In O'Connell-Pei(2013) a q-weighted version of the Robinson-Schensted algorithm was introduced. In this paper we show that this algorithm has a symmetry property analogous to the well known symmetry property of the normal Robinson-Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin(1979,1986,1994,1995). This approach, which uses "growth graphs", can also be applied to a wider class of insertion algorithms which have a branching structure, including some of the other q-weighted versions of the Robinson-Schensted algorithm which have recently been introduced by Borodin-Petrov(2013).
We prove a positivity theorem for a certain family of operators defined in terms of the stochastic six-vertex model. We explore connections of this result with other vertex models and $t$-deformed Schensted insertions.
We consider the Robinson-Schensted-Knuth algorithm applied to a random input and investigate the shape of the bumping route (in the vicinity of the $y$-axis) when a specified number is inserted into a large Plancherel-distributed tableau. We show that after a projective change of the coordinate system the bumping route converges in distribution to the Poisson process.
We present a probabilistic generalization of the Robinson--Schensted correspondence in which a permutation maps to several different pairs of standard Young tableaux with nonzero probability. The probabilities depend on two parameters $q$ and $t$, and the correspondence gives a new proof of the squarefree part of the Cauchy identity for Macdonald polynomials. By specializing $q$ and $t$ in various ways, one recovers both the row and column insertion versions of the Robinson--Schensted correspondence, as well as several $q$- and $t$-deformations of row and column insertion which have been intro…
Gelfand models and Robinson-Schensted correspondence
在[F. Caselli，Involutory reflection groups and their models，J. Algebra 24（2010），370- 393]，为所有非例外的不可约复反射群构造了一个统一的Gelfand模型，该复反射群是对合的。这样的模型可以自然地分解成由$S_n$ -共轭类所索引的子模的直和，并且我们在这里给出了一个将这些子模的不可约分解与投射Robinson-Schensted对应联系起来的一般结果.这种描述也以非常明确的方式反映了这些群的分裂表示的存在。
We construct new "standard modules" for the representations of general linear groups over a local non-archimedean field. The construction uses a modified Robinson-Schensted-Knuth correspondence for Zelevinsky's multisegments. Typically, the new class categorifies the basis of Doubilet, Rota, and Stein for matrix polynomial rings, indexed by bitableaux. Hence, our main result provides a link between the dual canonical basis (coming from quantum groups) and the DRS basis.
We investigate the evolution in time of the position of a fixed number in the insertion tableau when the Robinson-Schensted-Knuth algorithm is applied to a sequence of random numbers. When the length of the sequence tends to infinity, a typical trajectory after scaling converges uniformly in probability to some deterministic curve.
Matrix-Ball Construction of affine Robinson-Schensted correspondence
在他对仿射型$A$中的Kazhdan-Lusztig细胞的研究中，Shi引入了Robinson-Schensted对应的仿射模拟。我们推广了Viennot和Fulton的矩阵球构造，给出了Shi算法的一个更组合的实现。作为一种双积，我们也给出了一种通过通常的Robinson-Schensted碰撞算法实现仿射对应的方法。其次，受Lusztig和Xi的启发，我们将算法扩展到扩展仿射对称群和三元组$(P, Q, ρ)$之间的双射，其中$P$和$Q$是tabloid，$ρ$是主导权重。权值$ρ$在仿射矩阵球构造方面得到自然的解释。最后证明了逆映射的纤维具有外尔群对称性，从而解释了权的优势条件.
We consider Robinson-Schensted-Knuth algorithm applied to a random input and study the growth of the bottom rows of the corresponding Young diagrams. We prove multidimensional Poisson limit theorem for the resulting Plancherel growth process. In this way we extend the result of Aldous and Diaconis to more than just one row. This result can be interpreted as convergence of the multi-line Hammersley process to its stationary distribution which is given by a collection of independent Poisson point processes.
