Sliding Presentation of the Jeux de Taquin for Classical Lie Groups

The simple \(GL(n,\mathbb {C})\)-modules are described by using semistandard Young tableaux. Any semistandard skew tableau can be transformed into a well defined semistandard tableau by a combinatorial operation, the Schützenberger jeu de taquin. Associated to the classical Lie groups \(SP(2n,\mathbb {C})\), \(SO(2n+1,\mathbb {C})\), there are other notions of semistandard Young tableaux and jeux de taquin. In this paper, we study these various jeux de taquin, proving that each of them has a simple and explicit formulation as a step-by-step sliding. Any of these jeux de taquin is the restriction of the orthogonal one, associated to \(SO(2n+1,\mathbb {C})\).     

Tableau stabilization and rectangular tableaux fixed by promotion powers

Journal of Algebraic Combinatorics - We introduce tableau stabilization, a new phenomenon and statistic on Young tableaux based on jeu de taquin. We investigate bounds for tableau stabilization,...     

A Symmetry Theorem on a Modified Jeu De Taquin

For their bijective proof of the hook-length formula for the number of standard tableaux of a fixed shape Novelli et al. define a modified jeu de taquin which transforms an arbitrary filling of the Ferrers diagram with 1, 2, , n(tabloid) into a standard tableau. Their definition relies on a total order of the cells in the Ferrers diagram induced by a special standard tableau, however, this definition also makes sense for the total order induced by any other standard tableau. Given two standard tableaux P, Q of the same shape we show that the number of tabloids which result in P if we perform the modified jeu de taquin with respect to the total order induced by Q is equal to the number of tabloids which result in Q if we perform the modified jeu de taquin with respect to P. This symmetry theorem extends to skew shapes and shifted skew shapes.     

Lusztig Data of Kashiwara-Nakashima Tableaux in Type D

We describe the embedding from the crystal of Kashiwara-Nakashima tableaux in type D of an arbitrary shape into that of i-Lusztig data associated to a family of reduced expressions i which are compatible with the maximal Levi subalgebra of type A. The embedding is described explicitly in terms of well-known combinatorics of type A including the Schützenberger’s jeu de taquin and an analog of RSK algorithm.

     

Shifted tableaux,schur Q-functions,and a conjecture of R. Stanley

We present an analog of the Robinson-Schensted correspondence that applies to shifted Young tableaux and is considerably simpler than the one proposed in [B. E. Sagan, J. Combin. Theory Ser. A 27 (1979), 10–18]. In addition, this algorithm enjoys many of the important properties of the original Robinson-Schensted map including an interpretation of row lengths in terms of k-increasing sequences, a jeu de taquin, and a generalization to tableaux with repeated entries analogous to Knuth's construction (Pacific J. Math. 34 (1970), 709–727). The fact that the Knuth relations hold for our algorithm yields a simple proof of a conjecture of Stanley.     

Primitive elements of the Hopf algebras of tableaux

The character theory of symmetric groups, and the theory of symmetric functions, both make use of the combinatorics of Young tableaux, such as the Robinson–Schensted algorithm, Schützenberger’s “jeu de taquin”, and evacuation. In 1995 Poirier and the second author introduced some algebraic structures, different from the plactic monoid, which induce some products and coproducts of tableaux, with homomorphisms. Their starting point are the two dual Hopf algebras of permutations, introduced by the authors in 1995. In 2006 Aguiar and Sottile studied in more detail the Hopf algebra of permutations: among other things, they introduce a new basis, by Möbius inversion in the poset of weak order, that allows them to describe the primitive elements of the Hopf algebra of permutations. In the present Note, by a similar method, we determine the primitive elements of the Poirier–Reutenauer algebra of tableaux, using a partial order on tableaux defined by Taskin.     

Increasing Tableaux,Narayana Numbers and an Instance of the Cyclic Sieving Phenomenon

We give a counting formula for the set of rectangular increasing tableaux in terms of generalized Narayana numbers. We define small m–Schröder paths and give a bijection between the set of increasing rectangular tableaux and small m–Schröder paths, generalizing a result of Pechenik [4]. Using K–jeu de taquin promotion, we give a cyclic sieving phenomenon for the set of increasing hook tableaux.     

An explicit bijection between semistandard tableaux and non-elliptic sl 3 webs

The sl ₃ spider is a diagrammatic category used to study the representation theory of the quantum group U _q (sl ₃). The morphisms in this category are generated by a basis of non-elliptic webs. Khovanov–Kuperberg observe that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection via a recursive growth algorithm. Recently, Tymoczko gave a simple version of this bijection in the case that the tableaux are standard and used it to study rotation and join of webs. This article builds on Tymoczko’s bijection to give a simple and explicit algorithm for constructing all non-elliptic sl ₃ webs. As an application, we generalize results of Petersen–Pylyavskyy–Rhoades and Tymoczko proving that, for all non-elliptic sl ₃ webs, rotation corresponds to jeu de taquin promotion and join corresponds to shuffling.     

