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 a fixed linear operator M of the full flags V_1 \subsetneq V_2 >... \subsetneq V_n in GL_n with M V_i contained in V_{i+1} for all i.
In this paper I show that all Hessenberg varieties in type A_n and semisimple and regular nilpotent Hessenberg varieties in types B_n,C_n, and D_n can be paved by affine spaces. Moreover, this paving is the intersection of a particular Bruhat decomposition with the Hessenberg variety. In type A_n, an equivalent description of the cells of the paving in terms of certain fillings of a Young diagram can be used to compute the Betti numbers of Hessenberg varieties. As an example, I show that the Poincare polynomial of the Peterson variety in A_n is \sum_{i =0}^{n-1} \binom{n-1}{i} x^{2i}.
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 and vice versa. Our proof explicitly identifies these Schubert varieties by their permutation and computes their dimension. We use this to answer an open question by proving that Hessenberg varieties are not always pure dimensional. We give examples that neither semisimple nor nilpotent Hessenberg varieties need be pure; the latter are connected, non-pure-dime…
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 the Lie algebra $\mathfrak{g}$ and a Hessenberg subspace $H\subseteq \mathfrak{g}$. This paper considers when two Hessenberg spaces define the same Hessenberg variety when paired with $X$. To answer this question we present the containment poset $\mathcal{P}_X$ of type $A$ Hessenberg varieties with a fixed first parameter $X$ and prove directly that if $X$ i…
Recently Brosnan and Chow have proven a conjecture of Shareshian and Wachs describing a representation of the symmetric group on the cohomology of regular semisimple Hessenberg varieties for $GL_n(\mathbb{C})$. A key component of their argument is that the Betti numbers of regular Hessenberg varieties for $GL_n(\mathbb{C})$ are palindromic. In this paper, we extend this result to all reductive algebraic groups, proving that the Betti numbers of regular Hessenberg varieties are palindromic.
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 a theorem of Tymoczko to this setting, showing that these varieties have an affine paving obtained by intersecting with Schubert cells. Our method of proof constructs an an affine paving for each Springer fiber that restricts to an affine paving of the Hessenberg variety. We use the combinatorial properties of this paving to prove that Hessenberg varieties o…
It is well-known that the $T$-fixed points of a Schubert variety in the flag variety $GL_n(\mathbb{C})/B$ can be characterized purely combinatorially in terms of Bruhat order on the symmetric group $\mathfrak{S}_n$. In a recent preprint, Cho, Hong, and Lee give a combinatorial description of the $T$-fixed points of Hessenberg analogues of Schubert varieties (which we call Hessenberg Schubert varieties) in a regular semisimple Hessenberg variety. This note gives an interpretation of their result in terms of Bruhat order by making use of a partition of the symmetric group defined using so-called…
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 main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, and then show that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain families of Hessenberg varieties, whose generic fibers are regul…
This paper studies the geometry and combinatorics of three interrelated varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg varieties. We prove that each parabolic Hessenberg variety is the pullback of a Steinberg variety under the projection of the flag variety to an appropriate partial flag variety and we give three applications of this result. The first application constructs an explicit paving of all Steinberg varieties in Lie type $A$ in terms of semistandard tableaux. As a result, we obtain an elementary proof of a theorem of Steinberg and Shimomura that the well-kn…
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, we provide pattern avoidance criteria implying that the proportion of Schubert varieties that are adjoint Hessenberg varieties approaches zero as the rank of $G$ increases. We show also that in type A, some Schubert varieties are not isomorphic to any adjoint Hessenberg variety.
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 the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert ba…
This article surveys recent developments on Hessenberg varieties, emphasizing some of the rich connections of their cohomology and combinatorics. In particular, we will see how hyperplane arrangements, representations of symmetric groups, and Stanley's chromatic symmetric functions are related to the cohomology rings of Hessenberg varieties. We also include several other topics on Hessenberg varieties to cover recent developments.
In this paper we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We give a connectedness criterion for semisimple Hessenberg varieties generalizing a criterion given by Anderson and Tymoczko. We show that nilpotent Hessenberg varieties are rationally connected.
