  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.
