A q-rational Murnaghan-Nakayama rule

Abstract

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 L f and the general linear group GL( r, C ) centralize each other on the tensor space ⊗ f V 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 T m, n = (⊗ m V) ⊗ (⊗ n V∗, 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 B m, n r which is a subalgebra of the Brauer algebra B m + n r . If we let V q = V ⊗ C ( q), then T q m, n = (⊗ m V q ) ⊗ (⊗ n V q ∗ ) contains the rational representations of the quantum group U q ( gl( r, C )), and when r ⩾ m + n its centralizer is the two-parameter Iwahori-Hecke algebra H m, n r ( q). We derive a combinatorial rule for computing B m, n r -characters and H m, n r ( q)-characters, and use it to compute character tables. The Murnaghan-Nakayama rules for L m and the Iwahori-Hecke algebra H m ( q) are obtained as the special case when n = 0.