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Modular law through GKM theory

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arxiv 2310.16235 v1 pith:ASEEZYM3 submitted 2023-10-24 math.AT math.AGmath.COmath.RT

Tatsuya Horiguchi , Mikiya Masuda , Takashi Sato This is my paper · ORCID
classification math.ATmath.AGmath.COmath.RT
keywords modulargraphshessenbergregularresultsemisimpletheoryvarieties
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The solution of Shareshian-Wachs conjecture by Brosnan-Chow and Guay-Paquet tied the graded chromatic symmetric functions on indifference graphs (or unit interval graphs) and the cohomology of regular semisimple Hessenberg varieties with the dot action. A similar result holds between unicellular LLT polynomials and twins of regular semisimple Hessenberg varieties. A recent result by Abreu-Nigro enabled us to prove these results by showing the modular law for the geometrical objects, and this is indeed done by Precup-Sommers and Kiem-Lee. In this paper, we give elementary and simpler proofs to the modular law through GKM theory.

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  1. Divided difference operators for Hessenberg representations

    math.CO 2025-07 conditional novelty 6.0 of 10

    For si-stable and almost-si-stable condition sets, the dot-action module H_C decomposes as a direct sum of a fixed subspace and a multiply-shifted copy, realizing the modular relation at the representation level.

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