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

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 G-module in terms of words in a particular Cayley graph of G. To apply the method to a particular group, we require a surjective homomorphism from a subalgebra of the group algebra into the character algebra. In the case of 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.