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.

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