Springer 簇的 K-理论

The torus equivariant cohomology rings of Springer varieties

The Springer variety of type $A$ associated to a nilpotent operator on $\mathbb{C}^n$ in Jordan canonical form admits a natural action of the $\ell$-dimensional torus $T^{\ell}$ where $\ell$ is the number of the Jordan blocks. We give a presentation of the $T^{\ell}$-equivariant cohomology ring of the Springer variety through an explicit construction of an action of the $n$-th symmetric group on the $T^{\ell}$-equivariant cohomology group. The $T^{\ell}$-equivariant analogue of so called Tanisaki's ideal will appear in the presentation.

Geometry and cohomology of Khovanov-Springer varieties

The Khovanov-Springer variety X(n) is a certain subvariety of the variety of flags of length 2n, which has been studied from various different points of view. We give a new proof of the ring structure of the cohomology of X(n) and relate it to some interesting geometric, combinatorial and algebraic phenomena.

The $S^1$-equivariant cohomology rings of $(n-k,k)$ Springer varieties

The main result of this note gives an explicit presentation of the $S^1$-equivariant cohomology ring of the $(n-k,k)$ Springer variety (in type $A$) as a quotient of a polynomial ring by an ideal $I$, in the spirit of the well-known Borel presentation of the cohomology of the flag variety.

Springer representations on the Khovanov Springer varieties

Springer varieties are studied because their cohomology carries a natural action of the symmetric group $S_n$ and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties $X_n$ as subvarieties of the product of spheres $(S^2)^n$. We show that if $X_n$ is embedded antipodally in $(S^2)^n$ then the natural $S_n$-action on $(S^2)^n$ induces an $S_n$-representation on the image of $H_*(X_n)$. This representation is the Springer representation. Our construction admits an elementary (and geometrically natural) combinatoria…

Springer 簇的等变 $K$-理论

Equivariant $K$-theory of Springer Varieties

本文用生成元和关系描述了与幂零算子关联的 $A$ 型 Springer 簇 $\mathcal{F}_λ$ 的拓扑等变 $K$-环，平行并推广了已有的等变上同调环和普通拓扑 $K$-环的描述。

Poset pinball, GKM-compatible subspaces, and Hessenberg varieties

This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem o…