In 2015, Brosnan and Chow, and independently Guay-Paquet, proved the Shareshian-Wachs conjecture, which links the Stanley-Stembridge conjecture in combinatorics to the geometry of Hessenberg varieties through Tymoczko's permutation group action on the cohomology ring of regular semisimple Hessenberg varieties. In previous work, the authors exploited this connection to prove a graded version of the Stanley-Stembridge conjecture in a special case. In this manuscript, we derive a new set of linear relations satisfied by the multiplicities of certain permutation representations in Tymoczko's representation. We also show that these relations are upper-triangular in an appropriate sense, and in particular, they uniquely determine the multiplicities. As an application of these results, we prove an inductive formula for the multiplicity coefficients corresponding to partitions with a maximal number of parts.
@article{10_37236_10489,
author = {Megumi Harada and Martha Precup},
title = {Upper triangular linear relations on multiplicities and the {Stanley-Stembridge} conjecture},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {3},
doi = {10.37236/10489},
zbl = {1505.14104},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10489/}
}
TY - JOUR
AU - Megumi Harada
AU - Martha Precup
TI - Upper triangular linear relations on multiplicities and the Stanley-Stembridge conjecture
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/10489/
DO - 10.37236/10489
ID - 10_37236_10489
ER -
%0 Journal Article
%A Megumi Harada
%A Martha Precup
%T Upper triangular linear relations on multiplicities and the Stanley-Stembridge conjecture
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/10489/
%R 10.37236/10489
%F 10_37236_10489
Megumi Harada; Martha Precup. Upper triangular linear relations on multiplicities and the Stanley-Stembridge conjecture. The electronic journal of combinatorics, Tome 29 (2022) no. 3. doi: 10.37236/10489
