On Schur's Q -functions and the Primitive Idempotents of a Commutative Hecke Algebra

Abstract

Let B n denote the centralizer of a fixed-point free involution in the symmetric group S 2 n . Each of the four one-dimensional representations of B n induces a multiplicity-free representation of S 2 n , and thus the corresponding Hecke algebra is commutative in each case. We prove that in two of the cases, the primitive idempotents can be obtained from the power-sum expansion of Schur's Q -functions, from which follows the surprising corollary that the character tables of these two Hecke algebras are, aside from scalar multiples, the same as the nontrivial part of the character table of the spin representations of S n .