Elitza Hristova
Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences, Sofia, Bulgaria

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 $0$. For any integer $p\geq 2$, let $I_p$ denote the two-sided associative ideal in $K\left\langle X \right \rangle$ generated by all commutators of length $p$. The group $\mathrm{GL}(n, K)$ acts in a natural way on the quotient $K\left\langle X \right \rangle / I_{p+1}$ and the $\mathrm{GL}(n, K)$-module structure of $K\left\langle X \right \rangle / I_{p+1}$ is known for $p=1, 2, 3, 4$. In this talk, we give some results on the $\mathrm{GL}(n, K)$-module structure of $K\left\langle X \right \rangle / I_{p+1}$ for any $p \geq 1$. More precisely, we give a bound on the values of partitions $\lambda$ such that the irreducible $\mathrm{GL}(n, K)$-module with highest weight $\lambda$ appears in the decomposition of $K\left\langle X \right\rangle / I_{p+1}$ as a $\mathrm{GL}(n, K)$-module. We discuss also applications of these results related to the algebras of $G$-invariants in $K\left\langle X \right \rangle / I_{p+1}$, where we take $K = \mathbb{C}$ and $G$ to be one of the classical complex groups $\mathrm{SL}(n, \mathbb{C})$, $\mathrm{O}(n, \mathbb{C})$, $\mathrm{SO}(n, \mathbb{C})$, or $\mathrm{Sp}(2k, \mathbb{C})$ (for $n=2k$).