Torsion table for the Lie algebra $\frak{nil}_n$

Abstract: We study the Lie ring $\mathfrak{nil}n$ of all strictly upper-triangular $n!\times!n$ matrices with entries in $\mathbb{Z}$. Its complete homology for $n!\leq!8$ is computed. We prove that every $p^m$-torsion appears in $H\ast(\mathfrak{nil}n;\mathbb{Z})$ for $p^m!\leq!n!-!2$. For $m!=!1$, Dwyer proved that the bound is sharp, i.e. there is no $p$-torsion in $H\ast(\mathfrak{nil}n;\mathbb{Z})$ when prime $p!>!n!-!2$. In general, for $m!>!1$ the bound is not sharp, as we show that there is $8$-torsion in $H\ast(\mathfrak{nil}8;\mathbb{Z})$. As a sideproduct, we derive the known result, that the ranks of the free part of $H\ast(\mathfrak{nil}_n;\mathbb{Z})$ 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 $H^\ast(\mathfrak{nil}_n;\mathbb{Q})$.