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

TL;DR

This paper computes the homology of the Lie ring of strictly upper-triangular matrices over integers for small n, revealing torsion bounds, algebraic structures, and connections to permutation statistics.

Contribution

It provides complete homology computations for n ≤ 8, establishes bounds on torsion, and describes the algebra structure of cohomology, offering new insights and methods.

Findings

Complete homology for n ≤ 8 computed.

p^m-torsion appears for p^m ≤ n-2.

Algebra structure of cohomology over Q determined.

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 $H^\ast(\mathfrak{nil}_n;\mathbb{Q})$.