Related papers: Equivariant Chow 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…

Let $G$ be the group $GL_r(C) \times (C^\times)^n$. We conjecture that the finely-graded Hilbert series of a $G$ orbit closure in the space of $r$-by-$n$ matrices is wholly determined by the associated matroid. In support of this, we prove…

We compute the $GL_{r+1}$-equivariant Chow class of the $GL_{r+1}$-orbit closure of any point $(x_1, \ldots, x_n) \in (\mathbb{P}^r)^n$ in terms of the rank polytope of the matroid represented by $x_1, \ldots, x_n \in \mathbb{P}^r$. Using…

We give a presentation for the (integral) torus-equivariant Chow ring of the quot scheme, a smooth compactification of the space of rational curves of degree d in the Grassmannian. For this presentation, we refine Evain's extension of the…

Let $\mathbb{G}(d,n)$ be the complex Grassmannian of affine $d$-planes in $n$-space. We study the problem of characterizing the set of algebraic subvarieties of $\mathbb{G}(d,n)$ invariant under the action of the maximal torus $T$ and…

The space of all pencils of conics in the plane $\mathbb{P} V$ (where $\dim V = 3$) is a projective Grassmannian $\mathbb{G} (1, \mathbb{P} \mathrm{Sym}^2 V^*)$ and admits a natural $\mathrm{PGL}(V)$ action. It is a classical theorem that…

The Chow class of the closure of the torus orbit of a point in a Grassmannian only depends on the matroid associated to the point. The Chow class can be extended to a matroid invariant of arbitrary matroids. We call the coefficients…

Let $G/P$ be a complex cominuscule flag manifold. We prove a type independent formula for the torus equivariant Mather class of a Schubert variety in $G/P$, and for a Schubert variety pulled back via the natural projection $G/Q \to G/P$. We…

Let $G = GL(n)$ and $K = GL(p) \times GL(q)$ with $p+q=n$, where the groups are taken over $\C$. In this paper we study a certain family of $K$-orbit closures on the flag variety $X$ of $G$. The geometry of these orbit closures plays a…

It is known that for the natural algebraic torus actions on the Grassmannians, the closures of torus orbits are toric varieties, and that these toric varieties are smooth if and only if the corresponding matroid polytopes are simple. We…

The Chow ring of a matroid (or more generally, atomic latice) is an invariant whose importance was demonstrated by Adiprasito, Huh and Katz, who used it to resolve the long-standing Heron-Rota-Welsh conjecture. Here, we make a detailed…

Consider an integer associated with every subset of the set of columns of an $n\times k$ matrix. The collection of those matrices for which the rank of a union of columns is the predescribed integer for every subset, will be denoted by…

We introduce a certain compactification of the space of projective configurations i.e. orbits of the group $PGL(k)$ on the space of $n$ - tuples of points in $P^{k-1}$ in general position. This compactification differs considerably from…

Let $T$ be a torus acting on $\CC^n$ in such a way that, for all $1\leq k\leq n$, the induced action on the grassmannian $G(k,n)$ has only isolated fixed points. This paper proposes a natural, elementary, explicit description of the…

The closure of a generic torus orbit in the flag variety $G/B$ of type $A_{n-1}$ is known to be a permutohedral variety and well studied. In this paper we introduce the notion of a generic torus orbit in the Schubert variety $X_w$ $(w\in…

We study the Chow classes of arbitrary matroids in the Grassmannian. We develop a new combinatorial approach for computing them, by first focusing on snake matroids and then extending our results via valuativity to any matroid. Our main…

The main classical result of Schubert calculus is that multiplication rules for the basis of Schubert cycles inside the cohomology ring of the Grassmannian $G(n,m)$ are the same as multiplication rules for the basis of Schur polynomials in…

We give explicit formulas for torus-equivariant fundamental classes of closed $K$-orbits on the flag variety $G/B$ when $G$ is one of the classical groups $SL(n,\C)$, $SO(n,\C)$, or $Sp(2n,\C)$, and $K$ is a symmetric subgroup of $G$. We…

Let X be a smooth projective variety with the action of the n dimensional torus. The article describes the moduli space of torus equivariant morphisms from stable toric varieties into X as the inverse limit of the GIT quotients of X and…

Extending results of Wyser, we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of $GL_n$ on the flag variety $GL_n/B$. Putting this together with a slight extension of…