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.
A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus
我们介绍了一个理论的jeu de taquin增加tableaux，扩展基本工作[Schützenberger '77]标准杨tableaux。我们应用这个给一个新的组合规则的K-理论舒伯特演算的格拉斯曼通过K-理论jeu de taquin，提供了一个替代的规则[Buch '02]和其他人。这一规则自然地推广到对任何极小旗型品种G/P给出一个自然根系统一规则，从而推广了[Jias-Yong '06]。我们还提出了类似的Fomin，Haiman，Schensted和Schützenberger的结果。
The Wronskian associates to d linearly independent polynomials of degree at most n, a non-zero polynomial of degree at most d(n-d). This can be viewed as giving a flat, finite morphism from the Grassmannian Gr(d,n) to projective space of the same dimension. In this paper, we study the monodromy groupoid of this map. When the roots of the Wronskian are real, we show that the monodromy is combinatorially encoded by Schutzenberger's jeu de taquin; hence we obtain new geometric interpretations and proofs of a number of results from jeu de taquin theory, including the Littlewood-Richardson rule.
The direct sum map Gr(a,n) x Gr(b,m) -> Gr(a+b,m+n) on Grassmannians induces a K-theory pullback that defines the splitting coefficients. We geometrically explain an identity from [Buch '02] between the splitting coefficients and the Schubert structure constants for products of Schubert structure sheaves. This is related to the topic of product and splitting coefficients for Schubert boundary ideal sheaves. Our main results extend jeu de taquin for increasing tableaux [Thomas-Yong '09] by proving transparent analogues of [Schützenberger '77]'s fundamental theorems on well-definedness of rectif…
Schützenberger's jeu de taquin is an algorithm on the structure of tableaux, which transforms a skew tableau into a Young one by local transformation rules on the columns of the tableaux. This algorithm defines an equivalence relation on tableaux compatible with the plactic congruence, and gives a proof of the Littlewood-Richardson rule on Schur polynomials. In this article, we introduce the notion of string of columns rewriting system as mechanism of transformations of glued sequences of columns. We describe the execution of the jeu de taquin algorithm as rewriting paths of a string of column…
We give a backward jeu de taquin slide analogue on semistandard reverse composition tableaux. These tableaux were first studied by Haglund, Luoto, Mason and van Willigenburg when defining quasisymmetric Schur functions. Our algorithm for performing backward jeu de taquin slides on semistandard reverse composition tableaux results in a natural operator on compositions that we call the jdt operator. This operator in turn gives rise to a new poset structure on compositions whose maximal chains we enumerate. As an application, we also give a noncommutative Pieri rule for noncommutative Schur funct…
This paper presents a combinatorial study of the super plactic monoid of type A, which is related to the representations of the general linear Lie superalgebra. We introduce the analogue of the Schützenberger's jeu de taquin on the structure of super tableaux over a signed alphabet. We show that this procedure which transforms super skew tableaux into super Young tableaux is compatible with the super plactic congruence and it is confluent. We deduce properties relating the super jeu de taquin to insertion algorithms on super tableaux. Moreover, we introduce the super evacuation procedure as an…
We establish a combinatorial connection between the real geometry and the $K$-theory of complex Schubert curves $S(λ_\bullet)$, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map $ω$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of $\mathbb{RP}^1$, with $ω$ as the monodromy operator. We prov…
对于他们的双射证明的钩长公式的数量标准tableaux的一个固定的形状诺维利，帕克和Stoyanovskii定义了一个修改的jeu de taquin的转换任意填充的费勒斯图与$1,2,...,n$（tabloid）到一个标准tableau。它们的定义依赖于由特殊标准表诱导的费勒斯图中的细胞的全序，然而，这个定义对于由任何其他标准表诱导的全序也是有意义的。给定两个相同形状的标准tableaux $P,Q$，我们表明，如果我们相对于$Q$诱导的总顺序执行修改的jeu de taquin，则导致$P$的tabloids的数量等于如果我们相对于$P$执行修改的jeu de taquin，则导致$Q$的tabloids的数量。这个对称性定理扩展到歪斜形状和移位歪斜形状。
