L'alg\`ebre des invariants d'un groupe de Coxeter agissant sur un mutiple de sa repr\'esentation standard

TL;DR

This paper investigates the algebra of invariants under Coxeter and complex reflection groups acting on tensor powers of their standard representations, revealing its structure as a free module over the invariants of the base algebra.

Contribution

It establishes that the invariants form a free module of a specific rank over the base invariants, and shows that the tensor power algebra is not free over the invariants.

Findings

(S(V)^{ ensor k})^G is a free (S(V)^G)^{ ensor k}-module of rank |G|^{k-1}

S(V)^{ ensor k} is not a free (S(V)^{ ensor k})^G-module

The algebra of invariants has a well-defined module structure with explicit rank

Abstract

Let G be a Coxeter group of type A_n, B_n, D_n or I_2(N), or a complex reflection group of type G(de,e,n). Let V be its standard representation and let k be an integer greater than 2. Then G acts on S(V)^{\otimes k}. We show that the algebra of invariants (S(V)^{\otimes k})^G is a free (S(V)^G)^{\otimes k}-module of rank |G|^{k-1}, and that S(V)^{\otimes k} is not a free (S(V)^{\otimes k})^G-module.