Homology of Segre powers of Boolean and subspace lattices

TL;DR

This paper studies the homology representations of Segre powers of Boolean lattices, providing explicit formulas for their decomposition into symmetric group irreducibles and connecting these to invariants of subspace lattices.

Contribution

It offers new explicit formulas for the homology decomposition of Segre powers of Boolean lattices and relates these to symmetric group actions and subspace lattice invariants.

Findings

Explicit decomposition formulas for homology representations.

Connection between homology invariants and subspace lattice properties.

Stable principal specialisation matches rank-selected invariants.

Abstract

Segre products of posets were defined by Bj\"orner and Welker (2005). We investigate the homology representations of the $t$-fold Segre power $B_n^{(t)}$ of the Boolean lattice $B_n$. The direct product $\sym_n^{\times t}$ of the symmetric group $\sym_n$ acts on the homology of rank-selected subposets of $B_n^{(t)}$. We give an explicit formula for the decomposition into $\sym_n^{\times t}$-irreducibles of the homology of the full poset, as well as formulas for the diagonal action of the symmetric group $\sym_n$. For the rank-selected homology, we show that the stable principal specialisation of the product Frobenius characteristic of the $\sym_n^{\times t}$-module coincides with the corresponding rank-selected invariant of the $t$-fold Segre power of the subspace lattice.