The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand—Zetlin Polytope

Abstract

Regular semisimple Hessenberg varieties are subvarieties of the flag variety Flag(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 GZ(λ) for λ = (λ 1 , λ 2 , . . . , λn). 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 α i := λ i − λ 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 Flag(C n ) as a sum of the cohomology classes of a certain set of Richardson varieties.

The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials

“…Our first main result states that the a w are strictly positive, that is, the expansion in (1.1) has full support. This answers a problem posed by Harada et al [27,Problem 6.6].…”

“…Given that A 2,1,1,0 = 6 and A 1,2,1,0 = 12, we obtain a w = 1 24 (12 + 6 + 6) = 1. The following immediate corollary answers a question asked in[27, Problem 6.6].Corollary 5.4. For any w ∈ S n , a w > 0;…”

“…The last decade has seen an ever-increasing interest in their study with impetus coming from the study of chromatic quasisymmetric functions and the potential ramifications for the Stanley-Stembridge conjecture [28,54,55]. The study of the cohomology rings of Hessenberg varieties has been linked to the study of hyperplane arrangements and representations of the symmetric group [4,3,16,27]. We refer the reader to Abe and Horiguchi's excellent survey article [2] and references therein for more details on the rich vein of mathematics surrounding Hessenberg varieties.…”

Fano and weak Fano Hessenberg varieties

“…In this subsection, we give an example of a pair of nef Hessenberg functions which illustrate the argument in Case 2-b in Section 5.3. Let n = 20, and h : [20] → [20] the Hessenberg function depicted in Figure 12, that is, h = (9,10,10,11,12,12,13,15,17,18,18,19,19,20,20,20,20,20,20,20).…”

“…To give the above classifications, we first compute the anti-canonical bundles of regular semisimple Hessenberg varieties explicitly, and we study their volumes by using the theory of line bundles over Richardson varieties. We note that the method of using Richardson varieties for computations of volumes of line bundles over Hessenberg varieties is motivated by Anderson-Tymoczko [4] and Harada-Horiguchi-Masuda-Park [13].…”