Robinson-Schensted Correspondence Research Articles

The role of paths in Robinson–Schensted algorithm (RS), and Kerov, Kirillov and Reshetikhin bijection (KKR)

AbstractIn this work we want to amplify the RS algorithm in the language of paths. Thus, it is a continuation of the previous work [Lulek et al., to appear in Mol. Phys.], where the magnetic interpretation of RS algorithm was given. We also want to present the intermediative role of paths in bijection between standard Young tableaux and rigged configurations. We will treat the paths as a “bridge” which allow us to move between this two objects. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Linear extension of the Robinson–Schensted algorithm

AbstractThe Robinson–Schensted (RS) algorithm demonstrates a bijection between the set of magnetic configurations f and the set of pairs of tableaux: a semistandard Weyl tableau P(f) accompanied by a standard Young tableau Q(f). We show that it is not only a bijection between sets, but it can be extended to a linear unitary transformation within the space of all quantum states of the magnet. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Evacuation and a geometric construction for Fibonacci tableaux

Tableaux have long been used to study combinatorial properties of permutations and multiset permutations. Discovered independently by Robinson and Schensted and generalized by Knuth, the Robinson–Schensted correspondence has provided a fundamental tool for relating permutations to tableaux. In 1963, Schützenberger defined a process called evacuation on standard tableaux which gives a relationship between the pairs of tableaux ( P , Q ) resulting from the Schensted correspondence for a permutation and both the reverse and the complement of that permutation. Viennot gave a geometric construction for the Schensted correspondence and Fomin described a generalization of the correspondence which provides a bijection between permutations and pairs of chains in Young's lattice. In 1975, Stanley defined a Fibonacci lattice and in 1988 he introduced the idea of a differential poset. Roby gave an insertion algorithm, analogous to the Schensted correspondence, for mapping a permutation to a pair of Fibonacci tableaux. The main results of this paper are to give an evacuation algorithm for the Fibonacci tableaux that is analogous to the evacuation algorithm on Young tableaux and to describe a geometric construction for the Fibonacci tableaux that is similar to Viennot's geometric construction for Young tableaux.

A Two-Dimensional Pictorial Presentation of Berele's Insertion Algorithm for Symplectic Tableaux

We give the first two-dimensional pictorial presentation of Berele's correspondence, an analogue of the Robinson-Schensted (R-S) correspondence for the symplectic group $Sp(2n, {\Bbb C})$. From the standpoint of representation theory, the R-S correspondence combinatorially describes the irreducible decomposition of the tensor powers of the natural representation of $GL(n,{\Bbb C})$. Berele's insertion algorithm gives the bijection that describes the irreducible decomposition of the tensor powers of the natural representation of $Sp(2n,{\Bbb C})$. Two-dimensional pictorial presentations of the R-S correspondence via local rules (first given by S. Fomin) and its many variants have proven very useful in understanding their properties and creating new generalizations. We hope our new presentation will be similarly useful.

Permutations with short monotone subsequences

We consider permutations of $1,2,...,n^2$ whose longest monotone subsequence is of length $n$ and are therefore extremal for the Erdős-Szekeres Theorem. Such permutations correspond via the Robinson-Schensted correspondence to pairs of square $n \times n$ Young tableaux. We show that all the bumping sequences are constant and therefore these permutations have a simple description in terms of the pair of square tableaux. We deduce a limit shape result for the plot of values of the typical such permutation, which in particular implies that the first value taken by such a permutation is with high probability $(1+o(1))n^2/2$.

Permutation Sign under the Robinson-Schensted Correspondence

We show how the sign of a permutation can be deduced from the tableaux induced by the permutation under the Robinson-Schensted correspondence. The result yields a simple proof of a conjecture on the squares of imbalances raised recently by Stanley.

Note sur les permutations minimales

The Robinson–Schensted correspondence maps a permutation onto a pair of standard Young tableaux of the same shape. The shape of the two tableaux is referred to as the shape of the permutation. By using the theory of Kazhdan–Luszitg, Hohlweg has recently characterized the permutations with a fixed shape and a minimal inversion number. The present note provides a combinatorial proof of this result by using Viennot’s geometric algorithm of the Robinson–Schensted correspondence.

