On dihedral invariants of the free associative algebra of rank two

Abstract

Let $K⟨X_d⟩$ denote the free associative algebra of rank $d\ge 2$ over a field $K$. By results of Lane (1976) and Kharchenko (1978), the algebra of invariants $K⟨X_d{⟩}^G$ 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 $K⟨X_d⟩$, given by permuting the positions of the variables. This endows $K⟨X_d⟩$ with the structure of a $(K⟨X_d⟩,\circ )$-$S$-algebra. With respect to this action, Koryukin proved that the invariant algebra $K⟨X_d{⟩}^G$ is finitely generated for every reductive group $G$. In this paper we study the algebra $\mathbb{C}⟨u,v{⟩}^{D_{2n}}$ of invariants under the action of the dihedral group D_{2n} $onthefreeassociativealgebra${\mathbb C} \langle u,v\rangle$ofrank$2$.WecomputetheHilbertseriesof${\mathbb C}\langle u,v\rangle^{D_{2n}}$andconstructanexplicitsetofgeneratorsfor${\mathbb C}\langle u,v\rangle^{D_{2n}}$asafreealgebra.Furthermore,wedescribeafinitegeneratingsetforthe$S$-algebra${\mathbb C}\langle u,v\rangle^{D_{2n}}$.