Tensor products of Lie nilpotent associative algebras and applications to codimension sequences

Let $G$ and $H$ be unital associative algebras over a field $K$, such that $G$ satisfies the identity $[x_1,\dots ,x_p]=0$ for some integer $p\ge 3$ and $H$ satisfies the identities $[x_1,x_2,x_3]=0$ and $[x_1,x_2]\cdots [x_{2k-1},x_{2k}]=0$ for some $k\ge 2$. In this paper, extending results of Deryabina and Krasilnikov, we show that the tensor product $G\otimes H$ is again a Lie nilpotent associative algebra, i.e., it satisfies $[x_1,\dots ,x_q]=0$ for some $q\ge p$. We also determine an explicit value of $q$ in the case $k=2$, i.e., when $H$ satisfies the identity $[x_1,x_2][x_3,x_4]=0$. As a corollary, we reprove a result of Drensky saying that any product of Grassmann algebras of the form $E\otimes E_{i_1}\otimes \cdots \otimes E_{i_s}$ or $E_{j_1}\otimes E_{j_2}\otimes \cdots \otimes E_{j_t}$, where $E$ denotes the Grassmann algebra over a countable dimensional vector space and $E_r$ denotes the Grasmann algebra over an $r$-dimensional vector space, satisfies an identity of the form $[x_1,\dots ,x_q]=0$ for some integer $q\ge 3$. In addition, we show that for products of the form $E\otimes E_{i_1}\otimes \cdots \otimes E_{i_s}$ the minimal value of $q$ is always and odd integer. We also provide several particular cases in which a value of $q$ can be explicitly computed. As an application, we consider a field of characteristic zero, the variety ${\mathfrak{N}}_p$ of Lie nilpotent associative algebras of index at most $p$ and the corresponding relatively free algebras of finite rank, $F_n({\mathfrak{N}}_p)$. We exhibit many explicit irreducible $S_n$-modules in the $S_n$-module decomposition of the space of proper multilinear polynomials in $F_n({\mathfrak{N}}_p)$ for any $p$. This gives a lower bound for the dimensions of the spaces of multilinear and proper multilinear polynomials in $F_n({\mathfrak{N}}_p)$.