Spin characters of generalized symmetric groups

TL;DR

This paper extends the computation of spin character values to wreath products of symmetric groups with finite abelian groups, using Mackey-Wigner's little groups, and includes classical Weyl groups.

Contribution

It generalizes the projective character value computation to a broader class of groups, filling a gap in the existing theory.

Findings

Computed projective character values for wreath products with finite abelian groups

Constructed irreducible modules using Mackey-Wigner's little groups

Obtained spin character values for all classical Weyl groups

Abstract

In 1911 Schur computed the spin character values of the symmetric group using two important ingredients: the first one later became famously known as the Schur Q-functions and the second one was certain creative construction of the projective characters on Clifford algebras. In the context of the McKay correspondence and affine Lie algebras, the first part was generalized to all wreath products by the vertex operator calculus in \cite{FJW} where a large part of the character table was produced. The current paper generalizes the second part and provides the missing projective character values for the wreath product of the symmetric group with a finite abelian group. Our approach relies on Mackey-Wigner's little groups to construct irreducible modules. In particular, projective modules and spin character values of all classical Weyl groups are obtained.