On cyclic invariants of the free associative algebra
Let $K\langle X_d\rangle$ be the free associative algebra of rank $d \geq 2$ over a field $K$. Lane in 1976 and Kharchenko in 1978 proved that the algebra of invariants $K\langle X_d\rangle^G$ is free for any subgroup $G \leq \text{GL}_d(K)$ and any field $K$. Later, Kharchenko introduced an additional action of the symmetric group $\text{Sym}(n)$ on the homogeneous component of degree $n$ of $K\langle X_d\rangle$, given by permuting the positions of the variables. This equips $K\langle X_d\rangle$ with the structure of a $(K\langle X_d\rangle,\circ)$-$S$-algebra. Then Koryukin showed that the algebra of invariants $K\langle X_d\rangle^G$ is finitely generated for every reductive group $G$ with respect to this action. In our paper we study the algebra $K\langle x_1,\ldots,x_d\rangle^{C_d}$ of invariants of the cyclic group $C_d$, $d\geq 2$, where $K$ is an arbitrary field of characteristic 0. We compute the Hilbert series of $K\langle x_1,\ldots,x_d \rangle^{C_d}$. When $K=\mathbb C$ we find a vector space basis of ${\mathbb C}\langle x_1,\ldots,x_d \rangle^{C_d}$ and explicitly describe the generators of ${\mathbb C}\langle x_1,\ldots,x_d \rangle^{C_d}$ as a free algebra. Moreover, we describe a finite generating set for the $S$-algebra $({\mathbb C}\langle x_1,\ldots,x_d \rangle^{C_d},\circ)$. We also transfer the results for $K=\mathbb C$ to the case of an arbitrary field of characteristic 0 for the $S$-algebra $(K\langle x_1,x_2,x_3 \rangle^{C_3},\circ)$ and find a minimal generating set for it as an $S$-algebra.