A note on Hessenberg varieties

Fano and weak Fano Hessenberg varieties

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.

广义 Hessenberg 簇的双有理几何与广义 Shareshian-Wachs 猜想

Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture

本文引入广义 Hessenberg 簇，证明对称群作用的扩展性，通过双有理几何分析给出 Shareshian-Wachs 猜想及其推广的初等证明，并提出两种计算上同调表示的算法。

The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture

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-…

Geometry of regular Hessenberg varieties

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.

The Geometry and Combinatorics of Some Hessenberg Varieties Related to the Permutohedral Variety

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…

Bases of the equivariant cohomologies of regular semisimple Hessenberg varieties

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, Slodowy slices, and integrable systems

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, …

The second cohomology of regular semisimple Hessenberg varieties from GKM theory

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…

The cohomology rings of regular semisimple Hessenberg varieties for $h=(h(1),n,\ldots,n)$

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…

正则幂零 Hessenberg 簇上同调环的一组加法基

An additive basis for the cohomology rings of regular nilpotent Hessenberg varieties

本文通过延拓更小正则幂零 Hessenberg 子簇的 Poincaré 对偶，构造了正则幂零 Hessenberg 簇上同调环的加法基，并证明这些对偶线性无关。

正则半单Hessenberg簇的双流形几何与单细胞LLT多项式

Geometry of the twin manifolds of regular semisimple Hessenberg varieties and unicellular LLT polynomials

本文研究正则半单Hessenberg簇双流形的几何，证明二者通过显式爆破与纤维丛映射关联，取上同调后直接证明模律，确立LLT型Shareshian-Wachs猜想。

Hessenberg varieties and Poisson slices

This work pursues a circle of Lie-theoretic ideas involving Hessenberg varieties, Poisson geometry, and wonderful compactifications. In more detail, one may associate a symplectic Hamiltonian $G$-variety $μ:G\times\mathcal{S}\longrightarrow\mathfrak{g}$ to each complex semisimple Lie algebra $\mathfrak{g}$ with adjoint group $G$ and fixed Kostant section $\mathcal{S}\subseteq\mathfrak{g}$. This variety is one of Bielawski's hyperkähler slices, and it is central to Moore and Tachikawa's work on topological quantum field theories. It also bears a close relation to two log symplectic Hamiltonian …

分解 Hessenberg 簇的上同调与染色对称函数的 e-正性

Splitting the cohomology of Hessenberg varieties and e-positivity of chromatic symmetric functions

本文利用遗忘映射的分解定理拆分正则半单 Hessenberg 簇上同调，组合刻画各部分 Frobenius 特征，并证明无差别图染色对称函数中长度为 2 的 $e_λ$ 系数非负。

反向黑森伯格卷积簇的不动点

Fixed points of reverse Hessenberg convolution varieties

本研究针对Syu Kato引入的、与单位区间图相关且通过几何Satake对应给出特征为图的色拟对称多项式的分次GL_m表示的簇，通过分析其不动点重新证明了Kato的相关结论。

Perverse sheaves, nilpotent Hessenberg varieties, and the modular law

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…

Unit Interval Orders and the Dot Action on the Cohomology of Regular Semisimple Hessenberg Varieties

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…

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

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…

Hessenberg varieties and hyperplane arrangements

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 semisimple Hessenberg variety $\mbox{Hess}(S,I)$. We show that a certain graded ring derived from the logarithmic derivation module of $\mathcal{A}_I$ is isomorphic to $H^*(\mbox{Hess}(N,I))$ and $H^*(\mbox{Hess}(S,I))^W$, the invariants in $H^*(\mbox{Hess}(S,I))$ under an action of the Weyl group $W$ of $G$. This isomorphism is shown for general Lie type, an…

