On cyclic invariants of the free associative algebra

TL;DR

This paper investigates the invariants of the cyclic group acting on free associative algebras, providing explicit bases, generators, and Hilbert series, especially over complex numbers, and extends results to arbitrary fields of characteristic zero.

Contribution

It explicitly describes generators, bases, and Hilbert series of cyclic invariants in free associative algebras, including minimal generating sets for specific cases, advancing understanding of invariant structures.

Findings

Computed Hilbert series of invariants

Explicit basis and generators over complex numbers

Extended results to arbitrary characteristic zero fields

Abstract

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.