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 decomposes into a sum of tensor powers of the -irreducible indexed by the partition recovering, as a special case, a theorem of Björner and Stanley for words of length at most For arbitrary ranks we show that the homology is an integer combination of positive tensor powers of the reflection representation , 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 , and show that its Frobenius characteristic is -positive and supported on the set
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 as an integer combination of the set We conjecture that this combination is nonnegative, establishing this fact for particular cases.
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Keywords: Subword order, reflection representation, $h$-positivity, Whitney homology, Kronecker product, internal product, Stirling numbers.
@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 = {http://www.numdam.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 - http://www.numdam.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. http://www.numdam.org/articles/10.5802/alco.184/
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