 Sheila Sundaram
 On the homology of subword order. Zoom https://msu.zoom.us/j/5476724571
 09/09/2020
 4:10 PM  5:00 PM

In this talk we examine the homology representation of the symmetric group $S_n$ on rankselected subposets of subword order. We show that the action on the rankselected chains is a nonnegative integer combination of tensor powers of the reflection representation $S_{(n1,1)}$ indexed by the partition $(n1,1)$, and that its Frobenius characteristic is $h$positive and supported on the set $T_{1}(n)=\{h_\lambda: \lambda=(nr, 1^r), r\ge 1\}.$
We give an explicit formula for the homology module for words of bounded length, as a sum of tensor powers of $S_{(n1,1)}$. 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 rankselected homology modules, modulo one copy of the reflection representation, are $h$positive and supported on the set $T_{2}(n)=\{h_\lambda: \lambda=(nr, 1^r), r\ge 2\}.$
Our analysis of the homology also uncovers curious enumerative formulas that may be interesting to investigate combinatorially.