Torsion table for the Lie algebra $\frak{nil}_n$
Leon Lampret, Ale\v{s} Vavpeti\v{c}

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 of all strictly upper-triangular matrices with entries in . Its complete homology for is computed. We prove that every -torsion appears in for . For , Dwyer proved that the bound is sharp, i.e. there is no -torsion in when prime . In general, for the bound is not sharp, as we show that there is -torsion in . As a sideproduct, we derive the known result, that the ranks of the free part of are the Mahonian numbers (=number of permutations of with inversions), using a different approach than Kostant. Furthermore, we determine the algebra structure (cup products) of…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
