On the $T$-ideal generated by the identity $f=x^n$

Let $V_{n+K}=V_{n+K}(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_{n+K})$ denote the vector space of all multilinear polynomials in $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_{n+K}$ over $\mathbb{F},$ a field of characteristic zero. In this paper we investigate the structure of the $S_{n+K}$-module $W_{n,n+K}={\left(x^n\right)}^T\cap V_{n+K}$, where ${\left(x^n\right)}^T◃\mathbb{F}⟨x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.}⟩$ is the $T$-ideal generated by the identity $f=x^n\mathrm{.}$