Words and polynomial invariants of finite groups in non-commutative variables

TL;DR

This paper develops a combinatorial approach to decompose tensor algebras of non-commutative variables under finite group actions, providing explicit methods for symmetric and dihedral groups to understand invariants.

Contribution

It introduces a general combinatorial method using Cayley graphs for decomposing tensor algebras into simple modules under finite group actions, with specific applications to symmetric and dihedral groups.

Findings

Decomposition method for tensor algebra modules using Cayley graphs.

Explicit homomorphisms for symmetric and dihedral groups.

Interpretation of graded invariants in terms of words in Cayley graphs.

Abstract

Let V be a complex vector space with basis {x_1,x_2,...,x_n} and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x_1, x_2,..., x_n with complex coefficients. We want to give a combinatorial interpretation for the decomposition of T(V) into simple G-modules. In particular, we want to study the graded space of invariants in T(V) with respect to the action of G. We give a general method for decomposing the space T(V) into simple modules in terms of words in a Cayley graph of the group G. To apply the method to a particular group, we require a homomorphism from a subalgebra of the group algebra into the character algebra. In the case of G as the symmetric group, we give an example of this homomorphism from the descent algebra. When G is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions of the invariant space in term of those words.