Comparing Codimension and Absolute Length in Complex Reflection Groups

TL;DR

This paper compares two partial orders, reflection length and codimension, in complex reflection groups, revealing their equivalence only in Coxeter groups and G(m,1,n), and explores implications for algebraic cohomology.

Contribution

It provides a detailed comparison of reflection length and codimension functions, algorithms for their computation, and characterizes when these orders coincide in complex reflection groups.

Findings

Coxeter groups and G(m,1,n) are the only groups where the orders coincide.

Algorithms for computing reflection length and poset relations are developed.

Explicit descriptions of atoms and generators for cohomology are provided.

Abstract

Reflection length and codimension of fixed point spaces induce partial orders on a complex reflection group. While these partial orders are of independent combinatorial interest, our investigation is motivated by a connection between the codimension order and the algebraic structure of cohomology governing deformations of skew group algebras. In this article, we compare the reflection length and codimension functions and discuss implications for cohomology of skew group algebras. We give algorithms using character theory for computing reflection length, atoms, and poset relations. Using a mixture of theory, explicit examples, and computer calculations in GAP, we show that Coxeter groups and the infinite family G(m,1,n) are the only irreducible complex reflection groups for which the reflection length and codimension orders coincide. We describe the atoms in the codimension order for the infinite family G(m,p,n), which immediately yields an explicit description of generators for cohomology.