Regular semisimple Hessenberg varieties are smooth subvarieties of the flag variety, and their examples contain the flag variety itself and the permutohedral variety which is a toric variety. We give a complete classification of Fano and weak Fano regular semisimple Hessenberg varieties in type A in terms of combinatorics of Hessenberg functions. In particular, we show that if the anti-canonical bundle of a regular semisimple Hessenberg variety is nef, then it is in fact nef and big.
Although regular semisimple Hessenberg varieties are smooth and irreducible, semisimple Hessenberg varieties are not necessarily smooth in general. In this paper we determine the irreducible components of semisimple Hessenberg varieties corresponding to the standard Hessenberg space. We prove that these irreducible components are smooth and give an explicit description of their intersections, which constitute the singular locus. We conclude with an example of a semisimple Hessenberg variety corresponding to another Hessenberg space which is singular and irreducible, showing that results of thi…
Regular nilpotent Hessenberg varieties form an important family of subvarieties of the flag variety, which are often singular and sometimes not normal varieties. Like Schubert varieties, they contain distinguished points called permutation flags. In this paper, we give a combinatorial characterization for a permutation flag of a regular nilpotent Hessenberg variety to be a singular point. We also apply this result to characterize regular nilpotent Hessenberg varieties which are normal algebraic varieties.
We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson. We give an inductive formula for the $S_n$-representation on the cohomology of an abelian regular semisimple Hessenberg variety with respect to the action defined by Tymoczko. Our result implies that a graded version of the Stanley-Stembridge conjecture holds in the abelian case, and generalizes results obtained by Shareshian-Wachs and Teff. Our proof uses previous work of Stanley, Gasharov, Shareshian-Wachs, and Brosnan-…
In this paper we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular nilpotent and arbitrary elements of \mathfrak{gl}_n(\C) are paved by affines. For example, our results prove that any Hessenbe…
In this paper we study a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials $f_{i,j}$ were introduced by Abe-Harada-Horiguchi-Masuda. We show that every polynomial $f_{i,j}$ is an alternating sum of certain Schubert polynomials.
We consider bases for the cohomology space of regular semisimple Hessenberg varieties, consisting of the classes that naturally arise from the Bialynicki-Birula decomposition of the Hessenberg varieties. We give an explicit combinatorial description of the support of each class, which enables us to compute the symmetric group actions on the classes in our bases. We then successfully apply the results to the permutohedral varieties to explicitly write down each class and to construct permutation submodules that constitute summands of a decomposition of cohomology space of each degree. This reso…
Hessenberg varieties $\mathcal{H}(X,H)$ form a class of subvarieties of the flag variety $G/B$, parameterized by an operator $X$ and certain subspaces $H$ of the Lie algebra of $G$. We identify several families of Hessenberg varieties in type $A_{n-1}$ that are $T$-stable subvarieties of $G/B$, as well as families that are invariant under a subtorus $K$ of $T$. In particular, these varieties are candidates for the use of equivariant methods to study their geometry. Indeed, we are able to show that some of these varieties are unions of Schubert varieties, while others cannot be such unions. Amo…
The solution of Shareshian-Wachs conjecture by Brosnan-Chow linked together the cohomology of regular semisimple Hessenberg varieties and graded chromatic symmetric functions on unit interval graphs. On the other hand, it is known that unicellular LLT polynomials have similar properties to graded chromatic symmetric functions. In this paper, we link together the unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties introduced by Ayzenberg-Buchstaber. We prove their palindromicity from topological viewpoint. We also show that modules of a symmetric group generated by f…
Regular semisimple Hessenberg varieties admit actions of associated Weyl groups on their cohomology space of each degree. In this paper, we consider the module structure of the cohomology spaces of regular semisimple Hessenberg varieties of type $A$. We define a subset of the Bialynicki-Birula basis of the cohomology space so that they become a module generator set of the cohomology module of each degree. We then use those generators to construct permutation submodules of the degree two cohomology module and show that they form a permutation module decomposition. Our construction is consistent…
