Cédric Lecouvey

Schensted-Type Correspondences and Plactic Monoids for Types B n and D n

We use Kashiwaras theory of crystal bases to study plactic monoids for Uq(so2n+1) and Uq(so2n). Simultaneously we describe a Schensted type correspondence in the crystal graphs of tensor powers of vector and spin representations and we derive a Jeu de Taquin for type B from the Sheats sliding algorithm.

Combinatorics of crystal graphs and Kostka-Foulkes polynomials for the root systems B n , C n and D n

We use Kashiwara-Nakashima combinatorics of crystal graphs associated with the roots systems Bn and Dn to extend the results of Lecouvey [C. Lecouvey, Kostka-Foulkes polynomials, cyclage graphs and charge statistics for the root system Cn, J. Algebraic Combin. (in press)] and Morris [A.-O. Morris, The characters of the group GL(n, q), Math. Z. 81 (1963) 112-123] by showing that Morris-type recurrence formulas also exist for the orthogonal root systems. We derive from these formulas a statistic on Kashiwara-Nakashima tableaux of types Bn, Cn and Dn generalizing the Lascoux-Schutzenberger charge and from which it is possible to compute the Kostka-Foulkes polynomials Kλµ(q) under certain conditions on (λµ), This statistic is different from that obtained in Lecouvey [C. Lecouvey, Kostka-Foulkes polynomials, cyclage graphs and charge statistics for the root system Cn,, J. Algebraic Combin. (in press)] from the cyclage graph structure on tableaux of type Cn. We show that such a structure also exists for the tableaux of types Bn and Dn but cannot be related in a simple way to the Kostka-Foulkes polynomials. Finally we give explicit formulas for Kλµ(q) when λ≤ 3,orn = 2 and µ=0.

A duality between q -multiplicities in tensor products and q -multiplicities of weights for the root systems B, C or D

Starting from Jacobi-Trudi type determinantal expressions for the Schur functions of types B, C and D, we define a natural q-analogue of the multiplicity [V(λ) : M(µ)] when M(µ) is a tensor product of row or column shaped modules defined by µ. We prove that these q-multiplicities are equal to certain Kostka-Foulkes polynomials related to the root systems C or D. Finally we express the corresponding multiplicities in terms of Kostka numbers.

An Algorithm for Computing the Global Basis of a Finite Dimensional Irreducible U q ( so 2 n + 1 ) or U q ( so 2 n )-Module

Abstract We describe a simple algorithm for computing the canonical basis of any irreducible finite-dimensional U q (s o 2n+1) or U q (s o 2n )-module.

Stabilized plethysms for the classical Lie groups

The plethysms of the Weyl characters associated to a classical Lie group by the symmetric functions stabilize in large rank. In the case of a power sum plethysm, we prove that the coefficients of the decomposition of this stabilized form on the basis of Weyl characters are branching coefficients which can be determined by a simple algorithm. This generalizes in particular some classical results by Littlewood on the power sum plethysms of Schur functions. We also establish explicit formulas for the outer multiplicities appearing in the decomposition of the tensor square of any irreducible finite-dimensional module into its symmetric and antisymmetric parts. These multiplicities can notably be expressed in terms of the Littlewood-Richardson coefficients.