Using a modified version of jeu de taquin, Novelli, Pak and Stoyanovskii gave a bijective proof of the hook-length formula for counting standard Young tableaux of fixed shape. In this paper we consider a natural extension of jeu de taquin to arbitrary posets. Given a poset P, jeu de taquin defines a map from the set of bijective labelings of the poset elements with $\{1,2,...,|P|\}$ to the set of linear extensions of the poset. One question of particular interest is for which posets this map yields each linear extension equally often. We analyze the double-tailed diamond poset $D_{m,n}$ and sh…
Irreducible characters in the symmetric group are of special interest in combinatorics. They can be expressed either combinatorially with ribbon tableaux, or algebraically with contents. In this paper, these two expressions are related in a combinatorial way. We first introduce a fine structure in the famous jeu de taquin called "trace forest", with which we are able to count certain types of ribbon tableaux, leading to a simple bijective proof of a character evaluation formula in terms of contents that dates back to Frobenius (1901). Inspired by this proof, we give an inductive scheme that gi…
We investigate Robinson-Schensted-Knuth algorithm (RSK) and Schützenberger's jeu de taquin in the infinite setup. We show that the recording tableau in RSK defines an isomorphism of the following two dynamical systems: (i) a sequence of i.i.d. random letters equipped with Bernoulli shift, and (ii) a random infinite Young tableau (with the distribution given by Vershik-Kerov measure, corresponding to some Thoma character of the infinite symmetric group) equipped with jeu de taquin transformation. As a special case we recover the results on non-colliding random walks and multidimensional Pitman …
We establish a combinatorial connection between the real geometry and the $K$-theory of complex Schubert curves $S(λ_\bullet)$, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map $ω$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of $\mathbb{RP}^1$, with $ω$ as the monodromy operator. We prov…
We introduce edge labeled Young tableaux. Our main results provide a corresponding analogue of [Schützenberger '77]'s theory of jeu de taquin. These are applied to the equivariant Schubert calculus of Grassmannians. Reinterpreting, we present new (semi)standard tableaux to study factorial Schur polynomials, after [Biedenharn-Louck '89], [Macdonald '92] and [Goulden-Greene '94] and others. Consequently, we obtain new combinatorial rules for the Schubert structure coefficients, complementing work of [Molev-Sagan '99], [Knutson-Tao '03], [Molev '08] and [Kreiman '09]. We also describe a conjectur…
In this paper, we study tropical-theoretic aspects of the ``rectification algorithm'' on skew Young tableaux. It is shown that the algorithm is interpreted as a time evolution of some tropical integrable system. By using this fact, we construct a new combinatorial map that is essentially equivalent to the rectification algorithm. Some of properties of the rectification can be seen more clearly via this map. For example, the uniqueness of a rectification boils down to an easy combinatorial problem. Our method is mainly based on the two previous researches: the theory of geometric tableaux by No…
Based on Thomas and Yong's K-theoretic jeu de taquin algorithm, we prove a uniform Littlewood-Richardson rule for the K-theoretic Schubert structure constants of all minuscule homogeneous spaces. Our formula is new in all types. For the main examples of Grassmannians of type A and maximal orthogonal Grassmannians it has the advantage that the tableaux to be counted can be recognized without reference to the jeu de taquin algorithm.
