The Stanley-Stembridge conjecture associates a symmetric function to each natural unit interval order $\mathcal P$. In this paper, we define relations à la Knuth on the symmetric group for each $\mathcal P$ and conjecture that the associated $\mathcal P$-Knuth equivalence classes are Schur-positive, refining theorems of Gasharov, Brosnan-Chow, and Guay-Paquet. The resulting equivalence graphs fit into the framework of D graphs studied by Assaf. Furthermore, we conjecture that the Schur expansion is given by column-readings of $\mathcal P$-tableaux that occur in the equivalence class. We prove these conjectures for $\mathcal P$ avoiding two specific suborders by introducing $\mathcal P$-analog of Robinson-Schensted insertion, giving an answer to a long standing question of Chow.
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.
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…
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 an algorithm which computes the annihilator in ${\rm U}(\frak{sl}(\infty))$ of any simple highest weight $\frak{sl}(\infty)$-module $L_\frak b(λ)$. This algorithm is based on an infinite version of the Robinson-Schensted algorithm.
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…
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…
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.
In his study of Kazhdan-Lusztig cells in affine type $A$, Shi has introduced an affine analog of Robinson-Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combinatorial realization of Shi's algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Lusztig and Xi, we extend the algorithm to a bijection between extended affine symmetric group and triples $(P, Q, ρ)$ where $P$ and $Q$ are tabloids and $ρ$ is a dominant weight. The weights $ρ$ get …
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 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 …
Taking transposes of Standard Young Tableaux defines a natural involution on the set $I(n)$ of involutions of length $n$ via the the Robinson-Schensted correspondence. In some cases, this involution can be defined without resorting to the Robinson-Schensted correspondence. As a byproduct, we get an interesting generalization of layered permutations.
Kashiwara's crystal graphs have a natural monoid structure that arises by identifying words labelling vertices that appear in the same position of isomorphic components. The celebrated plactic monoid (the monoid of Young tableaux), arises in this way from the crystal graph for the $q$-analogue of the general linear Lie algebra $\mathfrak{gl}_{n}$, and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence. The authors previously constructed an analogous `quasi-crystal' structure for the related hypo…
Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. In the particular case of the crystal graph for the $q$-analogue of the special linear Lie algebra $\mathfrak{sl}_{n}$, this monoid is the celebrated plactic monoid, whose elements can be identified with Young tableaux. The crystal graph and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence and so prov…
We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to $[n,n,n]$: the {\em reduced web} basis associated to Kuperberg's combinatorial description of the spider category; and the {\em left cell basis} for the left cell construction of Kazhdan and Lusztig. In the case of $[n,n]$, the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the images of these bases under classical maps: the {\em Robinson-Schensted algorithm} between perm…
We prove a restriction of an analogue of the Robinson--Schensted--Knuth correspondence for semi-skyline augmented fillings, due to Mason, to multisets of cells of a staircase possibly truncated by a smaller staircase at the upper left end corner, or at the bottom right end corner. The restriction to be imposed on the pairs of semi-skyline augmented fillings is that the pair of shapes, rearrangements of each other, satisfies an inequality in the Bruhat order, w.r.t. the symmetric group, where one shape is bounded by the reverse of the other. For semi-standard Young tableaux the inequality means…
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 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 [Matveev-Petrov 2016](arXiv:1504.00666) a $q$-deformed Robinson-Schensted-Knuth algorithm ($q$RSK) was introduced. In this article we give reformulations of this algorithm in terms of the Noumi-Yamada description, growth diagrams and local moves. We show that the algorithm is symmetric, namely the output tableaux pairs are swapped in a sense of distribution when the input matrix is transposed. We also formulate a $q$-polymer model based on the $q$RSK, prove the corresponding Burke property, which we use to show a strong law of large numbers for the partition function given stationary bounda…
We introduce and study q-randomized Robinson-Schensted-Knuth (RSK) correspondences which interpolate between the classical (q=0) and geometric (q->1) RSK correspondences (the latter ones are sometimes also called tropical). For 0<q<1 our correspondences are randomized, i.e., the result of an insertion is a certain probability distribution on semistandard Young tableaux. Because of this randomness, we use the language of discrete time Markov dynamics on two-dimensional interlacing particle arrays (these arrays are in a natural bijection with semistandard tableaux). Our dynamics act nicely on a …
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 …
