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arXiv · 2604.06479

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

Patricia Hersh·Sheila Sundaram
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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.

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Patricia Hersh, Sheila Sundaram. 2026-04-07. Stability and ribbon bases for the rank-selected homology of geometric lattices. https://arxiv.org/abs/2604.06479

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