Anouk BergeronBrlek ; Christophe Hohlweg ; Mike Zabrocki

Words and polynomial invariants of finite groups in noncommutative variables
dmtcs:2720 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2009,
DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)

https://doi.org/10.46298/dmtcs.2720
Words and polynomial invariants of finite groups in noncommutative variables
Authors: Anouk BergeronBrlek ^{1}; Christophe Hohlweg ^{2}; Mike Zabrocki ^{1}
1 Department of Mathematics and Statistics [Toronto]
2 Laboratoire de combinatoire et d'informatique mathématique [Montréal]
Let $V$ be a complex vector space with basis $\{x_1,x_2,\ldots,x_n\}$ and $G$ be a finite subgroup of $GL(V)$. The tensor algebra $T(V)$ over the complex is isomorphic to the polynomials in the noncommutative variables $x_1, x_2, \ldots, x_n$ 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.