Jeu de taquin and a monodromy problem for Wronskians of polynomials

The Wronskian associates to d linearly independent polynomials of degree at most n, a non-zero polynomial of degree at most d(n−d). This can be viewed as giving a flat, finite morphism from the Grassmannian Gr(d,n) to projective space of the same dimension. In this paper, we study the monodromy groupoid of this map. When the roots of the Wronskian are real, we show that the monodromy is combinatorially encoded by Schützenberger's jeu de taquin; hence we obtain new geometric interpretations and proofs of a number of results from jeu de taquin theory, including the Littlewood-Richardson rule.     

Puzzles,tableaux, and mosaics

We define mosaics, which are naturally in bijection with Knutson-Tao puzzles. We define an operation on mosaics, which shows they are also in bijection with Littlewood-Richardson skew-tableaux. Another consequence of this construction is that we obtain bijective proofs of commutativity and associativity for the ring structures defined either of these objects. In particular, we obtain a new, easy proof of the Littlewood-Richardson rule. Finally we discuss how our operation is related to other known constructions, particularly jeu de taquin.     

Schur positivity of skew Schur function differences and applications to ribbons and Schubert classes

Some new relations on skew Schur function differences are established both combinatorially using Schützenberger’s jeu de taquin, and algebraically using Jacobi-Trudi determinants. These relations lead to the conclusion that certain differences of skew Schur functions are Schur positive. Applying these results to a basis of symmetric functions involving ribbon Schur functions confirms the validity of a Schur positivity conjecture due to McNamara. A further application reveals that certain differences of products of Schubert classes are Schubert positive. For Manfred Schocker 1970–2006. S.J. van Willigenburg was supported in part by the National Sciences and Engineering Research Council of Canada.     

Monodromy and K-theory of Schubert curves via generalized jeu de taquin

We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves \(S(\lambda _\bullet )\), which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In Levinson (One-dimensional Schubert problems with respect to osculating flags, 2016, doi: 10.4153/CJM-2015-061-1), it was shown that the real geometry of these curves is described by the orbits of a map \(\omega \) on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of \({\mathbb {RP}}^1\), with \(\omega \) as the monodromy operator. We provide a fast, local algorithm for computing \(\omega \) without rectifying the skew tableau and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong’s genomic tableaux (Pechenik and Yong in Genomic tableaux, 2016. arXiv:1603.08490), which enumerate the K-theoretic Littlewood–Richardson coefficient associated to the Schubert curve. We then give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of \(S(\lambda _\bullet )\).     

Growth diagrams for the Schubert multiplication

We present a partial generalization of the classical Littlewood-Richardson rule (in its version based on Schützenberger's jeu de taquin) to Schubert calculus on flag varieties. More precisely, we describe certain structure constants expressing the product of a Schubert and a Schur polynomial. We use a generalization of Fomin's growth diagrams (for chains in Young's lattice of partitions) to chains of permutations in the so-called k-Bruhat order. Our work is based on the recent thesis of Beligan, in which he generalizes the classical plactic structure on words to chains in certain intervals in k-Bruhat order. Potential applications of our work include the generalization of the S₃-symmetric Littlewood-Richardson rule due to Thomas and Yong, which is based on Fomin's growth diagrams.     

Genomic tableaux

We explain how genomic tableaux [Pechenik–Yong ’15] are a semistandard complement to increasing tableaux [Thomas–Yong ’09]. From this perspective, one inherits genomic versions of jeu de taquin, Knuth equivalence, infusion and Bender–Knuth involutions, as well as Schur functions from (shifted) semistandard Young tableaux theory. These are applied to obtain new Littlewood–Richardson rules for K-theory Schubert calculus of Grassmannians (after [Buch ’02]) and maximal orthogonal Grassmannians (after [Clifford–Thomas–Yong ’14], [Buch–Ravikumar ’12]). For the unsolved case of Lagrangian Grassmannians, sharp upper and lower bounds using genomic tableaux are conjectured.     

U-Turn Alternating Sign Matrices,Symplectic Shifted Tableaux and their Weighted Enumeration

Alternating sign matrices with a U-turn boundary (UASMs) are a recent generalization of ordinary alternating sign matrices. Here we show that variations of these matrices are in bijective correspondence with certain symplectic shifted tableaux that were recently introduced in the context of a symplectic version of Tokuyama’s deformation of Weyl’s denominator formula. This bijection yields a formula for the weighted enumeration of UASMs. In this connection use is made of the link between UASMs and certain square ice configuration matrices.     

A note on the straightening law for the symplectic group

A Berele has shown that the symplectic tableaux of R. C. King and N. G. I. El-Sharkaway form a basis of the Schur modules for the symplectic group over a field of characteristic zero. The purpose of this note is to extend the previous result to the characteristic-free case by exhibiting a suitable straightening law.     

A note on the straightening law for the symplectic group

A Berele has shown that the symplectic tableaux of R. C. King and N. G. I. El-Sharkaway form a basis of the Schur modules for the symplectic group over a field of characteristic zero. The purpose of this note is to extend the previous result to the characteristic-free case by exhibiting a suitable straightening law.