We present a probabilistic generalization of the Robinson--Schensted correspondence in which a permutation maps to several different pairs of standard Young tableaux with nonzero probability. The probabilities depend on two parameters $q$ and $t$, and the correspondence gives a new proof of the squarefree part of the Cauchy identity for Macdonald polynomials (i.e., the equality of the coefficients of $x_1 \cdots x_n y_1 \cdots y_n$ on either side, which are related to permutations and standard Young tableaux). By specializing $q$ and $t$ in various ways, one recovers the row and column inserti…
We study the Steinberg variety associated to matrix Schubert varieties, and develop a Robinson-Schensted type correspondence, $τ\leftrightarrow(Λ,\mathsf Q,\mathsf P)$. Here $τ$ is a partial permutation of size $p\times q$, $Λ$ an admissible signed Young diagram of size $p+q$, and $\mathsf P$ (resp. $\mathsf Q$) a standard Young tableau of size $p$ (resp. $q$) whose shape is determined by $Λ$. By embedding the matrix Schubert variety into a Schubert variety, we find a close relationship between the combinatorics of the classical Robinson-Schensted-Knuth correspondence and our bijection. We als…
Robinson-Schensted-Knuth algorithm, jeu de taquin and Kerov-Vershik measures on infinite tableaux
本文研究了无限情形下的Robinson-Schensted-Knuth算法（RSK）和Schützenberger的jeu de taquin算法。我们证明了RSK中的记录表定义了以下两个动力系统的同构：（i）一个i.i.d.随机字母配备Bernoulli移位，和（ii）一个随机的无限Young tableau（与Vershik-Kerov措施，对应于一些托马斯字符的无限对称群的分布）配备jeu de taquin变换。作为特例，我们恢复了非碰撞随机游动和多维Pitman变换的结果.
A path-transformation for random walks and the Robinson-Schensted correspondence
在[O 'Connell and Yor（2002）]中引入了一个路径变换G，它具有这样的性质：对于整数格上属于某类随机游动的X，变换后的游动G（X）与原游动G（X）具有相同的规律，条件是不离开A型Weyl腔。本文证明了G与Robinson-Schensted算法密切相关，并利用这种联系给出了上述表示定理的一个新证明。新的证明对更大类的随机游动是有效的，并产生关于X和G（X）的联合律的额外信息。利用Donsker定理恢复了布朗模型的相应结果。这些都与随机矩阵理论中的厄米布朗运动和高斯酉包络有关。我们在路径变换G和RS算法之间建立的联系也为后者提供了一个新的公式和解释。这可以用来研究RS算法的性质，而且，很容易扩展到连续设置。
The numbers $f_λ$ of standard tableaux of shape $λ\vdash n$ satisfy 2 fundamental recursions: $f_λ= \sum f_{λ^-}$ and $(n + 1)f_λ=\sum f_{λ^+}$, where $λ^-$ and $λ^+$ run over all shapes obtained from $λ$ by adding or removing a square respectively. The first of these recursions is trivial; the second can be proven algebraically from the first. These recursions together imply algebraically the dimension formula $n! =\sum f_λ^2$ for the irreducible representations of $S_n$. We show that a combinatorial analysis of this classical algebraic argument produces an infinite family of algorithms, amon…
The set of orbits of $GL(V)$ in $Fl(V)\times Fl(V)\times V$ is finite, and is parametrized by the set of certain decorated permutations in a work of Solomon. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in t…
In this paper, we introduce the ghost-box-ball system, which is an extended version of the classical soliton cellular automaton. It is initially motivated as a mechanism for making precise a connection between the Schensted insertion (of the Robinson-Schensted-Knuth correspondence) and the dynamical process of the box-ball system. In addition to this motivation, we explore generalisations of classical notions of the box-ball system, including the solitonic phenomenon, the asymptotic sorting property, and the invariant shape construction. We analyse the ghost-box-ball system beyond its initial …
