On the eigenvalues of representations of reflection groups and wreath products

John R. Stembridge

Vol. 140 (1989), No. 2, 353–396

DOI: 10.2140/pjm.1989.140.353

Abstract

Let G be a finite group. The eigenvalues of any g ∈ G of order m in a (complex) representation ρ may be expressed in the form ω^e₁,ω^e₂,…, with ω = e^2πi∕m. We call the integers e_j (mod m) the cyclic exponents of g with respect to ρ. We give explicit combinatorial descriptions of the cyclic exponents of the (irreducible) representations of the symmetric groups, the classical Weyl groups, and certain finite unitary reflection groups. We also show that for any finite group G, the cyclic exponents of the wreath product G≀S_n can be described in terms of the cyclic exponents of G. For each of the infinite families of finite unitary reflection groups W, we also provide explicit, combinatorial descriptions of the generalized exponents of W. These parameters arise in the symmetric algebra of the associated reflection representation, and by a theorem of Springer, are closely related to the cyclic exponents of W.

Mathematical Subject Classification 2000

Primary: 20C30

Secondary: 20C15

Milestones

Received: 6 April 1988

Published: 1 December 1989