Equivariant closure operators and trisp closure maps

Equivariant K-theory classes of matrix orbit closures

The group $G = GL_r(k) \times (k^\times)^n$ acts on $\mathbf{A}^{r \times n}$, the space of $r$-by-$n$ matrices: $GL_r(k)$ acts by row operations and $(k^\times)^n$ scales columns. A matrix orbit closure is the Zariski closure of a point orbit for this action. We prove that the class of such an orbit closure in $G$ equivariant $K$-theory of $\mathbf{A}^{r \times n}$ is determined by the matroid of a generic point. We present two formulas for this class. The key to the proof is to show that matrix orbit closures have rational singularities.

Equivariant Chow classes of matrix orbit closures

Let $G$ be the product $GL_r(C) \times (C^\times)^n$. We show that the $G$-equivariant Chow class of a $G$ orbit closure in the space of $r$-by-$n$ matrices is determined by a matroid. To do this, we split the natural surjective map from the $G$ equvariant Chow ring of the space of matrices to the torus equivariant Chow ring of the Grassmannian. The splitting takes the class of a Schubert variety to the corresponding factorial Schur polynomial, and also has the property that the class of a subvariety of the Grassmannian is mapped to the class of the closure of those matrices whose row span is …

对称箭图轨道闭包的等变几何

Equivariant geometry of symmetric quiver orbit closures

本文将有限型对称箭图表示簇的等变几何问题与对称簇GL(n)/K的相应问题统一，通过构造显式嵌入转换了奇点、组合学及等变上同调与K理论的结果。

K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles splitting as direct sums

We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$ for various symmetric pairs $(G,K)$. In type $A$, we realize the closures of $K=GL(p,\C) \times GL(q,\C)$-orbits on $GL(p+q,\C)/B$ as universal degeneracy loci for a vector bundle over a variety which is equipped with a single flag of subbundles and which splits as a direct sum of subbundles of ranks $p$ and $q$. The precise description of such a degeneracy locus relies upon knowing a set-theoretic descri…