Related papers: Hessenberg varieties of parabolic type
We study geometric and topological properties of Hessenberg varieties of codimension one in the type A flag variety. Our main results: (1) give a formula for the Poincar\'e polynomial, (2) characterize when these varieties are irreducible,…
We study a family of subvarieties of the flag variety defined by certain linear conditions, called Hessenberg varieties. We compare them to Schubert varieties. We prove that some Schubert varieties can be realized as Hessenberg varieties…
Hessenberg varieties are a family of subvarieties of the flag variety, including the Springer fibers, the Peterson variety, and the entire flag variety itself. The seminal example arises from a problem in numerical analysis and consists for…
Hessenberg varieties are a family of subvarieties of full flag varieties. This family contains well-known varieties such as Springer fibers, Peterson varieties, and permutohedral varieties. It was introduced by De Mari-Procesi-Shayman in…
In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof, with the additional goal of laying the groundwork for future computations of Newton-Okounkov bodies of Hessenberg varieties. Our…
We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical…
This paper investigates the geometry of regular Hessenberg varieties associated with the minimal indecomposable Hessenberg space in the flag variety of a complex reductive group. These varieties form a flat family of irreducible…
We study a collection of Hessenberg varieties in the type A flag variety associated to a nonzero semisimple matrix whose conjugacy class has minimal dimension. We prove each such minimal semisimple Hessenberg variety is a union Richardson…
Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis…
In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail…
Let $\mathfrak{g}$ be a complex semisimple Lie algebra. For a regular element $x$ in $\mathfrak{g}$ and a Hessenberg space $H\subseteq \mathfrak{g}$, we consider a regular Hessenberg variety $X(x,H)$ in the flag variety associated with…
We consider Hessenberg varieties in the flag variety of $GL_n(\mathbb{C})$ with the property that the corresponding Hessenberg function defines an ad-nilpotent ideal. Each such Hessenberg variety is contained in a Springer fiber. We extend…
Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a "Giambelli formula" expressing…
Flag varieties are well-known algebraic varieties with many important geometric, combinatorial, and representation theoretic properties. A Hessenberg variety is a subvariety of a flag variety identified by two parameters: an element $X$ of…
After proving that every Schubert variety in the full flag variety of a complex reductive group $G$ is a general Hessenberg variety, we show that not all such Schubert varieties are adjoint Hessenberg varieties. In fact, in types A and C,…
Given a semisimple complex linear algebraic group $G$ and a lower ideal $I$ in positive roots of $G$, three objects arise: the ideal arrangement $\mathcal{A}_I$, the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$, and the regular…
This survey paper describes Springer fibers, which are used in one of the earliest examples of a geometric representation. We will compare and contrast them with Schubert varieties, another family of subvarieties of the flag variety that…
We construct a concrete isomorphism from the permutohedral variety to the regular semisimple Hessenberg variety associated to the Hessenberg function $h_+(i)=i+1$, $1\le i\le n-1$. In the process of defining the isomorphism, we introduce a…
This article explores the relationship between Hessenberg varieties associated with semisimple operators with two eigenvalues and orbit closures of a spherical subgroup of the general linear group. We establish the specific conditions under…
Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator $X$ and a nondecreasing function $h$. The family of Hessenberg varieties for regular $X$ is particularly important: they are used in quantum…