The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this variety's cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action. Sixteen years later, Garsia and Procesi made Springer's work more transparent and accessible by presenting the cohomology ring as a graded quotient of a polynomial ring. They combinatorially describe an explicit basis for this quotient. The goal of this paper is to generalize their work. Our main result deepens their analysis of Springer varieties and extend…
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 analysis. We describe completely the homology of Hessenberg varieties over GL_n(C) and show that they have no odd-dimensional homology. We provide an explicit geometric construction which partitions each Hessenberg variety into pieces homeomorphic to affine space. We characterize these affine pieces by fillings of Young tableaux and show that the dimension of the af…
For an endomorphism $s:V\rightarrow V$ of a finite dimensional complex vector space and an action of a torus $T$ on the full flag variety $\text{GL}_n({\mathbb C})/B$, we give a description of its fixed point set when $s$ is semisimple or regular nilpotent. We also compute the one dimensional orbits of this action on the Hessenberg subvariety $\text{Hes}(s,h)\subseteq \text{GL}_n({\mathbb C})/B$ for any Hessenberg function $h$. For the action of the one dimensional torus $S$ and a regular nilpotent endomorphism $N:V\rightarrow V$, we give a new computation of the equivariant cohomology of the …
We use the Springer correspondence to give a partial characterization of the irreducible representations which appear in the Tymoczko dot-action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. In type A, we apply these techniques to prove that all irreducible summands which appear in the pushforward of the constant sheaf on the universal Hessenberg family have full support. We also observe that the recent results of Brosnan and Chow, which apply the local invariant cycle theorem to the family of regular Hessenberg varieties in type A, extend to arbitrary Li…
For a connected, simply-connected complex simple algebraic group $G$, we examine a class of Hessenberg varieties associated with the minimal nilpotent orbit. In particular, we compute the Poincaré polynomials and irreducible components of these varieties in Lie type $A$. Furthermore, we show these Hessenberg varieties to be GKM with respect to the action of a maximal torus $T\subseteq G$. The corresponding GKM graphs are then explicitly determined. Finally, we present the ordinary and $T$-equivariant cohomology rings of our varieties as quotients of those of the flag variety.
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 using the geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in [CVX2…
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 $\mathfrak{g}$. We take a Hessenberg space so that $X(x,H)$ is irreducible, and show that the higher cohomology groups of the structure sheaf of $X(x,H)$ vanish. We also study the flat family of regular Hessenberg varieties, and prove that the scheme-theoretic fibers over the closed points are reduced. We include applications of these results as well.
In this manuscript we develop the theory of poset pinball, a combinatorial game recently introduced by Harada and Tymoczko for the study of the equivariant cohomology rings of GKM-compatible subspaces of GKM spaces. Harada and Tymoczko also prove that in certain circumstances, a successful outcome of Betti poset pinball yields a module basis for the equivariant cohomology ring of the GKM-compatible subspace. Our main contributions are twofold. First we construct an algorithm (which we call the dimension pair algorithm) which yields the result of a successful outcome of Betti poset pinball for …
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 sequence of varieties which we call the prepermutohedral varieties. We first determine the toric structure of these varieties and compute the Euler characteristics and the Betti numbers using the theory of toric varieties. Then, we describe the cohomology of these varieties. We also find a natural way to encode the one-dimensional components of the cohomolog…
We consider generalizations of the Springer resolution of the nilpotent cone of a simple Lie algebra by replacing the cotangent bundle with certain other vector bundles over the flag variety. We show that the analogue of the Springer sheaf has as direct summands only intersection cohomology sheaves that arise in the Springer correspondence. The fibers of these general maps are nilpotent Hessenberg varieties, and we build on techniques established by De Concini, Lusztig, and Procesi to study their geometry. For example, we show that these fibers have vanishing cohomology in odd degrees. This le…
