Wednesday, September 9, 2020
4:10pm to 5:00pm


Math Seminar Series
(Conferences / Seminars / Lectures)
Sheila Sundaram will be speaking. The presentation is titles "On the homology of subword order."
The Zoom link can be found on the seminar page linked below.
In this talk we examine the homology representation of the symmetric group Sn 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(n−1,1) indexed by the partition (n−1,1), and that its Frobenius characteristic is hpositive and supported on the set T1(n)={hλ:λ=(n−r,1r),r≥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 rankselected homology modules, modulo one copy of the reflection representation, are hpositive and supported on the set T2(n)={hλ:λ=(n−r,1r),r≥2}. Our analysis of the homology also uncovers curious enumerative formulas that may be interesting to investigate combinatorially. more information...
