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 ${T}_{1}\left(n\right)=\{{h}_{\lambda}:\lambda =(n-r,{1}^{r}),r\ge 1\}.$

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

Revised:

Accepted:

Published online:

Keywords: Subword order, reflection representation, $h$-positivity, Whitney homology, Kronecker product, internal product, Stirling numbers.

Sundaram, Sheila ^{1}

@article{ALCO_2021__4_5_879_0, author = {Sundaram, Sheila}, title = {The reflection representation in the homology of subword order}, journal = {Algebraic Combinatorics}, pages = {879--907}, publisher = {MathOA foundation}, volume = {4}, number = {5}, year = {2021}, doi = {10.5802/alco.184}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.184/} }

TY - JOUR AU - Sundaram, Sheila TI - The reflection representation in the homology of subword order JO - Algebraic Combinatorics PY - 2021 SP - 879 EP - 907 VL - 4 IS - 5 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.184/ DO - 10.5802/alco.184 LA - en ID - ALCO_2021__4_5_879_0 ER -

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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