood-Richardson数，虽然我们给出了两个非常不同的Horn递归，但它们有相同的解集。这项工作的另一个组合副产品是一个新的霍恩型递归通常的Littlewood-Richardson系数。
A Combinatorial Approach to Multiplicity-Free Richardson Subvarieties of the Grassmannian
我们认为布赫规则的K-理论的格拉斯曼，在舒尔多重性自由的情况下分类的Stembridge。使用Knutson的结果，人们看到，布赫的系数有关的Moebius反演。通过考虑Grassmannian Grothendieck多项式的乘积展开式，给出了这一结论的一个直接的组合证明。最后，我们扩展到托马斯和杨的多重性自由的情况下。
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.
Inequalities between Littlewood-Richardson Coefficients
我们证明了一个猜想Fomin，富尔顿，李，潘，与序偶的分区，持有许多无限家庭这样的对。我们还表明，有界高度的情况下，可以减少到检查，猜想持有有限数量的对，任何给定的高度。此外，我们提出了将该猜想自然推广到倾斜形状的情况。
RC-graphs and a generalized Littlewood-Richardson rule
使用推广的Schensted插入算法的rc-图，我们提供了一个Littlewood-Richardson规则乘以某些舒伯特多项式的Schur多项式。
The Littlewood-Richardson rule, and related combinatorics
本文介绍了Littlewood-Richardson规则及其相关的组合构造，并给出了基于表切换、对偶等价和共格运算的证明。最后，我们将用一节将这些相当现代的技术与利特尔伍德-理查森规则的早期工作联系起来。
A geometric Littlewood-Richardson rule
我们描述了一个明确的几何Littlewood-Richardson规则，解释为变形的交叉两个舒伯特品种，使他们打破舒伯特品种。对基场没有任何限制，所有出现的多重数都是1;这对应用程序很重要。这条规则应该被看作是皮耶里规则的推广到任意舒伯特类，通过显式同伦。它与其他Littlewood-Richardson规则有一个直接的双射，例如tableaux，Knutson和Tao的谜题。这给出了Littlewood-Richardson规则的第一个几何证明和解释。它有一个主机的几何后果，在配套文件“舒伯特归纳”。这个规则在K理论中也有一个解释，由Buch提出，它给了K理论一个扩展的谜题。该规则建议一个自然的方法来解决一个悬而未决的问题，找到一个Littlewood-Richardson规则的旗帜品种，导致一个猜想，证明是真实的5维。最后，该规则为类似的开放问题提供了方法，例如辛格拉斯曼和双旗簇的Littlewood-Richardson规则。
A Richardson variety $X_\ga^\gc$ in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety $X^\gc$ in the Orthogonal Grassmannian and a opposite Schubert variety $X_\ga$ therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any $T$-fixed point of $X_\ga^\gc$, thus generalizing a result of Raghavan-Upadhyay \cite{Ra-Up2}. Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal bounded RSK (OBRSK). 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.
Littlewood-Richardson polynomials
我们引入一族依赖于参数的无穷序列的对称函数环。这样一个环的一个特殊的基是由舒尔函数的类似物组成的。相应的结构系数是参数的多项式，我们称之为Littlewood-Richardson多项式。通过修改B的早期结果，给出了它们计算的组合规则。萨根和作者。新的规则提供了一个公式，这些多项式是明显的积极意义上的W。格雷姆我们应用这个公式计算产品的等变Schubert类Grassmannian这意味着一个稳定的性质的结构系数。这种展开式的第一个明显的正公式是由A. Knutson和T.道使用组合数学的难题，而稳定性的性质是不明显的，从该公式。我们还在A. L.构造的量子固有子的基础上，用Littlewood-Richardson多项式描述了一般线性李代数的Casimir元代数中的乘法法则. Okounkov和G.奥尚斯基
Littlewood-Richardson semigroups
本文是我在1997年4月14日MSRI的表示论和对称函数研讨会上的发言的扩展摘要。讨论了长度为$\leq r$的三重划分半群$LR_r$的显式描述，使得相应的Littlewood-Richardson系数不为零.在讨论了这个问题的历史和以前已知的结果，我们提出了一种新的方法的基础上的“多面体”的组合表达式的Littlewood-Richardson系数。
We study toric degenerations of opposite Schubert and Richardson varieties inside degenerations of Grassmannians and flag varieties. These degenerations are parametrized by matching fields in the sense of Sturmfels and Zelevinsky. We construct so-called restricted matching field ideals whose generating sets are understood combinatorially through tableaux. We determine when these ideals are toric and coincide with Gröbner degenerations of Richardson varieties using the well established standard monomial theory for Grassmannians and flag varieties.
