A q-rational Murnaghan-Nakayama rule

The Murnaghan-Nakayama rule is a combinatorial rule for computing symmetric group characters. It has recently been extended to compute Iwahori-Hecke algebra characters and Brauer algebra characters. It is proved using the fact that the symmetric group Lf and the general linear group GL(r, C) centralize each other on the tensor space ⊗fV of f copies of the natural representation V of GL(r, C). This tensor space contains the polynomial representations of GL(r, C). The mixed tensor space Tm, n = (⊗m V) ⊗ (⊗nV∗, where V∗ is the dual to V, contains the rational representations of GL(r, C). When r ⩾ m + n, its centralizer is the complex algebra Bm, nr which is a subalgebra of the Brauer algebra Bm + nr. If we let Vq = V ⊗ C(q), then Tqm, n = (⊗m Vq) ⊗ (⊗nVq∗) contains the rational representations of the quantum group Uq(gl(r, C)), and when r ⩾ m + n its centralizer is the two-parameter Iwahori-Hecke algebra Hm, nr(q). We derive a combinatorial rule for computing Bm, nr-characters and Hm, nr(q)-characters, and use it to compute character tables. The Murnaghan-Nakayama rules for Lm and the Iwahori-Hecke algebra Hm(q) are obtained as the special case when n = 0.