Comparing Codimension and Absolute Length in Complex Reflection Groups

Abstract

Reflection length and codimension of fixed point spaces induce partial orders on a complex reflection group. Motivated by connections to the algebraic structure of cohomology governing deformations of skew group algebras, we show that Coxeter groups and the infinite family G(m, 1, n) are the only irreducible complex reflection groups for which reflection length and codimension coincide. We then discuss implications for the degrees of generators of Hochschild cohomology. Along the way, we describe the codimension atoms for the infinite family G(m, p, n), give algorithms using character theory, and determine two-variable Poincaré polynomials recording reflection length and codimension.