There exists a well-known hook-length formula for calculating the dimensions of 2D Young diagrams. Unfortunately, the analogous formula for 3D case is unknown. We introduce an approach for calculating the estimations of dimensions of three-dimensional Young diagrams also known as plane partitions. The most difficult part of this task is the calculation of co-transition probabilities for a central Markov process. We propose an algorithm for approximate calculation of these probabilities. It generates numerous random paths to a given diagram. In case the generated paths are uniformly distributed…
We study an infinite version of the "jeu de taquin" sliding game, which can be thought of as a natural measure-preserving transformation on the set of infinite Young tableaux equipped with the Plancherel probability measure. We use methods from representation theory to show that the Robinson-Schensted-Knuth ($\operatorname {RSK}$) algorithm gives an isomorphism between this measure-preserving dynamical system and the one-sided shift dynamics on a sequence of independent and identically distributed random variables distributed uniformly on the unit interval. We also show that the jeu de taquin …
It is known that the right key of a Kashiwara-Nakashima tableau can be computed using the Lecouvey-Sheats symplectic jeu de taquin. Motivated by Willis' direct way of computing type A right keys, we also give a way of computing symplectic right keys without the use of jeu de taquin.
We develop a combinatorial rule to compute the real geometry of type B Schubert curves $S(λ_\bullet)$ in the orthogonal Grassmannian $\mathrm{OG}_n$, which are one-dimensional Schubert problems defined with respect to orthogonal flags osculating the rational normal curve. Our results are natural analogs of results previously known only in type A. First, using the type B Wronski map, we show that the real locus of the Schubert curve has a natural covering map to $\mathbb{RP}^1$, with monodromy operator $ω$ defined as the commutator of jeu de taquin rectification and promotion on skew shifted se…
An increasing tableau is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a bijection between rectangular 2-row increasing tableaux and small Schröder paths. We demonstrate relations between the jeu de taquin for increasing tableaux developed by H. Thomas and A. Yong and the combinatorics of tropical frieze patterns. We then use this jeu de taquin to present new instances of the cyclic sieving phenomenon of V. Reiner, D. Stanton, a…
We introduce tableau stabilization, a new phenomenon and statistic on Young tableaux based on jeu de taquin. We investigate bounds for tableau stabilization, the shape of stabilized tableaux, and tableau stabilization as a permutation statistic. We apply tableau stabilization to construct the sufficiently large rectangular tableaux fixed by powers of promotion, which were counted by Brendon Rhoades via the cyclic sieving phenomenon
We give a counting formula for the set of rectangular increasing tableaux in terms of generalized Narayana numbers. We define small $m$-Schröder paths and give a bijection between the set of increasing rectangular tableaux and small $m$-Schröder paths, generalizing a result of Pechenik [3]. Using $K$-jeu de taquin promotion, which was defined by Thomas and Yong [10], we give a cyclic sieving phenomenon for the set of increasing hook tableaux.
我们定义马赛克，这是自然的双射与克努森-陶难题。我们定义了一个操作的马赛克，这表明他们也在双射与Littlewood-Richardson skew-tableaux。这种建设的另一个后果是，我们获得双射证明的交换性和结合性的环结构定义这些对象之一。特别是，我们得到了一个新的，简单的证明Littlewood-Richardson规则。最后，我们将讨论我们的操作是如何与其他已知的结构，特别是jeu de taquin。
We present an adaptation of jeu de taquin and promotion for arbitrary fillings of moon polyominoes. Using this construction we show various symmetry properties of such fillings taking into account the lengths of longest increasing and decreasing chains. In particular, we prove a conjecture of Jakob Jonsson. We also relate our construction to the one recently employed by Christian Krattenthaler, thus generalising his results.
