Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies

Abstract

In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton-Okounkov bodies of the two-dimensional Peterson variety.

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

“…which contains 6 elements. From Proposition 3.3 we obtain the formulas Vol(F (1, 1, 1)) = α (3,0,0) + α (2,1,0) + α (2,0,1) + α (1,2,0) + α (1,1,1) Vol(F (2, 1, 1)) = α (2,1,0) + 2α (1,2,0) + α (1,1,1) + 2α (0,3,0) + α (0,2,1)…”

“…Vol(F (1, 2, 1)) = α (2,0,1) + α (1,1,1) + α (1,0,2) Vol(F (3, 1, 1)) = α (2,0,1) + α (1,1,1) + α (1,0,2) Vol(F (2, 2, 1)) = α (1,2,0) + α (1,1,1) + 2α (0,3,0) + 2α (0,2,1) + α (0,1,2)…”

Singular string polytopes and functorial resolutions from Newton–Okounkov bodies

“…, λ n } associated to the word i and the multiplicity list m by setting In this special case we will use the notation (3.9) L i,m := L i (m 1 β1 , · · · , m n βn ). 1 The Killing form is naturally defined on the Lie algebra of G but its restriction to the Lie algebra h of H is positive-definite, so we may identify…”

“…Finally, recent work of e.g. Abe, Horiguchi, Murai, Masuda, Sato [3] and others [1,2] suggest that there are interesting relationships between: the cohomology rings of Hessenberg varieties (which are certain subvarieties of the flag variety) and their associated volume polynomials, on the one hand, and string polytopes and (the volumes of unions of) their faces, on the other hand. Moreover, as in the Schubert calculus considerations of e.g.…”

Geometry of Regular Hessenberg Varieties

“…A regular Hessenberg variety is the one defined by a regular element x in g, namely an element x ∈ g whose Lie algebra centralizer Z g (x) has the minimum possible dimension. There are two cases of regular Hessenberg varieties that are well-studied: (1) if s ∈ g is a regular semisimple element, then X(s, H) is called a regular semisimple Hessenberg variety; (2) if N 0 ∈ g is a regular nilpotent element, then X(N 0 , H) is called a regular nilpotent Hessenberg variety. For example, one can choose a Hessenberg space H so that X(s, H) is a toric variety and X(N 0 , H) is the Peterson variety (see Section 2.1 for details).…”