Related papers: Hessenberg varieties and hyperplane arrangements

Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction to grade one regular semisimple elements from non-equivalent Heisenberg subalgebras of a loop algebra $\G\otimes{\bf C}[\lambda,\lambda^{-1}]$ are studied. The graded…

It is conjectured (following the Stanley-Stembridge conjecture) that the cohomology rings of regular semisimple Hessenberg varieties yield permutation representations, but the decompositions of the modules are only known in some cases. For…

Let $B$ denote the upper triangular subgroup of $SL_2(C)$, $T$ its diagonal torus and $U$ its unipotent radical. A complex projective variety $Y$ endowed with an algebraic action of $B$ such that the fixed point set $Y^U$ is a single point,…

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability…

As an associative algebra, the Heisenberg-Weyl algebra $\mathcal{H}$ is generated by two elements $A$, $B$ subject to the relation $AB-BA=1$. As a Lie algebra, however, where the usual commutator serves as Lie bracket, the elements $A$ and…

We introduce generalized Hessenberg varieties and establish basic facts. We show that the Tymoczko action of the symmetric group $S_n$ on the cohomology of Hessenberg varieties extends to generalized Hessenberg varieties and that natural…

In this manuscript we consider the extent to which an irreducible representation for a reductive Lie group can be realized as the sheaf cohomolgy of an equivariant holomorphic line bundle defined on an open invariant submanifold of a…

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…

We construct an invariant of closed oriented $3$-manifolds using a finite dimensional, involutory, unimodular and counimodular Hopf algebra $H$. We use the framework of normal o-graphs introduced by R. Benedetti and C. Petronio, in which…

We study the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its…

Let $G$ be a connected semi-simple group defined over and algebraically closed field, $T$ a fixed Cartan, $B$ a fixed Borel containing $T$, $S$ a set of simple reflections associated to the simple positive roots corresponding to $(T,B)$,…

Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C|=BC. We study the ring homomorphism HH*(kC) --> H*(|C|,k) and prove it is split surjective. This…

We prove the additive version of the conjecture proposed by Ginzburg and Kaledin. This conjecture states that if X/G is an orbifold modeled on a quotient of a smooth affine symplectic variety X (over C) by a finite group G\subset Aut(X) and…

We aim in this manuscript to describe a specific notion of geometric positivity that manifests in cohomology rings associated to the flag variety $G/B$ and, in some cases, to subvarieties of $G/B$. We offer an exposition on the the…

An affine Hecke algebras can be realized as an equivariant K-group of the corresponding Steinberg variety. This gives rise naturally to some two-sided ideals of the affine Hecke algebra by means of the closures of nilpotent orbits of the…

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…

The Poincar\'e polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space $G/B$, while the $h$-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety.…

Let $G\subset SO(4)$ denote a finite subgroup containing the Heisenberg group. In these notes we classify all these groups, we find the dimension of the spaces of $G$-invariant polynomials and we give equations for the generators whenever…

For any connected complex reductive group $G$ and element $z$ of its Weyl group $W$, we use work of Lusztig and Abreu-Nigro to compute the graded $W$-character of the intersection cohomology of any closed Lusztig variety for $z$ over the…

Let $G$ be a simple complex Lie group, $\alg{g}$ be its Lie algebra, $K$ be a maximal compact form of $G$ and $\alg{k}$ be a Lie algebra of $K$. We denote by $X\rightarrow \overline{X}$ the anti-involution of $\alg{g}$ which singles out the…