Francois Descouens
Hall-Littlewood functions indexed by rectangular partitions, specialized at primitive roots of unity, can be expressed as plethysms. We propose a combinatorial proof of this formula using A. Schilling's bijection between ribbon tableaux and ribbon rigged configurations.
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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.
We show that the bijection from rigged configurations to tensor products of Kirillov-Reshetikhin crystals extends to a crystal isomorphism between the $B(\infty)$ models given by rigged configurations and marginally large tableaux.
In this paper, we extend work of the first author on a crystal structure on rigged configurations of simply-laced type to all non-exceptional affine types using the technology of virtual rigged configurations and crystals. Under the bijection between rigged configurations and tensor products of Kirillov-Reshetikhin crystals specialized to a single tensor factor, we obtain a new tableaux model for Kirillov-Reshetikhin crystals. This is related to the model in terms of Kashiwara-Nakashima tableaux via a filling map, generalizing the recently discovered filling map in type $D_n^{(1)}$.
Rigged configurations are combinatorial objects prominent in the study of solvable lattice models. Marginally large tableaux are semi-standard Young tableaux of special form that give a realization of the crystals ${\cal B}(\infty)$. We introduce cascading sequences to characterize marginally large tableaux. Then we use cascading sequences and a non-explicit crystal isomorphism between marginally large tableaux and rigged configurations to give a characterization of the latter set, and to give an explicit bijection between the two sets.
Extending the work arXiv:math/0508107, we introduce the affine crystal action on rigged configurations which is isomorphic to the Kirillov-Reshetikhin crystal B^{r,s} of type D_n^(1) for any r,s. We also introduce a representation of B^{r,s} (r not equal to n-1,n) in terms of tableaux of rectangular shape r x s, which we coin Kirillov-Reshetikhin tableaux (using a non-trivial analogue of the type A column splitting procedure) to construct a bijection between elements of a tensor product of Kirillov-Reshetikhin crystals and rigged configurations.
Recently, the analogue of the promotion operator on crystals of type A under a generalization of the bijection of Kerov, Kirillov and Reshetikhin between crystals (or Littlewood--Richardson tableaux) and rigged configurations was proposed. In this paper, we give a proof of this conjecture. This shows in particular that the bijection between tensor products of type A_n^{(1)} crystals and (unrestricted) rigged configurations is an affine crystal isomorphism.
We construct a natural bijection between the set of admissible pictures and the set of $U_q(gl(m,n))$-Littlewood-Richardson tableaux.
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.
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 …
For an affine algebra of nonexceptional type in the large rank we show the fermionic formula depends only on the attachment of the node 0 of the Dynkin diagram to the rest, and the fermionic formula of not type A can be expressed as a sum of that of type A with Littlewood-Richardson coefficients. Combining this result with math.CO/9901037 and arXiv:1002.3715 we settle the X=M conjecture under the large rank hypothesis.
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.
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 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.
