# The volume polynomial of regular semisimple Hessenberg varieties and the Gelfand-Zetlin polytope

## TL;DR

This paper provides an explicit combinatorial formula for the volume polynomials of regular semisimple Hessenberg varieties, linking them to Gelfand-Zetlin polytopes and analyzing the permutohedral case with geometric and combinatorial insights.

## Contribution

It generalizes existing tools to derive explicit formulas for volume polynomials and offers a positive combinatorial expression for their coefficients, including a detailed analysis of the permutohedral variety.

## Key findings

- Derived explicit volume polynomial formulas in terms of Gelfand-Zetlin polytope faces.
- Provided a positive combinatorial formula for polynomial coefficients.
- Decomposed the permutohedron into combinatorial cubes and related it to Richardson varieties.

## Abstract

Regular semisimple Hessenberg varieties are subvarieties of the flag variety $\mathrm{Flag}(\mathbb{C}^n)$ arising naturally in the intersection of geometry, representation theory, and combinatorics. Recent results of Abe-Horiguchi-Masuda-Murai-Sato and Abe-DeDieu-Galetto-Harada relate the volume polynomials of regular semisimple Hessenberg varieties to the volume polynomial of the Gelfand-Zetlin polytope $\mathrm{GZ}(\lambda)$ for $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_n)$. The main results of this manuscript use and generalize tools developed by Anderson-Tymoczko, Kiritchenko-Smirnov-Timorin, and Postnikov, in order to derive an explicit formula for the volume polynomials of regular semisimple Hessenberg varieties in terms of the volumes of certain faces of the Gelfand-Zetlin polytope, and also exhibit a manifestly positive, combinatorial formula for their coefficients with respect to the basis of monomials in the $\alpha_i := \lambda_i-\lambda_{i+1}$. In addition, motivated by these considerations, we carefully analyze the special case of the permutohedral variety, which is also known as the toric variety associated to Weyl chambers. In this case, we obtain an explicit decomposition of the permutohedron (the moment map image of the permutohedral variety) into combinatorial $(n-1)$-cubes, and also give a geometric interpretation of this decomposition by expressing the cohomology class of the permutohedral variety in $\mathrm{Flag}(\mathbb{C}^n)$ as a sum of the cohomology classes of a certain set of Richardson varieties.