Upper Triangular Linear Relations on Mmultiplicities and the Stanley-Stembridge Conjecture

In 2015, Brosnan and Chow, independently Guay-Paquet, proved the Shareshian-Wachs conjecture, which links Stanley-Stembridge conjecture in combinatorics to geometry of Hessenberg varieties through Tymoczko's permutation group action on cohomology ring regular semisimple varieties. previous work, authors exploited this connection prove a graded version special case. manuscript, we derive new set linear relations satisfied by multiplicities certain representations representation. We also show that these are upper-triangular an appropriate sense, particular, they uniquely determine multiplicities. As application results, inductive formula for multiplicity coefficients corresponding partitions with maximal number parts.