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。
We give a new description of the Pieri rule for k-Schur functions using the Bruhat order on the affine type-A Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and k-Schur functions.
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。
An S_3-symmetric Littlewood-Richardson rule
经典的Littlewood-Richardson系数C（lambda，mu，nu）通过指数的置换携带自然的$S_3$对称性。我们的“纸箱规则”计算这些数字透明和统一解释这六个对称性，以前制定的Littlewood-Richardson规则表现在最多的三个。
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 give a combinatorial description of the multiplicity at any torus fixed point on a Richardson variety in the Symplectic Grassmannian.
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.
A Littlewood-Richardson rule for Grassmannian Permutations
本文给出了一个计算旗流形上的相交数的组合规则，这些相交数来自于从格拉斯曼投影中拉回的Schubert类的乘积。这个规则推广了格拉斯曼的已知规则。
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.
Let $k$ and $n$ be positive coprime integers with $k<n$. Let $T$ denote the subgroup of diagonal matrices in $SL(n,\mathbb{C})$. We study the GIT quotient of Richardson varieties $X^v_w$ in the Grassmannian $\mathrm{Gr}_{k,n}$ by $T$ with respect to a $T$-linearised line bundle $\cal{L}$ corresponding to the Plücker embedding. We give necessary and sufficient combinatorial conditions for the quotient variety $T \backslash\mkern-6mu\backslash (X_w^v)^{ss}_T({\cal L})$ to be smooth.
Combinatorics and geometry of Littlewood-Richardson cones
我们提出了几个直接双射之间的不同组合解释的Littlewood-Richardson系数。双射是由显式线性映射定义的，它有其他的应用。
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.
Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials
证明了仿射对称群的Littlewood-Richardson系数是某些抛物型Kazhdan-Lusztig多项式在1处的值.这些q-类似物的Littlewood-Richardson多重性与以前介绍的丝带tableaux。
An algorithmic Littlewood-Richardson rule
我们引入了一个基于斜Young图算法变形的Littlewood-Richardson规则，并给出了一个具有经典规则的双射。其结果是一个直接的组合解释和证明的几何规则提出的Coskun。我们还提出了一个推论的Specht模块的中间图。
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.
Local Properties of Richardson Varieties in the Grassmannian via a Bounded Robinson-Schensted-Knuth Correspondence
本文给出了Grassmannian中Richardson簇的切锥理想在任意T-不动点处的显式Grobner基，从而推广了Kodiyalam-Raghavan和Kreiman-Lakshmibai的一个结果.我们的证明是基于广义的Robinson-Schensted-Knuth（RSK）对应，我们称之为有界RSK（BRSK）。利用Grobner基的结果，通过对不相交格路族的计数，导出了Richardson簇在任意T-不动点的重数公式，推广了Krattehthaler首先证明的一个结果.
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.
The computation of Kostka Numbers and Littlewood-Richardson Coefficients is #P-complete
Kostka数和Littlewood-Richardson系数在对称群和特殊线性群的表示论中起着重要的作用。人们对它们的计算很感兴趣。它们的计算复杂性问题一直是一个民间传说的问题，但E.拉萨特我们证明了这两个量的计算都是P-完全的。减少计算Kostka数，是从P-完全问题的数量计数的2 × k列联表具有给定的行和列的总和。这种还原的主要成分是D. E.高德纳计算Littlewood-Richardson系数的问题是从计算Kostka数的问题简化而来的。
A Littlewood-Richardson rule for the K-theory of Grassmannians
证明了Grassmann簇的Grothendieck环关于其Schubert结构层基的结构常数的一个显式组合公式。我们还涉及K-理论的格拉斯曼双代数稳定的Grothendieck多项式，这是一个K-理论平行的环对称函数。
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 between Littlewood-Richardson tableaux and rigged configurations
从Littlewood-Richardson表定义了一个双射到操纵配置。它表明，该映射保留了适当的统计，从而证明了广义Kostka多项式，这是q-模拟的重性不可约的一般线性群模块索引的矩形分区张量积的准粒子表达式。
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。
Initial ideals of tangent cones to Richardson varieties in the Symplectic Grassmannian
本文研究辛格拉斯曼流形中 Richardson 簇切锥的初始理想，构造其任意 T-不动点处切锥理想的显式 Gröbner 基，推广了已有结果。
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…
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