We address a unification of the Schubert calculus problems solved by [A. Buch '02] and [A. Knutson-T. Tao '03]. That is, we prove a combinatorial rule for the structure coefficients in the torus-equivariant K-theory of Grassmannians with respect to the basis of Schubert structure sheaves. We thereby deduce the conjectural rule of [H. Thomas-A. Yong '13] for the same coefficients. Both rules are positive in the sense of [D. Anderson-S. Griffeth-E. Miller '11] (and moreover in a stronger form). Our work is based on the combinatorics of genomic tableaux and a generalization of [M.-P. Schutzenberg…
A bijective proof of the hook-length formula for shifted standard tableaux
本文基于修正的jeu de taquin和Novelli，Pak和Stoyanovskii关于普通标准表的钩长公式的双射证明思想，给出了固定形状移位标准表的钩长公式的双射证明.在Novelli、Pak和Stoyanovskii的证明中，他们定义了费勒斯图的任意填充与整数$1,2,...,n$和标准tableaux和hook tabloids对之间的双射。在我们的移动版本的算法的映射从一组任意填充的移动费勒斯图到一组移动的标准tableaux是类似的建设Novelli，Pak和Stoyanovskii，然而，与他们的算法不同，我们被迫使用'rowwise'总顺序的细胞在移动费勒斯图，而不是'columnwise'总顺序的算法中的基本顺序。不幸的是，在移位的情况下，移位的钩子小报的构造更加复杂。作为一个副作用，我们得到了一个简单的随机算法生成一个给定形状的移位标准tableaux，它产生每一个这样的tableaux同样可能。
Rowmotion is an invertible operator on the order ideals of a poset which has been extensively studied and is well understood for the rectangle poset. In this paper, we show that rowmotion is equivariant with respect to a bijection of Hamaker, Patrias, Pechenik and Williams between order ideals of rectangle and trapezoid posets, thereby affirming a conjecture of Hopkins that the rectangle and trapezoid posets have the same rowmotion orbit structures. Our main tools in proving this are $K$-jeu-de-taquin and (weak) $K$-Knuth equivalence of increasing tableaux. We define $almost$ $minimal$ $tablea…
A key fact about M.-P. Schützenberger's (1972) promotion operator on rectangular standard Young tableaux is that iterating promotion once per entry recovers the original tableau. For tableaux with strictly increasing rows and columns, H. Thomas and A. Yong (2009) introduced a theory of $K$-jeu de taquin with applications to $K$-theoretic Schubert calculus. The author (2014) studied a $K$-promotion operator $\mathcal{P}$ derived from this theory, but showed that the key fact does not generally extend to $K$-promotion of such increasing tableaux. Here we show that the key fact holds for labels o…
Patrias and Pylyavskyy introduced shifted Hecke insertion as an application of their theory of dual filtered graphs. We use shifted Hecke insertion to construct symmetric function representatives for the K-theory of the orthogonal Grassmannian. These representatives are closely related to the shifted Grothendieck polynomials of Ikeda and Naruse. We then recover the K-theory structure coefficients of Clifford-Thomas-Yong/Buch-Samuel by introducing a shifted K-theoretic Poirier-Reutenauer algebra. Our proofs depend on the theory of shifted K-theoretic jeu de taquin and the weak K-Knuth relations…
The number of standard Young tableaux of a fixed shape is famously given by the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to jeu-de-taquin which transforms an arbitrary filling of a partition into a standard Young tableau by exchanging adjacent entries. Recently, Krattenthaler and Müller defined the complexity of this algorithm as the average number of performed exchanges, and Neumann and the author proved it fulfils some nice symmetry properties. In this paper we recall and extend the previous r…
我们提出了一个部分推广到Schubert演算旗品种的经典Littlewood-Richardson规则，在其版本的基础上Schuetzenberger的jeu de taquin。更确切地说，我们描述了某些结构常数表达的产品的舒伯特和舒尔多项式。我们使用的推广Fomin的增长图（在杨格的分区链）链的排列在所谓的k-Bruhat顺序。我们的工作是基于Beligan最近的论文，他在其中推广了经典的词的plastic结构的链在一定的时间间隔在k-Bruhat秩序。我们的工作的潜在应用包括推广的S_3对称Littlewood-Richardson规则，由于托马斯和杨，这是基于Fomin的增长图。
The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is proved by Lenart-Lubovsky that the quantum alcove model does not depend on the choice of a reduced alcove path, which is a shortest path of alcoves from the fundamental one to its translation by the given dominant weight. This is established through quantum Yang-Baxter moves, which biject the objects of the model associated with two such alcove paths…
