Branching rules, Kostka-Foulkes polynomials and q-multiplicities in tensor product for the root systems B-n, C-n and D-n

摘要

The Kostka-Foulkes polynomials K-lambda,mu(phi)(q) related to a root system phi can be defined as alternating sums running over the Weyl group associated to phi. By restricting these sums over the elements of the symmetric group when phi is of type B-n, C-n or D-n, we obtain again a class (K) over tilde (phi)(lambda,mu)(q) of Kostka - Foulkes polynomials. When phi is of type C-n or D-n there exists a duality between these polynomials and some natural q-multiplicities u(lambda,mu)(q) and U-lambda,U-mu(q) in tensor products [11]. In this paper we first establish identities for the (K) over tilde (phi)(lambda,mu)(q) which implies in particular that they can be decomposed as sums of Kostka - Foulkes polynomials K-lambda,mu(An-1)(q) with nonnegative integer coefficients. Moreover these coefficients are branching coefficients( This allows us to clarify the connection between the q-multiplicities u(lambda,mu)(q), U-lambda,U-mu(q) and the polynomials K-lambda,mu(lozenge)(q) defined by Shimozono and Zabrocki. Finally we show that u(lambda,mu)(q) and U-lambda,U-mu(q) coincide up to a power of q with the one dimension sum introduced by Hatayama and co-workers when all the parts of mu are equal to 1, which partially proves some conjectures of Lecouvey and Shimozono and Zabrocki.