Characterization of $\mathfrak{k}$-highest weight and $\widehat{\mathfrak{g}}$-dominant tableaux via $1$-$0$-slack recording tableaux in the quantum Littlewood-Richardson rule and lattice points in flagged hive polytopes
该研究针对n≤4的情形，扩展了量子Littlewood-Richardson规则中𝔨最高权表的刻画，通过Naito-Suzuki-Watanabe双射得到对应的ĝ优表，还建立了相关表与标记蜂巢格点的双射。
q-Supernomial coefficients are generalizations of the q-binomial coefficients. They can be defined as the coefficients of the Hall-Littlewood symmetric function in a product of the complete symmetric functions or the elementary symmetric functions. Hatayama et al. give explicit expressions for these q-supernomial coefficients. A combinatorial expression as the generating function of ribbon tableaux with (co)spin statistic follows from the work of Lascoux, Leclerc and Thibon. In this paper we interpret the formulas by Hatayama et al. in terms of rigged configurations and provide an explicit sta…
We introduce a new family of noncommutative analogues of the Hall-Littlewood symmetric functions. Our construction relies upon Tevlin's bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new Hall-Littlewood functions to permutation tableaux, and also give an exact formula for the q-enumeration of permutation tableaux of a fixed shape. This gives an explicit formula for: the steady state probability of each state in the partially asymmetric exclusion process (PASEP); the polynomial enumerating permutations with a fixed set of weak excedances according t…
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…
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。
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 …
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 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 give a bijection $Φ$ from rigged configurations to a tensor product of Kirillov--Reshetikhin crystals of the form $B^{r,1}$ and $B^{1,s}$ in type $D_4^{(3)}$. We show that the cocharge statistic is sent to the energy statistic for tensor products $\bigotimes_{i=1}^N B^{r_i,1}$ and $\bigotimes_{i=1}^N B^{1,s_i}$. We extend this bijection to a single $B^{r,s}$, show that it preserves statistics, and obtain the so-called Kirillov--Reshetikhin tableaux model for $B^{r,s}$. Additionally, we show that $Φ$ commutes with the virtualization map and that $B^{1,s}$ is naturally a virtual crystal in ty…
Level-restricted paths play an important role in crystal theory. They correspond to certain highest weight vectors of modules of quantum affine algebras. We show that the recently established bijection between Littlewood--Richardson tableaux and rigged configurations is well-behaved with respect to level-restriction and give an explicit characterization of level-restricted rigged configurations. As a consequence a new general fermionic formula for the level-restricted generalized Kostka polynomial is obtained. Some coset branching functions of type $A$ are computed by taking limits of these fe…
In an earlier work, the authors developed a rigged configuration model for the crystal $B(\infty)$ (which also descends to a model for irreducible highest weight crystals via a cutting procedure). However, the result obtained was only valid in finite types, affine types, and simply-laced indefinite types. In this paper, we show that the rigged configuration model proposed does indeed hold for all symmetrizable types. As an application, we give an easy combinatorial condition that gives a Littlewood-Richardson rule using rigged configurations which is valid in all symmetrizable Kac-Moody types.
We give a uniform description of the bijection $Φ$ from rigged configurations to tensor products of Kirillov--Reshetikhin crystals of the form $\bigotimes_{i=1}^N B^{r_i,1}$ in dual untwisted types: simply-laced types and types $A_{2n-1}^{(2)}$, $D_{n+1}^{(2)}$, $E_6^{(2)}$, and $D_4^{(3)}$. We give a uniform proof that $Φ$ is a bijection and preserves statistics. We describe $Φ$ uniformly using virtual crystals for all remaining types, but our proofs are type-specific. We also give a uniform proof that $Φ$ is a bijection for $\bigotimes_{i=1}^N B^{r_i,s_i}$ when $r_i$, for all $i$, map to $0$…
The recording tableaux in the quantum Littlewood-Richardson map, the orthogonal transpose symmetry map, and the computation of $\mathfrak{k}$-highest weight tableaux
本文组合证明了 Watanabe 给出的量子 LR 映射的满射性，并利用其逆映射计算和刻画了一类 $\mathfrak{k}$-最高权半标准杨表。
We define mosaics, which are naturally in bijection with Knutson-Tao puzzles. We define an operation on mosaics, which shows they are also in bijection with Littlewood-Richardson skew-tableaux. Another consequence of this construction is that we obtain bijective proofs of commutativity and associativity for the ring structures defined either of these objects. In particular, we obtain a new, easy proof of the Littlewood-Richardson rule. Finally we discuss how our operation is related to other known constructions, particularly jeu de taquin.
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。
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.
We study the Gaussent-Littelmann formula for Hall-Littlewood polynomials and we develop combinatorial tools to describe the formula in a purely combinatorial way for type A_n, B_n and C_n. This description is in terms of Young tableaux and arises from identifying one-skeleton galleries that appear in the Gaussent-Littelmann formula with Young tableaux. Furthermore, we show by using these tools that the Gaussent-Littelmann formula and the well-known Macdonald formula for Hall-Littlewood polynomials for type A_n are the same.
