The reflection representation in the homology of subword order

Sheila Sundaram

doi:10.5802/alco.184

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.

Highlights

Let A∗ denote the free monoid of words of finite length in an alphabet A
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 T1(n) = {hλ : λ = (n − r, 1r), r 1}
We show in Theorem 5.2 that for any nonempty subset S of ranks [1, k], the homology representation of Sn may be written as an integer combination of positive tensor powers of the reflection representation

Summary

Introduction

Let A∗ denote the free monoid of words of finite length in an alphabet A. The Sn-homology module of any finite nonempty rank-selected subposet of subword order on A∗ is a nonnegative integer combination of positive tensor powers of the irreducible indexed by the partition (n − 1, 1). These considerations lead us to examine the tensor powers of the reflection representation (see Section 6), and the question of how many tensor powers are linearly independent characters. We are led to a decomposition (Theorem 7.7) showing that the kth tensor power of S(n−1,1) plus or minus one copy of S(n−1,1), has Frobenius characteristic equal to a nonnegative integer combination of the homogeneous symmetric functions {h(n−r,1r) : r 2} It is “almost” an h-positive permutation module. A salient feature of Theorems 1.3 and 1.4 is that only Young subgroups indexed by hooks appear

Subword order

Tensor powers of the reflection representation I

Tensor powers of the reflection representation II

Findings

The subposet of normal words