Robinson-Schensted-Knuth Correspondence Research Articles

Mahonian STAT on rearrangement class of words

In 2000, Babson and Steingrímsson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed as sums of vincular pattern occurrence statistics. STAT is one of such Mahonian statistics discovered by them. In 2016, Kitaev and the third author introduced a words analogue of STAT and proved a joint equidistribution result involving two sextuple statistics on the whole set of words with fixed length and alphabet. Moreover, their computer experiments hinted at a finer involution on R(w), the rearrangement class of a given word w. We construct such an involution in this paper, which yields a comparable joint equidistribution between two sextuple statistics over R(w). Our involution builds on Burstein’s involution and Foata–Schützenberger’s involution that utilizes the celebrated RSK algorithm.

R-systems

Birational toggling on Gelfand–Tsetlin patterns appeared first in the study of geometric crystals and geometric Robinson–Schensted–Knuth correspondence. Based on these birational toggle relations, Einstein and Propp introduced a discrete dynamical system called birational rowmotion associated with a partially ordered set. We generalize birational rowmotion to the class of arbitrary strongly connected directed graphs, calling the resulting discrete dynamical system the R-system. We study its integrability from the points of view of singularity confinement and algebraic entropy. We show that in many cases, singularity confinement in an R-system reduces to the Laurent phenomenon either in a cluster algebra, or in a Laurent phenomenon algebra, or beyond both of those generalities, giving rise to many new sequences with the Laurent property possessing rich groups of symmetries. Some special cases of R-systems reduce to Somos and Gale-Robinson sequences.

A new link between the descent algebra of type B, domino tableaux and Chow’s quasisymmetric functions

Introduced by Solomon in his 1976 paper, the descent algebra of a finite Coxeter group received significant attention over the past decades. As proved by Gessel, in the case of the symmetric group its structure constants give the comultiplication table for the fundamental basis of quasisymmetric functions. We show that this latter property actually implies several well known relations linked to the Robinson–Schensted–Knuth correspondence and some of its generalisations. This provides a new link between these results and the theory of quasisymmetric functions and allows to derive more advanced formulae involving Kronecker coefficients. Using the theory of type B quasisymmetric functions introduced by Chow, we extend this connection to the hyperoctahedral group and derive new formulae relating the structure constants of the descent algebra of type B, the numbers of domino tableaux of given descent set and the Kronecker coefficients of the hyperoctahedral group.

Hall–Littlewood RSK field

We introduce a randomized Hall–Littlewood RSK algorithm and study its combinatorial and probabilistic properties. On the probabilistic side, a new model—the Hall–Littlewood RSK field—is introduced. Its various degenerations contain known objects (the stochastic six vertex model, the asymmetric simple exclusion process) as well as a variety of new ones. We provide formulas for a rich class of observables of these models, extending existing results about Macdonald processes. On the combinatorial side, we establish analogs of properties of the classical RSK algorithm: invertibility, symmetry, and a “bijectivization” of the skew-Cauchy identity.

Crystals and trees: Quasi-Kashiwara operators, monoids of binary trees, and Robinson–Schensted-type correspondences

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 gln, 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 hypoplactic monoid (the monoid of quasi-ribbon tableaux), which has similarly neat combinatorial properties. This paper constructs an analogous ‘crystal-type’ structure for the sylvester and Baxter monoids (the monoids of binary search trees and pairs of twin binary search trees, respectively). Both monoids are shown to arise from this structure just as the plactic monoid does from the usual crystal graph. The interaction of the structure with the sylvester and Baxter versions of the Robinson–Schensted–Knuth correspondence is studied. The structure is then applied to prove results on the number of factorizations of elements of these monoids, and to prove that both monoids satisfy non-trivial identities.

Variants of the RSK algorithm adapted to combinatorial Macdonald polynomials

We introduce variations of the Robinson–Schensted correspondence parametrized by positive integers p. Each variation gives a bijection between permutations and pairs of standard tableaux of the same shape. In addition to sharing many of the properties of the classical Schensted algorithm, the new algorithms are designed to be compatible with certain permutation statistics introduced by Haglund in the study of Macdonald polynomials. In particular, these algorithms provide an elementary bijective proof converting Haglund's combinatorial formula for Macdonald polynomials to an explicit combinatorial Schur expansion of Macdonald polynomials indexed by partitions μ satisfying μ1≤3 and μ2≤2. We challenge the research community to extend this RSK-based approach to more general classes of partitions.

