The Symmetric Group Action on Rank-selected Posets of Injective Words

Abstract

The symmetric group \(\mathfrak {S}_{n}\) acts naturally on the poset of injective words over the alphabet {1, 2,…,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.