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

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 $n\le 8$ is computed. We prove that every $p^m$-torsion appears in $H_{\ast }({\mathfrak{nil}}_n;\mathbb{Z})$ for $p^m\le n-2$. 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 $H^{\ast }({\mathfrak{nil}}_n;\mathbb{Q})$.

10 pages, 1 table