Related papers: Total positivity in Springer fibres

Using Lusztig's total positivity in split real Lie groups V. Fock and A. Goncharov have introduced spaces of positive (framed) representations. For general semisimple Lie groups a generalization of Lusztig's total positivity was recently…

We prove that the prime torsion in the local integral intersection cohomology of Schubert varieties in the flag variety of the general linear group grows exponentially in the rank. The idea of the proof is to find a highly singular point in…

We show that for an algebraic reductive group $G$, the partition of a double Schubert cell in the flag variety $G/B$ defined by Deodhar, and coming from a Bialynicki-Birula decomposition, is not a stratification in general. We give a…

We show that the total space of the Milnor fibration associated with any cusp or simple elliptic singularity in complex three variables admits an $S^1$-parametric genus-one Lefschetz fibration structure over the $2$-disk. As a consequence,…

Let $G$ be a connected reductive group over $\mathbb{C}$ with Weyl group $W$. Following a suggestion of Bezrukavnikov, we define a map from two-sided cells to conjugacy classes in $W$ using the geometry of the affine flag variety. This is…

We consider the Springer fiber over a nilpotent endomorphism. Fix a Jordan basis and consider the standard torus relative to this. We deal with the problem to describe the flags fixed by the torus which belong to a given component of the…

In this paper, we investigate quantum Fourier analysis on subfactors and unitary fusion categories. We prove the complete positivity of the comultiplication for subfactors and derive a primary $n$-criterion of unitary categorifcation of…

Motivated by the study of Springer fibers and their totally nonnegative counterparts, we define a new subset of standard tableaux called Richardson tableaux. We characterize Richardson tableaux combinatorially using evacuation as well as in…

For the flag variety G/B of a reductive algebraic group G we define a certain (set-theoretical) cross-section phi from G/B to G, which depends on a choice of reduced expression for the longest element in the Weyl group. This cross-section…

We develop a novel perspective on reflection positivity (RP) on the strip by systematically developing the analogies with the unit disc and the upper half plane in the complex plane. These domains correspond to the three conjugacy classes…

This is a survey article on Richardson varieties and their combinatorics. A Richardson variety is the intersection, inside the flag manifold GL_n/B_+, of a Schubert cell (B_- u B_+)/B_+ and an opposite Schubert cell (B_+ w B_+)/B_+ (or the…

The purpose of this paper is to describe certain natural 4-vector fields on quaternionic flag manifolds, which geometrically determine the Bruhat cell decomposition. This structure naturally descends from the symplectic group, where it is…

We show that every complete noetherian local commutative ring R with residue field k can be realized as a universal deformation ring of a continuous linear representation of a profinite group. More specifically, R is the universal…

We obtain the topological obstructions to existence of a bundle of irreducible real Clifford modules over a pseudo-Riemannian manifold $(M,g)$ of arbitrary dimension and signature and prove that bundles of Clifford modules are associated to…

A complete flag in $\mathbb{R}^n$ is a sequence of nested subspaces $V_1 \subset \cdots \subset V_{n-1}$ such that each $V_k$ has dimension $k$. It is called totally nonnegative if all its Pl\"ucker coordinates are nonnegative. We may view…

Using an intuition from metric geometry, we prove that any flag and normal simplicial complex satisfies the non-revisiting path conjecture. As a consequence, the diameter of its facet-ridge graph is smaller than the number of vertices minus…

We study the totally nonnegative part of the Peterson variety in arbitrary Lie type and establish its connection to the strongly dominant weight polytope. In particular, we prove that the totally nonnegative part of the Peterson variety is…

We prove a conjecture of Dale Peterson on positivity in the multiplication in the T-equivariant cohomology of the flag variety. The theorem follows from a more general positivity result about the equivariant cohomology of varieties with…

A split of a polytope $P$ is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of $P$ admits a unique decomposition as a linear combination of weight functions corresponding to…

In this note, starting with any group homomorphism $f\colon\Gamma\to G$, which is surjective upon abelianization, we construct a universal central extension $u\colon U\twoheadrightarrow G,$ UNDER $\Gamma$ with the same surjective property,…