The reflection representation in the homology of subword order
Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 879-907.

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 S n -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 T 1 (n)={h λ :λ=(n-r,1 r ),r1}.

Our most definitive result describes the Frobenius characteristic of the homology for an arbitrary set of ranks, plus or minus one copy of the Schur function s (n-1,1) , as an integer combination of the set T 2 (n)={h λ :λ=(n-r,1 r ),r2}. We conjecture that this combination is nonnegative, establishing this fact for particular cases.

Received:
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Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.184
Classification: 05E10,  20C30
Keywords: Subword order, reflection representation, h-positivity, Whitney homology, Kronecker product, internal product, Stirling numbers.
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Sundaram, Sheila. The reflection representation in the homology of subword order. Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 879-907. doi : 10.5802/alco.184. https://alco.centre-mersenne.org/articles/10.5802/alco.184/

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