Bounded Littlewood identities with fixed number of odd rows or odd columns
本文研究了有界Littlewood恒等式的细化形式，通过固定奇数行或列数目，提出新的公式并探讨了与标准Young表的关联。
A super Robinson-Schensted-Knuth correspondence with symmetry and the super Littlewood-Richardson rule
本文提出了一种超 Robinson-Schensted-Knuth 对应关系，用于超表aux 上的符号字母，通过 Schensted 插入算法的超版本进行构建，并给出了超 RSK 对应关系的几何解释，最终推导出超 Schur 函数在有限符号字母上的组合版本的超 Littlewood-Richardson 规则。
Bounded Littlewood identities for cylindric Schur functions
本文提出有界Littlewood恒等式的仿射类似物，给出柱形Schur函数求和的行列式公式，并揭示柱形标准杨表与r-非交叉、s-非嵌套匹配间的新联系。
Littlewood-Richardson (LR) coefficients $c_{μν}^λ$ may be evaluated by means of several combinatorial models, including the original LR tableaux of skew shape $λ/μ$ and weight $ν$ and the LR hives with boundary edge labels $λ$, $μ$ and $ν$. Unfortunately, neither of these reveal in any obvious way the well-known symmetry property $c_{μν}^λ=c_{νμ}^λ$. Here we introduce two maps, $ρ^{(n)}$ on LR tableaux and $σ^{(n)}$ on LR hives, that each interchange contributions to $c_{μν}^λ$ and $c_{νμ}^λ$ for any partitions $λ$, $μ$, $ν$ of lengths no greater than $n$, and then prove not only that each of …
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 introduce a new combinatorial object, semistandard increasing decomposition tableau and study its relation to a semistandard decomposition tableau introduced by Kraśkiewicz and developed by Lam and Serrano. We also introduce generalized Littlewood--Richardson coefficients for products of Schur's symmetric functions and give combinatorial descriptions in terms of tableau words. The insertion algorithms play central roles for proofs. A new description of shifted Littlewood--Richardson coefficients is given in terms of semistandard increasing decomposition tableaux. We show that a "big" Schur …
The connection between the generating functions of various sets of tableaux and the appropriate families of quasisymmetric functions is a significant tool to give a direct analytical proof of some advanced bijective results and provide new combinatorial formulas. In this paper we focus on two kinds of type B Littlewood-Richardson coefficients and derive new formulas using weak composition quasisymmetric functions and Chow's quasisymmetric functions. In the type A case these coefficients give the multiplication table for Schur functions, i.e. the generating functions for classical Young tableau…
We will look at the Catalan numbers from the {\it Rigged Configurations} point of view originated \cite{Kir} from an combinatorial analysis of the Bethe Ansatz Equations associated with the higher spin anisotropic Heisenberg models . Our strategy is to take a combinatorial interpretation of Catalan numbers $C_n$ as the number of standard Young tableaux of rectangular shape $(n^2)$, or equivalently, as the Kostka number $K_{(n^2),1^{2n}}$, as the starting point of research. We observe that the rectangular (or multidimensional) Catalan numbers $ C(m,n)$ introduced and studied by P. MacMahon \cit…
The monomial expansion formula for Hall-Littlewood $P$-polynomials
本文通过Hecke代数方法推导了Hall-Littlewood多项式的Macdonald展开公式，并推广了Klostermann递归，为相关组合恒等式提供了新证明。
We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of `doubled crystal' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are…
Using a 0/1 encoding of Young diagrams and its consequences for rim hook tableaux, we prove a reduction formula of Littlewood for arbitrary characters of the symmetric group, evaluated at elements with all cycle lengths divisible by a given integer. As an application, we find explicitly the coefficients in a formula of Kostant for certain powers of the Dedekind $η$-function, avoiding most of the original machinery.
We present a family of analogs of the Hall-Littlewood symmetric functions in the $Q$-function algebra. The change of basis coefficients between this family and Schur's $Q$-functions are $q$-analogs of numbers of marked shifted tableaux. These coefficients exhibit many parallel properties to the Kostka-Foulkes polynomials.
