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no. 5
Modular law through GKM theory
Horiguchi, Tatsuya1; Masuda, Mikiya2; Sato, Takashi2
1 National institute of technology Ube College 2-14-1 Tokiwadai Ube Yamaguchi 755-8555 Japan
2 Osaka Central Advanced Mathematical Institute Sumiyoshi-ku Osaka 558-8585 Japan
Algebraic Combinatorics, Volume 7 (2024) no. 5, pp. 1433-1451

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.380
Classification: 57S12, 05A05, 14M15
Keywords: Hessenberg variety, torus action, twin, GKM theory, equivariant cohomology, modular law, chromatic symmetric function, unicellular LLT polynomial

Horiguchi, Tatsuya 1; Masuda, Mikiya 2; Sato, Takashi 2

1 National institute of technology Ube College 2-14-1 Tokiwadai Ube Yamaguchi 755-8555 Japan
2 Osaka Central Advanced Mathematical Institute Sumiyoshi-ku Osaka 558-8585 Japan
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Horiguchi, Tatsuya; Masuda, Mikiya; Sato, Takashi. Modular law through GKM theory. Algebraic Combinatorics, Volume 7 (2024) no. 5, pp. 1433-1451. doi: 10.5802/alco.380

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