On dihedral invariants of the free associative algebra of rank two
Silvia Boumova, Vesselin Drensky, Şehmus Fındık
TL;DR
The paper addresses the problem of describing invariants of the free associative algebra under the dihedral group action $D_{2n}$. It employs a Molien-type averaging approach to compute the Hilbert series $H_{2n}(t)$, proves freeness of the invariants, and provides explicit homogeneous generators; it then analyzes the corresponding S-algebra structure under position-permutation and shows finite generation via a reduction to a symmetric subalgebra, culminating in generators $s_1$ and $p_n$. Key results include concrete expressions for $H_{2n}(t)$, a Fibonacci-growth example in the $n=3$ case, and a constructive, finite generating set for the S-algebra. These findings advance explicit noncommutative invariant theory for dihedral symmetries and furnish practical generators for computational use.
Abstract
Let $K\langle X_d\rangle$ denote the free associative algebra of rank $d \geq 2$ over a field $K$. By results of Lane (1976) and Kharchenko (1978), the algebra of invariants $K\langle X_d\rangle ^G$ is free for any subgroup $G \leq \GL_d(K)$ and any field $K$. Koryukin (1984) introduced an additional action of the symmetric group $Sym(n)$ on the homogeneous component of degree $n$ of $K\langle X_d\rangle$, given by permuting the positions of the variables. This endows $K\langle X_d\rangle $ with the structure of a $(K\langle X_d\rangle,\circ)$-$S$-algebra. With respect to this action, Koryukin proved that the invariant algebra $K\langle X_d\rangle ^G$ is finitely generated for every reductive group $G$. In this paper we study the algebra ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ of invariants under the action of the dihedral group D_{2n} $ on the free associative algebra ${\mathbb C} \langle u,v\rangle$ of rank $2$. We compute the Hilbert series of ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ and construct an explicit set of generators for ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ as a free algebra. Furthermore, we describe a finite generating set for the $S$-algebra ${\mathbb C}\langle u,v\rangle^{D_{2n}}$.