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Geometry of the twin manifolds of regular semisimple Hessenberg varieties and unicellular LLT polynomials

Young-Hoon Kiem, Donggun Lee

Abstract

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.

Geometry of the twin manifolds of regular semisimple Hessenberg varieties and unicellular LLT polynomials

Abstract

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.
Paper Structure (19 sections, 20 theorems, 122 equations)

This paper contains 19 sections, 20 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Unicellular LLT polynomials
  3. Definitions
  4. Modular law and characterization of $\mathrm{LLT}_{(-)}$
  5. Twin manifolds and their cohomology
  6. Isospectral manifolds
  7. Hessenberg varieties and their twins
  8. Goresky-Kottwitz-MacPherson theory
  9. Equivariant cohomology of $Y_h$
  10. An $S_n$-action on $H^*(Y_h)$
  11. Connectedness and multiplicativity
  12. Geometry of twin manifolds
  13. Roof of a triple
  14. Maps from the roof
  15. Cohomology of the roof
  16. ...and 4 more sections

Key Result

Proposition 2.5

ANAleLee Unicellular LLT polynomials and chromatic quasisymmetric functions satisfy the modular law.

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Modular triple
  • Definition 2.4: Modular law
  • Proposition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 45 more