ALGEBRAIC COMBINATORICS

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 ${S}_{\left(n-1,1\right)}$ indexed by the partition $\left(n-1,1\right),$ 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 ${S}_{\left(n-1,1\right)}$, 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 ${S}_{\left(n-1,1\right)}$, and show that its Frobenius characteristic is $h$-positive and supported on the set ${T}_{1}\left(n\right)=\left\{{h}_{\lambda }:\lambda =\left(n-r,{1}^{r}\right),r\ge 1\right\}.$

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 ${s}_{\left(n-1,1\right)},$ as an integer combination of the set ${T}_{2}\left(n\right)=\left\{{h}_{\lambda }:\lambda =\left(n-r,{1}^{r}\right),r\ge 2\right\}.$ We conjecture that this combination is nonnegative, establishing this fact for particular cases.

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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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