Title: On the eigenvalues of representations of reflection groups and wreath products
Abstract:Let Gbea finite group.The eigenvalues of any g e G of order m in a (complex) representation p may be expressed in the form α/ 1 , α/ 2 ,..., with ω = e 2πilm .We call the integers βj (mod m) the cycli...Let Gbea finite group.The eigenvalues of any g e G of order m in a (complex) representation p may be expressed in the form α/ 1 , α/ 2 ,..., with ω = e 2πilm .We call the integers βj (mod m) the cyclic exponents of g with respect to p.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 GI 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.Read More