Variants of Geometric RSK, Geometric PNG, and the Multipoint Distribution of the Log-Gamma Polymer

We show that the reformulation of the geometric Robinson–Schensted–Knuth correspondence via local moves, introduced in [29], can be extended to cases where the input matrix is replaced by more general polygonal, Young-diagram-like, arrays of the form ⁠. We also show that a rearrangement of the sequence of the local moves gives rise to a geometric version of the polynuclear growth model. These reformulations are used to obtain integral formulae for the Laplace transform of the joint distribution of the point-to-point partition functions of the log-gamma polymer at different space–time points. In the case of two points at equal time |$N$| and space at distance of order |$N^{2/3}$|⁠, we show formally that the joint law of the partition functions, scaled by |$N^{1/3}$|⁠, converges to the two-point function of the Airy process.

Representation theory: a combinatorial viewpoint

This book discusses the representation theory of symmetric groups, the theory of symmetric functions and the polynomial representation theory of general linear groups. The first chapter provides a detailed account of necessary representation-theoretic background. An important highlight of this book is an innovative treatment of the Robinson–Schensted–Knuth correspondence and its dual by extending Viennot's geometric ideas. Another unique feature is an exposition of the relationship between these correspondences, the representation theory of symmetric groups and alternating groups and the theory of symmetric functions. Schur algebras are introduced very naturally as algebras of distributions on general linear groups. The treatment of Schur–Weyl duality reveals the directness and simplicity of Schur's original treatment of the subject. In addition, each exercise is assigned a difficulty level to test readers' learning. Solutions and hints to most of the exercises are provided at the end.

An analogue of the Robinson–Schensted–Knuth correspondence and non-symmetric Cauchy kernels for truncated staircases

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 that the pair of their right keys is such that one key is bounded by the Schützenberger evacuation of the other. This bijection is then used to obtain an expansion formula of the non-symmetric Cauchy kernel, over staircases or truncated staircases, in the basis of Demazure characters of type A, and the basis of Demazure atoms. The expansion implies Lascoux expansion formula, when specialized to staircases or truncated staircases, and make explicit, in the latter, the Young tableaux in the Demazure crystal by interpreting Demazure operators via elementary bubble sorting operators acting on weak compositions.

An extension of Brualdi’s algorithm for the construction of [formula omitted]-matrices with prescribed row and column sum vectors

Given partitions R and S with the same weight and S⪯R∗, the Robinson–Schensted–Knuth correspondence establishes a bijection between the class A(R,S) of (0,1)-matrices with row-sum R and column-sum S, and pairs of Young tableaux with conjugate shape λ and λ∗, with S⪯λ⪯R∗. We give a canonical construction for matrices in A(R,S) whose insertion tableau has a prescribed shape λ, with S⪯λ⪯R∗. This algorithm generalizes some recent constructions due to R. Brualdi for the extremal cases λ=S and λ=R∗ (using a Ryser-like algorithm), and due to C.M. da Fonseca and R. Mamede for particular cases of λ.

Properties of the nonsymmetric Robinson–Schensted–Knuth algorithm

We introduce a generalization of the Robinson–Schensted–Knuth insertion algorithm for semi-standard augmented fillings whose basement is an arbitrary permutation σ∈S n . If σ is the identity, then our insertion algorithm reduces to the insertion algorithm introduced by the second author (Sémin. Lothar. Comb. 57:B57e, 2006) for semi-standard augmented fillings and if σ is the reverse of the identity, then our insertion algorithm reduces to the original Robinson–Schensted–Knuth row insertion algorithm. We use our generalized insertion algorithm to obtain new decompositions of the Schur functions into nonsymmetric elements called generalized Demazure atoms (which become Demazure atoms when σ is the identity). Other applications include Pieri rules for multiplying a generalized Demazure atom by a complete homogeneous symmetric function or an elementary symmetric function, a generalization of Knuth’s correspondence between matrices of non-negative integers and pairs of tableaux, and a version of evacuation for composition tableaux whose basement is an arbitrary permutation σ.

RSK correspondence and classically irreducible Kirillov–Reshetikhin crystals

We give a new combinatorial model of the Kirillov–Reshetikhin crystals of type An(1) in terms of non-negative integral matrices based on the classical RSK algorithm, which has a simple description of the affine crystal structure without using the promotion operator. We have a similar description of the Kirillov–Reshetikhin crystals associated to exceptional nodes in the Dynkin diagrams of classical affine or non-exceptional affine type, which are called classically irreducible together with those of type An(1).

Classic and Mirabolic Robinson–Schensted–Knuth Correspondence for Partial Flags

Abstract In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line.

