Branching rules, Kostka-Foulkes polynomials and qq-multiplicities in tensor product for the root systems B_n,C_nB\_{n},C\_{n} and D_nD\_{n}

The Kostka-Foulkes polynomials $K$ related to a root system $\phi$ can be defined as alternated sums running over the Weyl group associated to $\phi .$ By restricting these sums over the elements of the symmetric group when $$ is of type $B,C$ or $D$, we obtain again a class $\widetilde{K}$ of Kostka-Foulkes polynomials. When $\phi$ is of type $C$ or $D$ there exists a duality beetween these polynomials and some natural $q$-multiplicities $U$ in tensor product \cite{lec}. In this paper we first establish identities for the $\widetilde{K}$ which implies in particular that they can be decomposed as sums of Kostka-Foulkes polynomials related to the root system of type $A$ with nonnegative integer coefficients. Moreover these coefficients are branching rule coefficients. This allows us to clarify the connection beetween the $q$-multiplicities $U$ and the polynomials defined by Shimozono and Zabrocki in \cite{SZ}. Finally we establish that the $q$-multiplicities $U$ defined for the tensor powers of the vector representation coincide up to a power of $q$ with the one dimension sum $X$ introduced in \cite{Ok} This shows that in this case the one dimension sums $$ are affine Kazhdan-Lusztig polynomials

Additive combinatorics methods in associative algebras

We adapt methods coming from additive combinatorics in groups to the study of linear span in associative unital algebras. In particular, we establish for these algebras analogues of Diderrich-Kneser's and Hamidoune's theorems on sumsets and Tao's theorem on sets of small doubling. In passing we classify the finite-dimensional algebras over infinite fields with finitely many subalgebras. These algebras play a crucial role in our linear version of Diderrich-Kneser's theorem. We also explain how the original theorems for groups we linearize can be easily deduced from our results applied to group algebras. Finally, we give lower bounds for the Minkowski product of two subsets in finite monoids by using their associated monoid algebras.Comment: In this second version, we clarify and extend the domain of validity of Diderrich-Kneser's theorem for associative algebras. We simplify the proofs and we also add a section on Kneser's and Hamidoune's theorem in monoi

On the Mullineux involution for Ariki-Koike algebras

This note is concerned with a natural generalization of the Mullineux involution for Ariki-Koike algebras. Using a result of Fayers together with previous results by the authors, we give an efficient algorithm for computing this generalized Mullineux involution. Our algorithm notably does not involve the determination of paths in affine crystals.Comment: 17 page

Random walks in Weyl chambers and crystals

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 by Biane, Bougerol and O'Connell 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. 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.Comment: The second version presents minor modifications to the previous on

Springer basic sets and modular Springer correspondence for classical types

We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicitly the modular Springer correspondence for classical types (for representations in odd characteristic). In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence.Comment: 31 page

Pl\"unnecke and Kneser type theorems for dimension estimates

Given a division ring K containing the field k in its center and A,B two finite subsets of K\{0}, we give some analogues of Pl\"unnecke and Kneser theorems for the dimension of the k-linear span of the Minkowski product AB in terms of the dimensions of the k-linear spans of A and B. These Pl\"unnecke type estimates are then generalized to the case of associative algebras. We also obtain an analogue in the context of division rings of a theorem by Tao classifying the sets of small doubling in a group.Comment: 21 page