### Colored Descent Representations for Complex Reflection Groups

### Riccardo Biagioli (UQAM)

Abstract: Let V be a complex vector space of dimension n. A
pseudo-reflection on V is a linear transformation on V of finite order
which fixes a hyperplane in V pointwise. A complex reflection group on
V is a finite subgroup W < GL(V) generated by pseudo-reflections.
Such groups are characterized by the structure of their invariant ring.
Irreducible finite complex reflection groups have been classified by
Shephard-Todd. In a way similar to Coxeter groups (real reflection
groups), they have presentations in terms of generators and relations,
that can be visualized by Dynkin type diagrams.

We start by giving an introduction to this class of groups and by
showing their classification. Then by considering them as subgroups of
the wreath products $\mathbb{Z}_r \wr S_n$, and by using Clifford
theory, we define combinatorial statistics and (colored) descent
representations for complex reflection groups. These representations
were previously known for classical Weyl groups. One of the statistics
introduced, is the flag major index, which has an important role in the
decomposition of the descent representations into irreducible
components. Finally, we present a combinatorial identity relating the
combinatorial statistics with the degrees of the group.

Algebraic Combinatorics Seminar home