The reflection representation in the homology of subword order

TL;DR

This paper explores the homology representation of subword order in symmetric groups, revealing tensor power decompositions, dualities, and positivity properties, with conjectures on nonnegativity of certain combinations.

Contribution

It introduces new decompositions and positivity results for homology representations of subword order, extending previous theorems and proposing conjectures on nonnegativity.

Findings

Homology decomposes into tensor powers of the reflection representation.

Frobenius characteristic is h-positive and supported on specific partitions.

Established nonnegativity for chains and conjectured for general rank sets.

Abstract

We investigate the homology representation of the symmetric group on rank-selected subposets of subword order. We show that the homology module for words of bounded length, over an alphabet of size $n,$ decomposes into a sum of tensor powers of the $S_n$-irreducible $S_{(n-1,1)}$ indexed by the partition $(n-1,1),$ recovering, as a special case, a theorem of Bj\"orner and Stanley for words of length at most $k.$ For arbitrary ranks we show that the homology is an integer combination of positive tensor powers of the reflection representation $S_{(n-1,1)}$, and conjecture that this combination is nonnegative. We uncover a curious duality in homology in the case when one rank is deleted. We prove that the action on the rank-selected chains of subword order is a nonnegative integer combination of tensor powers of $S_{(n-1,1)}$, and show that its Frobenius characteristic is $h$-positive and supported on the set $T_{1}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 1\}.$ Our most definitive result describes the Frobenius characteristic of the homology for an arbitrary set of ranks, plus or minus one copy of the Schur function $s_{(n-1,1)},$ as an integer combination of the set $T_{2}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$ We conjecture that this combination is nonnegative, establishing this fact for particular cases.