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In 2015, Brosnan and Chow, and independently Guay-Paquet, proved theShareshian-Wachs conjecture, which links the Stanley-Stembridge conjecture incombinatorics to the geometry of Hessenberg varieties through Tymoczko'spermutation group action on the cohomology ring of regular semisimpleHessenberg varieties. In previous work, the authors exploited this connectionto prove a refined (graded) version of the Stanley-Stembridge conjecture in aspecial case. In this manuscript, we derive a new set of linear relationssatisfied by the multiplicities of certain permutation representations inTymoczko's representation. We also show that these relations areupper-triangular in an appropriate sense, and in particular, they uniquelydetermine the multiplicities. As an application of these results, we prove aninductive formula for the multiplicity coefficients corresponding to partitionswith a maximal number of parts. It follows from our formula that thesecoefficients are non-negative, thus giving additional positive evidence for thegraded Stanley--Stembridge conjecture in the general case.

Upper-triangular linear relations on multiplicities and the Stanley-Stembridge conjecture