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Parabolic Lusztig varieties and chromatic symmetric functions

Alex AbreuAntonio Nigro
Dec 2022
摘要
The characters of Kazhdan--Lusztig elements of the Hecke algebra over $S_n$(and in particular, the chromatic symmetric function of indifference graphs)are completely encoded in the (intersection) cohomology of certain subvarietiesof the flag variety. Considering the forgetful map to some partial flagvariety, the decomposition theorem tells us that this cohomology splits as asum of intersection cohomology groups with coefficients in some local systemsof subvarieties of the partial flag variety. We prove that these local systemscorrespond to representations of subgroups of $S_n$. An explicitcharacterization of such representations would provide a recursive formula forthe computation of such characters/chromatic symmetric functions, which couldsettle Haiman's conjecture about the positivity of the monomial characters ofKazhdan--Lusztig elements and Stanley--Stembridge conjecture about$e$-positivity of chromatic symmetric function of indifference graphs. We alsofind a connection between the character of certain homology groups ofsubvarieties of the partial flag varieties and the Grojnowski--Haiman hybridbasis of the Hecke algebra.
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