正则半单Lusztig变种的几何学

Geometry of regular semisimple Lusztig varieties

研究正则半单Lusztig变种的几何性质及其与Hessenberg变种的关系，证明其上同构性质及平坦退化结构。

A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra

This is a set of working notes which give a second proof of the Shareshian--Wachs conjecture, the first (and recent) proof being by Brosnan and Chow in November 2015. The conjecture relates some symmetric functions constructed combinatorially out of unit interval graphs (their $q$-chromatic quasisymmetric functions), and some symmetric functions constructed algebro-geometrically out of Tymoczko's representation of the symmetric group on the equivariant cohomology ring of a family of subvarieties of the complex flag variety, called regular semisimple Hessenberg varieties. Brosnan and Chow's pro…

Springer correspondence, hyperelliptic curves, and cohomology of Fano varieties

In \cite{CVX3}, we have established a Springer theory for the symmetric pair $(\operatorname{SL}(N),\operatorname{SO}(N))$. In this setting we obtain representations of (the Tits extension) of the braid group rather than just Weyl group representations. These representations arise from cohomology of families of certain (Hessenberg) varieties. In this paper we determine the Springer correspondence explicitly for IC sheaves supported on order 2 nilpotent orbits. In this process we encounter universal families of hyperelliptic curves. As an application we calculate the cohomolgy of Fano varieties…

Springer correspondence for the split symmetric pair in type $A$

In this paper we establish Springer correspondence for the symmetric pair $(\mathrm{SL}(N),\mathrm{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaves construction due to Grinberg. As applications, we obtain results on cohomology of Hessenberg varieties and geometric constructions of irreducible representations of Hecke algebras of symmetric groups at $q=-1$.

Upper-triangular linear relations on multiplicities and the Stanley-Stembridge conjecture

In 2015, Brosnan and Chow, and independently Guay-Paquet, proved the Shareshian-Wachs conjecture, which links the Stanley-Stembridge conjecture in combinatorics to the geometry of Hessenberg varieties through Tymoczko's permutation group action on the cohomology ring of regular semisimple Hessenberg varieties. In previous work, the authors exploited this connection to prove a refined (graded) version of the Stanley-Stembridge conjecture in a special case. In this manuscript, we derive a new set of linear relations satisfied by the multiplicities of certain permutation representations in Tymocz…

通过 GKM 理论的模律

Modular law through GKM theory

本文通过 GKM 理论，为正则半单 Hessenberg 簇等几何对象的模律提供了初等且更简单的证明，进而支持了 Shareshian-Wachs 猜想等相关结论。

Note on affine Deligne-Lusztig varieties

This note is based on my talk at ICCM 2013, Taipei. We give an exposition of the group-theoretic method and recent results on the questions of non-emptiness and dimension of affine Deligne-Lusztig varieties in affine flag varieties.

A note on the canonical divisor of quasi-homogeneous affine algebraic varieties

We give a necessary and sufficient condition for the canonical divisor to vanish on a quasi-homogeneous affine algebraic variety.

关于辛流形的泊松变形注释

Notes on Poisson deformations of symplectic varieties

本文对先前关于辛流形泊松变形的研究进行了修正和补充。

Almost commuting varieties for symplectic Lie algebras

In this note we introduce and study the almost commuting varieties for the symplectic Lie algebras.

A note on the Schur-finiteness of linear sections

Making use of the recent theory of noncommutative motives, we prove that Schur-finiteness in the setting of Voevodsky's mixed motives is invariant under homological projective duality. As an application, we show that the mixed motives of smooth linear sections of certain (Lagrangian) Grassmannians, spinor varieties, and determinantal varieties, are Schur-finite. Finally, we upgrade our applications from Schur-finiteness to Kimura-finiteness.

A note on L-packets and abelian varieties over local fields

A polarized abelian variety (X,λ) of dimension g over a local field K determines an admissible representation of GSpin_{2g+1}(K). We show that the restriction of this representation to Spin_{2g+1}(K) is reducible if and only if X is isogenous to its twist by the quadratic unramified extension of K. When g=1 and K = Q_p, we recover the well-known fact that the admissible GL_2(K) representation attached to an elliptic curve E is reducible upon restriction to SL_2(K) if and only if E has supersingular reduction.

Invariant tubular neighborhood theorem for affine varieties

The aim of this note is to prove the algebraic geometry analogue of the Invariant tubular neighborhood theorem which concerns the actions of compact Lie groups on smooth manifolds.