We use Kashiwara's theory of crystal bases to study the plactic monoid for U q sp 2n . Then we describe the corresponding insertion and sliding algorithms. The sliding algorithm is essentially the symplectic Jeu de Taquin defined by Sheats and our construction gives the proof of its compatibility with plactic relations.  2002Elsevier Science

We use Kashiwara crystal basis theory to associate a random walk W to each irreducible representation V of a simple Lie algebra. This is achieved by endowing the crystal attached to V with a (possibly non uniform) probability distribution compatible with its weight graduation. We then prove that the generalized Pitmann transform defined in [1] for similar random walks with uniform distributions yields yet a Markov chain. When the representation is minuscule, and the associated random walk has a drift in the Weyl chamber, we establish that this Markov chain has the same law as W conditionned to never exit the cone of dominant weights. For the defining representation V of gl n , we notably recover the main result of [19]. At the heart of our proof is a quotient version of a renewal theorem that we state in the context of general random walks in a lattice. This theorem also have applications in representation theory since it permits to precise the behavior of some outer multiplicities for large dominant weights.

In this paper, we formulate and prove linear analogues of results concerning matchings in groups. A matching in a group G is a bijection ϕ between two finite subsets A, B of G with the property, motivated by old questions on symmetric tensors, that aϕ(a) / ∈ A for all a ∈ A. Necessary and sufficient conditions on G, ensuring the existence of matchings under appropriate hypotheses, are known. Here we consider a similar question in a linear setting.Given a skew field extension K ⊂ L, where K commutative and central in L, we introduce analogous notions of matchings between finite-dimensional K -subspaces A, B of L, and obtain existence criteria similar to those in the group setting. Our tools mix additive number theory, combinatorics and algebra.

No abstract

No abstract

Abstract. Random paths are time continuous interpolations of random walks. By using Littelmann path model, we associate to each irreducible highest weight module of a Kac Moody algebra g a random path W. Under suitable hypotheses, we make explicit the probability of the event E: "W never exits the Weyl chamber of g". We then give the law of the random walk defined by W conditioned by the event E and prove this law can be recovered by applying to W a path transform of Pitman type. This generalizes the main results of [15] and [10] to Kac Moody root systems and arbitrary highest weight modules. Our approach here is new and more algebraic that in [15] and [10]. We indeed fully exploit the symmetry of our construction under the action of the Weyl group of g which permits to avoid delicate generalizations of the results of [10] on renewal theory.

This paper is concerned with one-dimensional sums in classical affine types. We prove a conjecture of Shimozono and Zabrocki (J Algebra 299:33-61, 2006) by showing they all decompose in terms of one-dimensional sums related to affine type A provided the rank of the root system considered is sufficiently large. As a consequence, any onedimensional sum associated to a classical affine root system with sufficiently large rank can be regarded as a parabolic Lusztig q-analogue.

For the affine Hecke algebra of type A at roots of unity, we make explicit the correspondence between geometrically constructed simple modules and combinatorially constructed simple modules and prove the modular branching rule. The latter generalizes work by Vazirani.