We continue work begun in \cite{ptab} which introduced \emph{perforated tableaux} as a combinatorial model for crystals of type $A_{n-1}$, emphasizing connections to the classical Robinson-Schensted-Knuth (RSK) correspondence and Lusztig involutions, and, more generally, exploring the role of insertion schemes in the analysis of crystal graphs. An essential feature of our work is the role of \emph{dual} crystals (\cite{GerberLecouvey,vanLeeuwen}) from which we obtain new results within and beyond the classic RSK theory.
For each fully commutative permutation, we construct a "boolean core," which is the maximal boolean permutation in its principal order ideal under the right weak order. We partition the set of fully commutative permutations into the recently defined crowded and uncrowded elements, distinguished by whether or not their RSK insertion tableaux satisfy a sparsity condition. We show that a fully commutative element is uncrowded exactly when it shares the RSK insertion tableau with its boolean core. We present the dynamics of the right weak order on fully commutative permutations, with particular in…
We present a combinatorial model, called \emph{perforated tableaux}, to study $A_{n-1}$ crystals, unifying several previously studied combinatorial models. We identify nodes in the $k$-fold tensor product of the standard crystal with length $k$ words in $[n]= \{ 1, \ldots n\}$. We model this crystal with perforated tableaux (ptableaux), extending this identification isomorphically to biwords, RSK $(P,Q)$ tableaux pairs, and matrix models. In the ptableaux setting, crystal operators are more simply defined and we can identify highest weights visually without computation. We generalize the tenso…
The RSK correspondence generalises the Robinson-Schensted correspondence by replacing permutation matrices by matrices with entries in ${\bf N}$, and standard Young tableaux by semistandard ones. For $r>0$, the Robinson-Schensted correspondence can be trivially extended, using the $r$-quotient map, to one between coloured permutations and pairs of standard $r$-ribbon tableaux built on a fixed $r$-core (the Stanton-White correspondence). This correspondence can also be generalised to arbitrary matrices with entries in ${\bf N}^r$ and pairs of semistandard $r$-ribbon tableaux built on a fixed $r…
We define a class L_{n, k} of permutations that generalizes alternating (up-down) permutations and give bijective proofs of certain pattern-avoidance results for this class. As a special case of our results, we give two bijections between the set A_{2n}(1234) of alternating permutations of length 2n with no four-term increasing subsequence and standard Young tableaux of shape (3^n), and between the set A_{2n + 1}(1234) and standard Young tableaux of shape (3^{n - 1}, 2, 1). This represents the first enumeration of alternating permutations avoiding a pattern of length four. We also extend previ…
Iterating the skew RSK correspondence discovered by Sagan and Stanley in the late '80s, we define a deterministic dynamics on the space of pairs of skew Young tableaux $(P,Q)$. We find that this skew RSK dynamics displays conservation laws which, in the picture of Viennot's shadow line construction, identify generalizations of Greene invariants. The introduction of a novel realization of $0$-th Kashiwara operators reveals that the skew RSK dynamics possesses symmetries induced by an affine bicrystal structure, which, combined with connectedness properties of Demazure crystals, leads to its lin…
We give combinatorial proofs of two identities from the representation theory of the partition algebra $C A_k(n), n \ge 2k$. The first is $n^k = \sum_λf^λm_k^λ$, where the sum is over partitions $λ$ of $n$, $f^λ$ is the number of standard tableaux of shape $λ$, and $m_k^λ$ is the number of "vacillating tableaux" of shape $λ$ and length $2k$. Our proof uses a combination of Robinson-Schensted-Knuth insertion and jeu de taquin. The second identity is $B(2k) = \sum_λ(m_k^λ)^2$, where $B(2k)$ is the number of set partitions of $\{1, >..., 2k\}$. We show that this insertion restricts to work for th…
We consider two examples of a fully decodable combinatorial encoding of Bernoulli schemes: the encoding via Weyl simplices and the much more complicated encoding via the RSK (Robinson--Schensted--Knuth) correspondence. In the first case, the decodability is a quite simple fact, while in the second case, this is a nontrivial result obtained by D.~Romik and P.~Śniady and based on the papers~ \cite{KV}, \cite{VK}, and others. We comment on the proofs from the viewpoint of the theory of measurable partitions; another proof, using representation theory and generalized Schur--Weyl duality, will be p…
In 1976, King defined certain tableaux model, called King tableaux in this paper, counting weight multiplicities of irreducible representation of the symplectic group $Sp(2m)$ for a given dominant weight. Since Kashiwara defined crystals, it is an open problem to provide a crystal structure on King tableaux. In this paper, we present crystal structures on King tableaux and semistandard oscillating tableaux. The semistandard oscillating tableaux naturally appear as $Q$-tableaux in the symplectic version of RSK algorithms. As an application, we discuss Littlewood-Richardson coefficients for $Sp(…
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 …
The crystal base of the modified quantized enveloping algebras of type $A_{+\infty}$ or $A_\infty$ is realized as a set of integral bimatrices. It is obtained by describing the decomposition of the tensor product of a highest weight crystal and a lowest weight crystal into extremal weight crystals, and taking its limit using a tableaux model of extremal weight crystals. This realization induces in a purely combinatorial way a bicrystal structure of the crystal base of the modified quantized enveloping algebras and hence its Peter-Weyl type decomposition generalizing the classical RSK correspon…
We define a set of operations called crystal operations on matrices with entries either in {0,1} or in N. There are horizontal and vertical crystal operations, giving rise to two commuting structures of a crystal graph on these matrices. They provide a new perspective on many aspects of the RSK correspondence and its dual, and related constructions. Under a straightforward encoding of semistandard tableaux by matrices, the operations correspond to crystal operations on tableaux, respectively to individual moves occurring during a jeu de taquin slide. We show that the (dual) RSK correspondence …
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 …