Left cells in type 𝐵_{𝑛} with unequal parameters

Kazhdan and Lusztig have shown that the partition of the symmetric group S n {\mathfrak {S}}_n into left cells is given by the Robinson–Schensted correspondence. The aim of this paper is to provide a similar description of the left cells in type B n B_n for a special class of choices of unequal parameters. This is based on a generalization of the Robinson–Schensted correspondence in type B n B_n . We provide an explicit description of the left cell representations and show that they are irreducible and constructible.

Posets of Matrices and Permutations with Forbidden Subsequences

The enumeration of permutations with specific forbidden subsequences has applications in areas ranging from algebraic geometry to the study of sorting algorithms. We consider a ranked poset of permutation matrices whose global structure incorporates the solution to the equivalent problem of enumerating permutations which contain a required subsequence. We describe this structure completely for saturated chains of lengths one and two, so settling several new and general instances of the original problem, and conclude with a superficial asymptotic investigation of arbitrary chains whose length is small by comparison with the rank of its constituent permutations. The value of this approach is reflected in the appearance of closed polynomial formulae (related to the Robinson-Schensted correspondence) and of a framework for the systematic analysis of associated combinatorial questions; indeed, we begin by studying a simpler poset of 0-1 sequences as the natural environment in which to introduce our insertion and deletion operators.

A path-transformation for random walks and the Robinson-Schensted correspondence

The author and Marc Yor recently introduced a path-transformationG(k)G^{(k)}with the property that, forXXbelonging to a certain class of random walks onZ+k\mathbb {Z}_+^k, the transformed walkG(k)(X)G^{(k)}(X)has the same law as the original walk conditioned never to exit the Weyl chamber{x: x1≤⋯≤xk}\{x:\ x_1\le \cdots \le x_k\}. In this paper, we show thatG(k)G^{(k)}is closely related to the Robinson-Schensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law ofXXandG(k)(X)G^{(k)}(X). The corresponding results for the Brownian model are recovered by Donsker’s theorem. These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory. The connection we make between the path-transformationG(k)G^{(k)}and the Robinson-Schensted algorithm also provides a new formula and interpretation for the latter. This can be used to study properties of the Robinson-Schensted algorithm and, moreover, extends easily to a continuous setting.

Cellules de Kazhdan–Lusztig et correspondance de Robinson–Schensted

Kazhdan and Lusztig have introduced (left, right and two-sided) cells in an arbitrary Coxeter group. For the symmetric group, they showed that these cells are given by the Robinson–Schensted correspondence. Here, we describe a Robinson–Schensted correspondence for the complex reflection groups G( e,1, n). In a recent joint work with C. Bonnafé, we have shown that, in the case e=2 (where G(2,1, n) is the Coxeter group of type B n ), this correspondence determines the Kazhdan–Lusztig cells with respect to certain unequal parameters. To cite this article: L. Iancu, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

A geometric version of the Robinson–Schensted correspondence for skew oscillating tableaux

We consider an analogue of the Robinson–Schensted correspondence for skew oscillating tableaux and we propose a geometric version of this correspondence, extending similar constructions for standard (Combinatoire et représentation du groupe symétrique, Lecture Notes in Mathematics, Vol. 579, Springer, Berlin, 1977, pp. 29–58) and oscillating tableaux (Formal Power Series and Algebraic Combinatorics, FPSAC’99, Univ. Politecnicà de Catalunya, 1999, pp. 141–152). We deduce from this geometric construction new proofs of some combinatorial properties of this correspondence.

A proof of Pieri’s formula using the generalized Schensted insertion algorithm for rc-graphs

We provide a generalization of the Schensted insertion algorithm for rc-graphs of Bergeron and Billey. The new algorithm is used to give a new proof of Pieri's formula.

On the Hypersurface Orbital Varieties of sl(N, [formula omitted

We study the structure of hypersurface orbital varieties of sl(N,C) (those that are hypersurfaces in the nilradical of some parabolic subalgebra) and how information about this structure is encoded in the standard Young tableau associated to it by the Robinson–Schensted algorithm. We present a conjecture for the exact form of the unique non-linear defining equations of hypersurface orbital varieties and proofs of the conjecture in certain cases.