Let $n$ be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $GL(n,{\mathbb{C}})/B$ such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $GL(n-1,{\mathbb{C}})/B$, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations…
Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety arising in the study of quantum cohomology, geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent Hessenberg varieties for all classical types, generalizing results of de Concini-Lusztig-Procesi and Kostant. This paving is in fact the intersection of a particular Bruhat decomposition with the Hessenberg variety. The nonempty cells of the paving and their dimensions are identified by combinatorial conditions on roots. We use the paving t…
Motivated by a 1993 conjecture of Stanley and Stembridge, Shareshian and Wachs conjectured that the characteristic map takes the dot action of the symmetric group on the cohomology of a regular semisimple Hessenberg variety to $ωX_G(t)$, where $X_G(t)$ is the chromatic quasisymmetric function of the incomparability graph $G$ of the corresponding natural unit interval order, and $ω$ is the usual involution on symmetric functions. We prove the Shareshian--Wachs conjecture. Our proof uses the local invariant cycle theorem of Beilinson-Bernstein-Deligne to obtain a surjection from the cohomology o…
Recent work of Shareshian and Wachs, Brosnan and Chow, and Guay-Paquet connects the well-known Stanley-Stembridge conjecture in combinatorics to the dot action of the symmetric group $S_n$ on the cohomology rings $H^*(Hess(S,h))$ of regular semisimple Hessenberg varieties. In particular, in order to prove the Stanley-Stembridge conjecture, it suffices to construct (for any Hessenberg function $h$) a permutation basis of $H^*(Hess(S,h))$ whose elements have stabilizers isomorphic to Young subgroups. In this manuscript we give several results which contribute toward this goal. Specifically, in s…
We consider effective actions of a compact torus $T^{n-1}$ on an even-dimensional smooth manifold $M^{2n}$ with isolated fixed points. We prove that under certain conditions on weights of tangent representations, the orbit space is a manifold with corners. Given that the action is Hamiltonian, the orbit space is homeomorphic to $S^{n+1} \setminus (U_1 \sqcup \ldots \sqcup U_l)$ where $S^{n+1}$ is the $(n+1)$--sphere and $U_1, \ldots, U_l$ are open domains. We apply the results to regular Hessenberg varieties and manifolds of isospectral Hermitian matrices of staircase form.
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 $\mathrm{GZ}(λ)$ for $λ=(λ_1,λ_2,\ldots,λ_n)$. The main results of this manuscript use and generalize tools developed by Anderson-Tymoczko, Kiritchenko-Smirnov-Timorin, and Postnikov, in orde…
This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties. We fix a simply-connected complex semisimple linear algebraic group $G$ and restrict our attention to a particular family of Hessenberg varieties, …
We describe the second cohomology of a regular semisimple Hessenberg variety by generators and relations explicitly in terms of GKM theory. The cohomology of a regular semisimple Hessenberg variety becomes a module of a symmetric group $\mathfrak{S}_n$ by the dot action introduced by Tymoczko. As an application of our explicit description, we give a formula describing the isomorphism class of the second cohomology as an $\mathfrak{S}_n$-module. Our formula is not exactly the same as the known formula by Chow or Cho-Hong-Lee but they are equivalent. We also discuss its higher degree generalizat…
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 cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and $K$-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends o…
We investigate the cohomology rings of regular semisimple Hessenberg varieties whose Hessenberg functions are of the form $h=(h(1),n\dots,n)$ in Lie type $A_{n-1}$. The main result of this paper gives an explicit presentation of the cohomology rings in terms of generators and their relations. Our presentation naturally specializes to Borel's presentation of the cohomology ring of the flag variety and it is compatible with the representation of the symmetric group $\mathfrak{S}_n$ on the cohomology constructed by J. Tymoczko. As a corollary, we also give an explicit presentation of the $\mathfr…
This paper describes a paving by affines for regular nilpotent Hessenberg varieties in all Lie types, namely a kind of cell decomposition that can be used to compute homology despite its weak closure conditions. Precup recently proved a stronger result; we include ours because we use different methods. We then use this paving to prove that the homology of the Peterson variety injects into the homology of the full flag variety. The proof uses intersection theory and expands the class of the Peterson variety in the homology of the flag variety in terms of the basis of Schubert classes. We explic…
