Related papers: Complexity of the usual torus action on Kazhdan-Lu…

We investigate the toric geometry of two families of generalised determinantal varieties arising from permutations: Matrix Schubert varieties ($\overline{X_w}$) and Kazhdan-Lusztig varieties ($\mathcal{N}_{v,w}$). Matrix Schubert varieties…

$T$-varieties are normal varieties equipped with an action of an algebraic torus $T$. When the action is effective, the complexity of a $T$-variety $X$ is $\dim(X)-\dim(T)$. Matrix Schubert varieties, introduced by Fulton in 1992, are…

Let $B$ be a Borel subgroup of $\mathrm{GL}_n(\mathbb{C})$ and $\mathbb{T}$ a maximal torus contained in $B$. Then $\mathbb{T}$ acts on $\mathrm{GL}_{n}(\mathbb{C})/B$ and every Schubert variety is $\mathbb{T}$-invariant. We say that a…

The goal of this paper is twofold. Firstly, we provide a type-uniform formula for the torus complexity of the usual torus action on a Richardson variety, by developing the notion of algebraic dimensions of Bruhat intervals, strengthening a…

This thesis is devoted to the study of geometric properties of affine algebraic varieties endowed with an action of an algebraic torus. It comes from three preprints which correspond to the indicated points (1), (2), (3). Let $X$ be an…

We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring…

It is well known that many geometric properties of Schubert varieties of type $A$ can be interpreted combinatorially. Given two permutations $w,x\in S_n$ we give a combinatorial consequence of the property that the smooth locus of the…

Let $G$ be a simply connected semi-simple complex algebraic group. Fix a maximal torus $T$ and a Borel subgroup $B$ such that $T\subset B\subset G$. Let $W$ the Weyl group of $G$ relative to $T$. For any $w$ in $W$, let $X_w=\bar {BwB/B}$…

A Kazhdan-Lusztig atlas, introduced by He, Knutson and Lu, on a stratified variety (V,Y) is a way of modeling the stratification Y of V locally on the stratification of Kazhdan-Lusztig varieties X^w_o \cap X_v by their intersection with…

The paper is based on a talk. Complete exposition is given in "Equivariant Hirzebruch class for singular varieties". Starting from the classical theory we describe Hirzebruch class and the related Todd genus of a complex singular algebraic…

Let w_0 denote the permutation [n,n-1,...,2,1]. We give two new explicit formulae for the Kazhdan-Lusztig polynomials P_{w_0w,w_0x} in S_n when x is a maximal element in the singular locus of the Schubert variety X_w. To do this, we utilize…

Given a matrix Schubert variety $\overline{X_\pi}$, it can be written as $\overline{X_\pi}=Y_\pi\times \mathbb{C}^q$ (where $q$ is maximal possible). We characterize when $Y_{\pi}$ is toric (with respect to a $(\mathbb{C}^*)^{2n-1}$-action)…

We propose a method to compute a desingularization of a normal affine variety X endowed with a torus action in terms of a combinatorial description of such a variety due to Altmann and Hausen. This desingularization allows us to study 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…

Tangent spaces to Schubert varieties of type A were characterized by Lakshmibai and Seshadri. This result was extended to the other classical types by Lakshmibai. We give a uniform characterization of tangent spaces to Schubert varieties in…

We introduce certain torus-equivariant classes on permutohedral varieties which we call "tautological classes of matroids" as a new geometric framework for studying matroids. Using this framework, we unify and extend many recent…

We characterize all varieties with a torus action of complexity one that admit iteration of Cox rings.

Let $W$ be a Weyl group, and let $\CT_W$ be the complex toric variety attached to the fan of cones corresponding to the reflecting hyperplanes of $W$, and its weight lattice. The real locus $\CT_W(\R)$ is a smooth, connected, compact…

The thagomizer matroid, realized as the graphic matroid of the complete tripartite graph $K_{1,1,n}$, has full automorphism group isomorphic to the hyperoctahedral group whenever $n \ge 2$. In the equivariant setting for this action, we…

Let X be a normal affine T-variety of complexity at most one over a perfect field k, where T stands for the split algebraic torus. Our main result is a classification of additive group actions on X that are normalized by the T-action. This…