The symmetric group action on rank-selected posets of injective words

TL;DR

This paper generalizes previous work by computing the symmetric group action on the homology of all rank-selected subposets of injective words, including a new case involving r-colored injective words.

Contribution

It extends the known symmetric group representation results to all rank-selected subposets and introduces a generalization to r-colored injective words.

Findings

Computed the symmetric group representation on homology of all rank-selected subposets.

Extended results to the poset of r-colored injective words.

Provided a unified framework for these group actions.

Abstract

The symmetric group $\mathfrak{S}_n$ acts naturally on the poset of injective words over the alphabet $\{1, 2,\dots,n\}$. The induced representation on the homology of this poset has been computed by Reiner and Webb. We generalize their result by computing the representation of $\mathfrak{S}_n$ on the homology of all rank-selected subposets, in the sense of Stanley. A further generalization to the poset of $r$-colored injective words is given.