Brauer Diagrams, Updown Tableaux and Nilpotent Matrices

We interpret geometrically a variant of the Robinson-Schensted correspondence which links Brauer diagrams with updown tableaux, in the spirit of Steinberg's result [32] on the original Robinson-Schensted correspondence. Our result uses the variety of all $$(N,\omega ,V)$$ where $$V$$ is a complete flag in $$\mathbb{C}^{2n} ,\omega $$ is a nondegenerate alternating bilinear form on $$\mathbb{C}^{2n} $$ and N is a nilpotent element of the Lie algebra of the simultaneous stabilizer of both ω and $$V$$ instead of Steinberg's variety of $$(N,V,V')$$ where $$V {\text{and}} V'$$ are two complete flags in $$\mathbb{C}^n $$ and N is a nilpotent element of the Lie algebra of the simultaneous stabilizer of both $$V {\text{and}} V'$$ .

The Skeleton of a Reduced Word and a Correspondence of Edelman and Greene

Stanley conjectured that the number of maximal chains in the weak Bruhat order of $S_n$, or equivalently the number of reduced decompositions of the reverse of the identity permutation $ w_0 = n,n-1,n-2,\ldots,2,1$, equals the number of standard Young tableaux of staircase shape $s=\{n-1,n-2,\ldots,1\}$. Originating from this conjecture remarkable connections between standard Young tableaux and reduced words have been discovered. Stanley proved his conjecture algebraically, later Edelman and Greene found a bijective proof. We provide an extension of the Edelman and Greene bijection to a larger class of words. This extension is similar to the extension of the Robinson-Schensted correspondence to two line arrays. Our proof is inspired by Viennot's planarized proof of the Robinson-Schensted correspondence. As it is the case with the classical correspondence the planarized proofs have their own beauty and simplicity.

Flag Varieties and Interpretations of Young Tableau Algorithms

The conjugacy class of nilpotent n×n matrices can be parameterized by partitions λ of n, and for a nilpotent η in the class parameterized by λ, the variety Fη of η-stable flags has its irreducible components parameterized 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 Fη parameterized by tableaux P and Q is the permutation associated to (P,Q) under the Robinson–Schensted correspondence. Other constructions for which we give interpretations are Schützenberger's involution of the set of Young tableaux, jeu de taquin (leading also to an interpretation of Littlewood–Richardson coefficients), and the transpose Robinson–Schensted correspondence (defined using column insertion). In each case we use a doubly indexed family of partitions, defined in terms of the flag (or pair of flags) determined by a point chosen in the variety under consideration, and we show that for generic choices, the family satisfies combinatorial relations that make it correspond to an instance of the algorithmic operation being interpreted (as described in M. A. A. van Leeuwen, Electron. J. Combin.3, No. 2 (1996), R15).

Descents, Quasi-Symmetric Functions, Robinson-Schensted for Posets, and the Chromatic Symmetric Function

We investigate an apparent hodgepodge of topics: a Robinson-Schensted algorithm for (3 + 1)-free posets, Chung and Graham's G-descent expansion of the chromatic polynomial, a quasi-symmetric expansion of the path-cycle symmetric function, and an expansion of Stanley's chromatic symmetric function X G in terms of a new symmetric function basis. We show how the theory of P-partitions (in particular, Stanley's quasi-symmetric function expansion of the chromatic symmetric function X G ) unifies them all, subsuming two old results and implying two new ones. Perhaps our most interesting result relates to the still-open problem of finding a Robinson-Schensted algorithm for (3 + 1)-free posets. (Magid has announced a solution but it appears to be incorrect.) We show that such an algorithm ought to “respect descents”, and that the best partial algorithm so far—due to Sundquist, Wagner, and West—respects descents if it avoids a certain induced subposet.

The Potts model representation and a Robinson–Schensted correspondence for the partition algebra

We construct a correspondence between the set of partitions of a finite set M and the set of pairs of walks to the same vertex on a graph giving the Bratteli diagram of the partition algebra on M. This is the precise analogue of the correspondence between the set of permutations of a finite set and the set of pairs of Young tableaux of the same shape, called the Robinson–Schensted correspondence.

KRS and Powers of determinantal ideas

The goal of this paper is to determine Gobner bases of powers of determinantal ideals and to show that the Rees algebras of (products of) determinantal ideals are normal and Cohen–Macaulay if the characteristic of the base field is non-exceptional. Our main combinatorial result is a generalization of Schensted's Theorem on the Knuth–Robinson–Schensted correspondence.