Nagoya Mathematical Journal

Twisted invariant theory for reflection groups

C. Bonnafé, G. I. Lehrer, and J. Michel

Full-text: Open access

Abstract

Let $G$ be a finite reflection group acting in a complex vector space $V = \mathbb{C}^{r}$, whose coordinate ring will be denoted by $S$. Any element $\gamma \in \mathrm{GL}(V)$ which normalises $G$ acts on the ring $S^{G}$ of $G$-invariants. We attach invariants of the coset $G\gamma$ to this action, and show that if $G'$ is a parabolic subgroup of $G$, also normalised by $\gamma$, the invariants attaching to $G'\gamma$ are essentially the same as those of $G\gamma$. Four applications are given. First, we give a generalisation of a result of Springer-Stembridge which relates the module structures of the coinvariant algebras of $G$ and $G'$ and secondly, we give a general criterion for an element of $G\gamma$ to be regular (in Springer's sense) in invariant-theoretic terms, and use it to prove that up to a central element, all reflection cosets contain a regular element. Third, we prove the existence in any well-generated group, of analogues of Coxeter elements of the real reflection groups. Finally, we apply the analysis to quotients of $G$ which are themselves reflection groups.

Article information

Source
Nagoya Math. J., Volume 182 (2006), 135-170.

Dates
First available in Project Euclid: 20 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1150810005

Mathematical Reviews number (MathSciNet)
MR2235340

Zentralblatt MATH identifier
1172.20030

Subjects
Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 20H15: Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25] 51F15: Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55]

Citation

Bonnafé, C.; Lehrer, G. I.; Michel, J. Twisted invariant theory for reflection groups. Nagoya Math. J. 182 (2006), 135--170. https://projecteuclid.org/euclid.nmj/1150810005


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