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Geometry of the twin manifolds of regular semisimple Hessenberg varieties and unicellular LLT polynomials
Kiem, Young-Hoon1; Lee, Donggun2
1 School of Mathematics Korea Institute for Advanced Study 85 Hoegiro Dongdaemun-gu Seoul 02455 Korea
2 Center for Complex Geometry Institute for Basic Science 55 Expo-ro Yuseong-gu Daejeon 34126 Korea
Algebraic Combinatorics, Volume 7 (2024) no. 3, pp. 861-885.

Recently, Masuda-Sato and Precup-Sommers independently proved an LLT version of the Shareshian-Wachs conjecture, which says that the Frobenius characteristics of the cohomology of the twin manifolds of regular semisimple Hessenberg varieties are unicellular LLT polynomials. The purpose of this paper is to study the geometry of twin manifolds and we prove that they are related by explicit blowups and fiber bundle maps. Upon taking their cohomology, we obtain a direct proof of the modular law which establishes the LLT Shareshian-Wachs conjecture.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.349
Classification: 14M15, 05E10
Keywords: Hessenberg varieties, twin manifolds, unicellular LLT polynomials, Shareshian-Wachs conjecture, modular law

Kiem, Young-Hoon 1; Lee, Donggun 2

1 School of Mathematics Korea Institute for Advanced Study 85 Hoegiro Dongdaemun-gu Seoul 02455 Korea
2 Center for Complex Geometry Institute for Basic Science 55 Expo-ro Yuseong-gu Daejeon 34126 Korea
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Kiem, Young-Hoon; Lee, Donggun. Geometry of the twin manifolds of regular semisimple Hessenberg varieties and unicellular LLT polynomials. Algebraic Combinatorics, Volume 7 (2024) no. 3, pp. 861-885. doi : 10.5802/alco.349. https://alco.centre-mersenne.org/articles/10.5802/alco.349/

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