Title: On the homology of subword order. Zoom https://msu.zoom.us/j/5476724571

Date: 09/09/2020

Time: 4:10 PM - 5:00 PM

Place:

In this talk we examine the homology representation of the symmetric group $S_n$ on rank-selected subposets of subword order. We show that the action on the rank-selected chains is a nonnegative integer combination of tensor powers of the reflection representation $S_{(n-1,1)}$ indexed by the partition $(n-1,1)$, and that its Frobenius characteristic is $h$-positive and supported on the set $T_{1}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 1\}.$
We give an explicit formula for the homology module for words of bounded length, as a sum of tensor powers of $S_{(n-1,1)}$. This recovers, as a special case, a theorem of Bj\"orner and Stanley for words of length at most $k.$ We exhibit a curious duality in homology in the case when one rank is deleted. We also show that in many cases, the rank-selected homology modules, modulo one copy of the reflection representation, are $h$-positive and supported on the set $T_{2}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$
Our analysis of the homology also uncovers curious enumerative formulas that may be interesting to investigate combinatorially.