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 introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of `doubled crystal' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are…
We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall-Littlewood polynomials P_lambda/mu(x;t) and Hivert's quasisymmetric Hall-Littlewood polynomials G_gamma(x;t). More specifically, we provide: 1) the G-expansions of the Hall-Littlewood polynomials P_lambda, the monomial quasisymmetric polynomials M_alpha, the quasisymmetric Schur polynomials S_alpha, and the peak quasisymmetric functions K_alpha; 2) an expansion of P_lambda/mu in terms of the F_alpha's. The F-expansion of P_lambda/mu is facilitated by introducing sta…
We study k-Schur functions characterized by k-tableaux, proving combinatorial properties such as a k-Pieri rule and a k-conjugation. This new approach relies on developing the theory of k-tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender-Knuth involution. This work lays the groundwork needed to prove that the set of k-Schur Littlewood-Richardson coefficients contains the 3-point Gromov-Witten invariants; structure constants for the quantum cohomology ring.
We present a proof of a Littlewood-Richardson rule for the K-theory of odd orthogonal Grassmannians OG(n,2n+1), as conjectured in [Thomas-Yong '09]. Specifically, we prove that rectification using the jeu de taquin for increasing shifted tableaux introduced there, is well-defined and gives rise to an associative product. Recently, [Buch-Ravikumar '09] proved a Pieri rule for OG(n,2n+1) that [Feigenbaum-Sergel '09] showed confirms a special case of the conjecture. Together, these results imply the aforementioned conjecture.
Straight polyomino tilings of rectangles and special rim-hook tableaux
通过故障线分解与Hadamard积框架，结合Littlewood恒等式和Graham定理，推导了$2k\ imes n$矩形被$k\ imes 1$直线瓷砖覆盖的加权生成函数显式表达式。
A poset on a certain class of partitions known as k-shapes was recently introduced to provide a combinatorial rule for the expansion of a (k-1)-Schur functions into k-Schur functions at t=1. The main ingredient in this construction was a bijection, which we call the weak bijection, that associates to a k-tableau a pair made out of a (k-1)-tableau and a path in the poset of k-shapes. We define here a concept of charge on k-tableaux (which conjecturally gives a combinatorial interpretation for the expansion coefficients of Hall-Littlewood polynomials into k-Schur functions), and show that it is …
A new combinatorial approach to the ribbon tableaux generating functions and q-Littlewood Richardson coefficients of Lascoux, Leclerc and Thibon is suggested. We define operators which add ribbons to partitions and following Fomin and Greene study non-commutative symmetric functions in these operators. This allows us to give combinatorial interpretations for some (skew) q-Littlewood Richardson coefficients whose non-negativity appears not to be known. Our set up also leads to a new proof of the action of the Heisenberg algebra on the Fock space of U_q(sl^_n) due to Kashiwara, Miwa and Stern.
