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

Branching Rules, Kostka–Foulkes Polynomials and q-multiplicities in Tensor Product for the Root Systems B n , C n and D n