Torsion table for the Lie algebra ${\mathfrak{n}\mathfrak{i}\mathfrak{l}}_n$

We study the Lie ring ${\mathfrak{n}\mathfrak{i}\mathfrak{l}}_n$ of all strictly upper-triangular $n\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}n$ matrices with entries in $\mathbb{Z}$. Its complete homology for $n\text{\hspace{0.17em}⁣}\le \text{\hspace{0.17em}⁣}8$ is computed. We prove that every $p^m$-torsion appears in $H_{\ast }({\mathfrak{n}\mathfrak{i}\mathfrak{l}}_n;\mathbb{Z})$ for $p^m\text{\hspace{0.17em}⁣}\le \text{\hspace{0.17em}⁣}n\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}2$. For $m\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}1$, Dwyer proved that the bound is sharp, i.e. there is no $p$-torsion in $H_{\ast }({\mathfrak{n}\mathfrak{i}\mathfrak{l}}_n;\mathbb{Z})$ when prime $p\text{\hspace{0.17em}⁣}>\text{\hspace{0.17em}⁣}n\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}2$. In general, for $m\text{\hspace{0.17em}⁣}>\text{\hspace{0.17em}⁣}1$ the bound is not sharp, as we show that there is $8$-torsion in $H_{\ast }({\mathfrak{n}\mathfrak{i}\mathfrak{l}}_8;\mathbb{Z})$. As a sideproduct, we derive the known result, that the ranks of the free part of $H_{\ast }({\mathfrak{n}\mathfrak{i}\mathfrak{l}}_n;\mathbb{Z})$ are the Mahonian numbers (=number of permutations of $[n]$ with $k$ inversions), using a different approach than Kostant. Furthermore, we determine the algebra structure (cup products) of $H^{\ast }({\mathfrak{n}\mathfrak{i}\mathfrak{l}}_n;\mathbb{Q})$.