Séminaire Lotharingien de Combinatoire, 85B.1 (2021), 12 pp.
Sheila Sundaram
The Homology Representation of Subword Order
Abstract.
We investigate the homology representation of the symmetric group on rank-selected subposets of subword order. We show that the homology module for words of bounded length, over an alphabet of size n, decomposes into a sum of tensor powers of the Sn-irreducible
S(n-1,1) indexed by the partition (n-1,1), recovering, as a special case, a theorem of Björner and Stanley for words of length at most k. For arbitrary ranks we show that the homology is an integer combination of positive tensor powers of the reflection representation S(n-1,1), and conjecture that this combination is nonnegative. We uncover a curious duality in homology in the case when one rank is deleted.
We prove that the action on the rank-selected chains of subword order is a nonnegative integer combination of tensor powers of S(n-1,1), and show that its Frobenius characteristic is h-positive and supported on the set
T1(n) = {hλ:
λ = (n-r, 1r), r ≥ 1}.
Our most definitive result describes the Frobenius characteristic of the homology for an arbitrary set of ranks, plus or minus one copy of S(n-1,1), as an integer combination of the set
T2(n) = {hλ:
λ = (n-r, 1r), r ≥ 2}. We conjecture that
this combination is nonnegative, establishing this fact for particular cases.
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
The following versions are available:
Correction: The author in reference [1] should be "K. Bacławski"
instead of "K. BacLawski".