On the $\mathrm{GL}(n)$-module structure of Lie nilpotent associative relatively free algebras

TL;DR

This paper investigates the structure of certain Lie nilpotent associative algebras as modules over the general linear group, providing bounds on partitions and degrees of invariants, with implications for classical invariant theory.

Contribution

It characterizes the $ ext{GL}(n)$-module structure of quotients of free associative algebras by Lie nilpotent ideals and bounds the degrees of generators of their invariants under classical groups.

Findings

Bound on partitions $ ext{GL}(n)$-modules appear in the algebra.

Upper bounds on degrees of generators of invariants under classical groups.

Provides criteria for when invariants belong to certain Lie ideals.

Abstract

Let $K\left\langle X \right\rangle$ denote the free associative algebra generated by a set $X = \{x_1, \dots, x_n\}$ over a field $K$ of characteristic $0$. Let $I_p$, for $p \geq 2$, denote the two-sided ideal in $K\left\langle X \right\rangle$ generated by all commutators of the form $[u_1, \dots, u_p]$, where $u_1, \dots, u_p \in K\left\langle X \right\rangle$. We discuss the $\mathrm{GL}(n, K)$-module structure of the quotient $K\left\langle X \right\rangle / I_{p+1}$ for all $p \geq 1$ under the standard diagonal action. We give a bound on the values of partitions $\lambda$ such that the irreducible $\mathrm{GL}(n, K)$-module $V_{\lambda}$ appears in the decomposition of $K\left\langle X \right\rangle / I_{p+1}$ as a $\mathrm{GL}(n, K)$-module. As an application, we take $K = \mathbb{C}$ and we consider the algebra of invariants $(\mathbb{C}\left\langle X \right\rangle / I_{p+1})^G$ for $G = \mathrm{SL}(n, \mathbb{C})$, $\mathrm{O}(n, \mathbb{C})$, $\mathrm{SO}(n, \mathbb{C})$, or $\mathrm{Sp}(2s, \mathbb{C})$ (for $n=2s$). By a theorem of Domokos and Drensky, $(\mathbb{C}\left\langle X \right\rangle / I_{p+1})^G$ is finitely generated. We give an upper bound on the degree of generators of $(\mathbb{C}\left\langle X \right\rangle / I_{p+1})^G$ in a minimal generating set. In a similar way, we consider also the algebra of invariants $(\mathbb{C}\left\langle X \right\rangle / I_{p+1})^{G}$, where $G=\mathrm{UT}(n, \mathbb{C})$, and give an upper bound on the degree of generators in a minimal generating set. These results provide useful information about the invariants in $\mathbb{C}\left\langle X \right\rangle^G$ from the point of view of Classical Invariant Theory. In particular, for all $G$ as above we give a criterion when a $G$-invariant of $\mathbb{C}\left\langle X \right\rangle$ belongs to $I_p$.