Related papers: Total positivity in Springer fibres

We study the intersection of the totally positive part of a split semisimple group over the real numbers with a totally positive parabolic subgroup.

In this paper we reduce the problem of counting the number of connected components in the intersection of two opposite open Schubert cells in the variety of real complete flags to a purely combinatorial question of counting the number of…

We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in $GL_n$ form a real semi-algebraic cell of dimension $n-1$. Furthermore we prove a natural cell decomposition for its closure. The proof uses properties…

The existence of acyclic complete matchings on the face poset of a regular CW complex implies that the underlying topological space of the CW complex is contractible by discrete Morse theory. In this paper, we construct explicitly acyclic…

Generalizing Schubert cells in type A and a cell decomposition if Springer fibres in type A found by L. Fresse we prove that varieties of complete flags in nilpotent representations of an oriented cycle admit an affine cell decomposition…

We show that the totally nonnegative part of the twisted product of flag varieties of a Kac-Moody group admits a cellular decomposition, and the closure of each cell is a topological manifold with boundary. We also establish explicit…

In this paper we prove a direct geometric relation between Deligne--Lusztig varieties and Springer fibres in type $\mathsf{A}$: For any rational unipotent element, the Springer fibre cuts out a unique component of a specific…

We compute the Euler characteristics of the individual connected components of the intersection of two opposed big cells in the real flag variety of type $G_2$, verifying a conjecture of Rietsch.

Let a split element of a connected semisimple Lie group act on one of its flag manifolds. We prove that each connected set of fixed points of this action is itself a flag manifold. With this we can obtain the generalized Bruhat…

Let $u\in\mathrm{End}(\mathbb{C}^n)$ be nilpotent. The variety of $u$-stable complete flags is called the Springer fiber over $u$. Its irreducible components are parameterized by a set of standard Young tableaux. The Richardson (resp.…

The Peterson variety is a subvariety of the flag manifold $G/B$ equipped with an action of a one-dimensional torus, and a torus invariant paving by affine cells, called Peterson cells. We prove that the equivariant pull-backs of Schubert…

We study the nonnegative part \bar{G_{>0}} of the De Concini-Procesi compactification of a reductive algebraic group G, as defined by Lusztig. Using positivity properties of the canonical basis and parametrization of flag varieties, we will…

We show that a compact manifold admitting a Killing foliation with positive transverse curvature fibers over finite quotients of spheres or weighted complex projective spaces, provided that the singular foliation defined by the closures of…

Double Bruhat cells in a semisimple group are intersections of cells in two Bruhat decompositions corresponding to two opposite Borel subgroups. They form a geometric framework for the study of total positivity in semisimple groups; they…

We study the nonnegative part B_{\ge 0} of the flag variety of a reductive algebraic group G, as defined by Lusztig. Using positivity properties of the canonical basis it is shown that B_{\ge 0} has an algebraic cell decomposition indexed…

The totally nonnegative part of a partial flag variety G/P is known to have a decomposition into semi-algebraic cells. We show that the closure of a cell is again a union of cells and give a combinatorial description of the closure…

Let $G$ be a connected reductive group over an algebraically closed field $k$, and let $Fl$ be the affine flag variety of $G$. For every regular semisimple element $\gamma$ of $G(k((t)))$, the affine Springer fiber $Fl_{\gamma}$ can be…

In a seminal 1994 paper, Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_{\geq 0} of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a "remarkable…

Let $G$ be the special linear group of degree $2$ over an algebraically closed field $K$. Let $E$ be the natural module and $S^rE$ the $r$th symmetric power. We consider here, for $r,s\geq 0$, the tensor product of $S^rE$ and the dual of…

In this paper we study the partially ordered set Q^J of cells in Rietsch's cell decomposition of the totally nonnegative part of an arbitrary flag variety P^J_{\geq 0}. Our goal is to understand the geometry of P^J_{\geq 0}: Lusztig has…