Bijective Proof of a Symplectic Dual Pair Identity

We provide a combinatorial proof of a symplectic character identity relating the sum of a product of symplectic Schur functions to the product . This formula owes its origin to the existence of a dual pair of symplectic groups acting on spinors, as pointed out by Hasegawa. The first combinatorial proof, based on symplectic tableaux and a variation of the Robinson–Schensted–Knuth correspondence, was due to Terada. Here we use Schützenberger’s jeu de taquin, augmented by two simple zero weight transformations. The identity itself generalizes a well-known identity expressing as a sum of products of Schur functions that was due to Littlewood and proved combinatorially by Remmel. We offer an alternative combinatorial proof of this identity by means of the jeu de taquin, as a precursor to the proof of the symplectic identity.

Affine insertion and Pieri rules for the affine Grassmannian

We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n,C). Our main results are: 1) Pieri rules for the Schubert bases of H^*(Gr) and H_*(Gr), which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. 2) A new combinatorial definition for k-Schur functions, which represent the Schubert basis of H_*(Gr). 3) A combinatorial interpretation of the pairing between homology and cohomology of the affine Grassmannian. These results are obtained by interpreting the Schubert bases of Gr combinatorially as generating functions of objects we call strong and weak tableaux, which are respectively defined using the strong and weak orders on the affine symmetric group. We define a bijection called affine insertion, generalizing the Robinson-Schensted Knuth correspondence, which sends certain biwords to pairs of tableaux of the same shape, one strong and one weak. Affine insertion offers a duality between the weak and strong orders which does not seem to have been noticed previously. Our cohomology Pieri rule conjecturally extends to the affine flag manifold, and we give a series of related combinatorial conjectures.

Interlaced processes on the circle

When two Markov operators commute, it suggests that we can couple two copies of one of the corresponding processes. We explicitly construct a number of couplings of this type for a commuting family of Markov processes on the set of conjugacy classes of the unitary group, using a dynamical rule inspired by the RSK algorithm. Our motivation for doing this is to develop a parallel programme, on the circle, to some recently discovered connections in random matrix theory between reflected and conditioned systems of particles on the line. One of the Markov chains we consider gives rise to a family of Gibbs measures on bead configurations on the infinite cylinder. We show that these measures have determinantal structure and compute the corresponding space-time correlation kernel.

Demazure crystals of generalized Verma modules and a flagged RSK correspondence

We prove that the Robinson–Schensted–Knuth correspondence is a gl ∞ -crystal isomorphism between two realizations of the crystal graph of a generalized Verma module with respect to a maximal parabolic subalgebra of gl ∞ . A flagged version of the RSK correspondence is derived in a natural way by computing a Demazure crystal graph of a generalized Verma module. As an application, we discuss a relation between a Demazure crystal and plane partitions with a bounded condition.

Rational semistandard tableaux and character formula for the Lie superalgebra [formula omitted

A new combinatorial interpretation of the Howe dual pair ( gl ˆ ∞ | ∞ , gl n ) acting on an infinite-dimensional Fock space is presented. The character of a quasi-finite irreducible highest weight representation of gl ˆ ∞ | ∞ occurring in the Fock space is realized in terms of certain bitableaux of skew shapes. We study a general combinatorics of these bitableaux, including the Robinson–Schensted–Knuth correspondence and the Littlewood–Richardson rule, and then its dual relation with rational semistandard tableaux for gl n . This result also explains other Howe dual pairs ( g , gl n ) , where g is a Lie superalgebra.

Signed differential posets and sign-imbalance

We study signed differential posets, a signed version of differential posets. These posets satisfy enumerative identities which are signed analogues of those satisfied by differential posets. Our main motivations are the sign-imbalance identities for partition shapes originally conjectured by Stanley, now proven in [T. Lam, Growth diagrams, domino insertion and sign-imbalance, J. Combin. Theory Ser. A 107 (2004) 87–115; A. Reifergerste, Permutation sign under the Robinson–Schensted–Knuth correspondence, Ann. Comb. 8 (2004) 103–112; J. Sjöstrand, On the sign-imbalance of partition shapes, J. Combin. Theory Ser. A 111 (2005) 190–203]. We show that these identities result from a signed differential poset structure on Young's lattice, and explain similar identities for Fibonacci shapes.

Permutations Without Long Decreasing Subsequences and Random Matrices

We study the shape of the Young diagram $\lambda$ associated via the Robinson–Schensted–Knuth algorithm to a random permutation in $S_n$ such that the length of the longest decreasing subsequence is not bigger than a fixed number $d$; in other words we study the restriction of the Plancherel measure to Young diagrams with at most $d$ rows. We prove that in the limit $n\to\infty$ the rows of $\lambda$ behave like the eigenvalues of a certain random matrix (namely the traceless Gaussian Unitary Ensemble random matrix) with $d$ rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asymptotically like the largest eigenvalue of the corresponding random matrix.