2 00 4 Branching rules , Kostka - Foulkes polynomials and q - multiplicities in tensor product for the root systems

The Kostka-Foulkes polynomials K λ,μ(q) related to a root system φ can be defined as alternated sums running over the Weyl group associated to φ. By restricting these sums over the elements of the symmetric group when φ is of type Bn, Cn orDn, we obtain again a class K̃ φ λ,μ(q) of Kostka-Foulkes polynomials. When φ is of type Cn or Dn there exists a duality beetween these polynomials and some natural q-multiplicities uλ,μ(q) and Uλ,μ(q) in tensor product [14]. In this paper we first establish identities for the K̃ λ,μ(q) which implies in particular that they can be decomposed as sums of Kostka-Foulkes polynomials K An−1 λ,μ (q) with nonnegative integer coefficients. Moreover these coefficients are branching rule coefficients. This allows us to clarify the connection beetween the q-multiplicities uλ,μ(q), Uλ,μ(q) and the polynomials K ♦ λ,μ(q) defined in [25]. Finally we show that uλ,μ(q) and Uλ,μ(q) coincide up to a power of q with the one dimension sum introduced in [4] when all the parts of μ are equal to 1 which partially proves some conjectures of [14] and [25].