Stability and ribbon bases for the rank-selected homology of geometric lattices

TL;DR

This paper establishes sharp stability bounds for the homology of geometric lattices, introduces a new homology basis, and connects these concepts to Specht modules in a matroid context.

Contribution

It proves a conjecture on stability bounds, introduces a new homology basis, and links homology of geometric lattices to Specht modules.

Findings

Proved sharp uniform stability bounds for rank-selected homology.

Introduced a new homology basis with properties similar to polytabloids.

Resolved an open question of Björner regarding homology bases.

Abstract

This paper analyzes the representation theoretic stability, in the sense of Thomas Church and Benson Farb, of the rank-selected homology of the Boolean lattice and the partition lattice, proving sharp uniform representation stability bounds in both cases. It proves a conjecture of the first author and Reiner by giving the sharp stability bound for general rank sets for the partition lattice. Along the way, a new homology basis sharing useful features with the polytabloid basis for Specht modules is introduced for the rank-selected homology and for the rank-selected Whitney homology of any geometric lattice, resolving an old open question of Bj\"orner. These bases give a matroid theoretic analogue of Specht modules.