Poset homology of Rees products, and $q$-Eulerian polynomials

TL;DR

This paper proves a refined and $q$-analog version of a conjecture relating the homology of Rees products of posets to Eulerian numbers, extending results to vector space lattices and Coxeter group types.

Contribution

It introduces a $q$-analog and refinements of the Rees product homology conjecture, connecting poset topology with $q$-Eulerian polynomials and Coxeter group types.

Findings

Proved the original conjecture relating homology dimension to derangements.

Established a $q$-analog involving subspace lattices and $( ext{maj}, ext{exc})$-$q$-Eulerian polynomials.

Extended results to equivariant and type BC versions.

Abstract

The notion of Rees product of posets was introduced by Bj\"orner and Welker, where they study connections between poset topology and commutative algebra. Bj\"orner and Welker conjectured and Jonsson proved that the dimension of the top homology of the Rees product of the truncated Boolean algebra $B_n \setminus \{0\}$ and the $n$-chain $C_n$ is equal to the number of derangements in the symmetric group $\mathfrak S_n$. Here we prove a refinement of this result, which involves the Eulerian numbers, and a $q$-analog of both the refinement and the original conjecture, which comes from replacing the Boolean algebra by the lattice of subspaces of the $n$-dimensional vector space over the $q$ element field, and involves the $(\maj,\exc)$-$q$-Eulerian polynomials studied in previous papers of the authors. Equivariant versions of the refinement and the original conjecture are also proved, as are type BC versions (in the sense of Coxeter groups) of the original conjecture and its $q$-analog.