We present two symmetric function operators $H_3^{qt}$ and $H_4^{qt}$ that have the property $H_{3}^{qt} H_{(2^a1^b)}[X;q,t] = H_{(32^a1^b)}[X;q,t]$ and $H_4^{qt} H_{(2^a1^b)}[X;q,t] = H_{(42^a1^b)}[X;q,t]$. These operators are generalizations of the analogous operator $H_2^{qt}$ and also have expressions in terms of Hall-Littlewood vertex operators. We also discuss statistics, $a_μ(T)$ and $b_μ(T)$, on standard tableaux such that the $q,t$ Kostka polynomials are given by the sum over standard tableaux of shape $\la$, $K_{\laμ}(q,t) = \sum_T t^{a_μ(T)} q^{b_μ(T)}$ for the case when when $μ$ is…
In work with A. Yong, the author introduced genomic tableaux to prove the first positive combinatorial rule for the Littlewood-Richardson coefficients in torus-equivariant $K$-theory of Grassmannians. We then studied the genomic Schur function $U_λ$, a generating function for such tableaux, showing that it is non-trivially a symmetric function, although generally not Schur-positive. Here we show that $U_λ$ is, however, positive in the basis of fundamental quasisymmetric functions. We give a positive combinatorial formula for this expansion in terms of gapless increasing tableaux; this is, more…
A uniform action of the dihedral group $ Z_2\times D_3$ on Littlewood--Richardson coefficients
本文证明十二阶二面体群 $Z_2\times D_3$ 忠实地作用于 Littlewood–Richardson 表、伴随表、蜂巢和拼图，并将隐藏的交换与共轭对称性在线性时间内归约到已知 LR 交换子、转置子及 Luzstig–Schützenberger 对合。
We define a solitary game, the Yamanouchi toppling game, on any connected graph of n vertices. The game arises from the well-known chip-firing game when the usual relation of equivalence defined on the set of all configurations is replaced by a suitable partial order. The set all firing sequences of length m that the player is allowed to perform in the Yamanouchi toppling game is shown to be in bijection with all standard Young tableaux whose shape is a partition of the integer m with at most n-1 parts. The set of all configurations that a player can obtain from a starting configuration is enc…
This paper makes precise the close connection between the affine Hecke algebra, the path model, and the theory of crystals. Section 2 is a basic pictorial exposition of Weyl groups and affine Weyl groups and Section 5 is an exposition of the theory of (a) symmetric functions, (b) crystals and (c) the path model. Sections 3 and 4 give an exposition of the affine Hecke algebra and recent results regarding the combinatorics of spherical functions on p-adic groups (Hall-Littlewood polynomials). The $q$-analogue of the theory of crystals developed in Section 4 specializes to the path model version …
Let $Λ$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_λ[X/(1-t)]\}_{λ_1\leq k}$. We introduce a new family of symmetric polynomials, $\{A_λ^{(k)}[X;t]\}_{λ_1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A_λ^{(k)}[X;t]$ form a basis for $V_k$ and that the Macdonald polynomials indexed by partitions whose first part is not larger than $k$ expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but wo…
We give a new characterization of Littlewood-Richardson-Stembridge tableaux for Schur $P$-functions by using the theory of $\mf{q}(n)$-crystals. We also give alternate proofs of the Schur $P$-expansion of a skew Schur function due to Ardila and Serrano, and the Schur expansion of a Schur $P$-function due to Stembridge using the associated crystal structures.
We present a sum-product formula for the classical Hall polynomial which is based on tableaux that have been introduced by T. Klein in 1969. In the formula, each summand corresponds to a Klein tableau, while the product is taken over the cardinalities of automorphism groups of short exact sequences which are derived from the tableau. For each such sequence, one can read off from the tableau the summands in an indecomposable decomposition, and the size of their homomorphism and automorphism groups. Klein tableaux are refinements of Littlewood-Richardson tableaux in the sense that each entry $\e…
We introduce the structure of vacillating Hecke tableaux, and establish a one-to-one correspondence between vacillating Hecke tableaux and linked partitions by using the Hecke insertion algorithm developed by Buch, Kresch, Shimozono, Tamvakis and Yong. Linked partitions arise in free probability theory. Motivated by the Hecke insertion algorithm, we define a Hecke diagram as a Young diagram possibly with a marked corner. A vacillating Hecke tableau is defined as a sequence of Hecke diagrams subject to certain addition and deletion of rook strips. The notion of a rook strip was introduced by Bu…
The conjugacy classes of nilpotent $n\times n$ matrices can be parametrised by partitions $λ$ of $n$, and for a nilpotent $η$ in the class parametrised by $λ$, the variety $F_η$ of $η$-stable flags has its irreducible components parametrised by the standard Young tableaux of shape $λ$. We indicate how several algorithmic constructions defined for Young tableaux have significance in this context, thus extending Steinberg's result that the relative position of flags generically chosen in the irreducible components of $F_η$ parametrised by tableaux $P$ and $Q$, is the permutation associated to $(…
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