Related papers: The M\"obius function on Affine Grassmannian eleme…

For a fixed finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION problem asks, given an $n$-vertex input graph $G,$ for the minimum number of vertices that intersect all minor models in $G$ of the graphs in ${\cal F}$. by…

Motivated by the geometry of certain hyperplane arrangements, Manin and Schechtman defined for each positive integer n a hierarchy of finite partially ordered sets B(n, k), indexed by positive integers k, called the higher Bruhat orders.…

We extend (scheme-theoretic) Bruhat-Tits theory to quasi-reductive groups i.e. with trivial split unipotent radical over discretely valued henselian non-archimedean fields $K$, whose ring of integers is excellent and residue field is…

Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B be a Borel subgroup of G. Then B acts with finitely many orbits on the variety N_2 of the nilpotent elements in…

We consider two-dimensional N=(2,2) supersymmetric gauge theory on discretized Riemann surfaces. We find that the discretized theory can be efficiently described by using graph theory, where the bosonic and fermionic fields are regarded as…

For any Kac-Moody group $\mathbf{G}$, we prove that the Bruhat order on the semidirect product of the Weyl group and the Tits cone for $\mathbf{G}$ is strictly compatible with a $\mathbb{Z}$-valued length function. We conjecture in general…

Let $Q$ be a quiver, $M$ a representation of $Q$ with an ordered basis $\cB$ and $\ue$ a dimension vector for $Q$. In this note we extend the methods of \cite{L12} to establish Schubert decompositions of quiver Grassmannians $\Gr_\ue(M)$…

Let G be a quasi simple algebraic group over an algebraically closed field k whose characteristic is not very bad for G, and let B be a Borel subgroup of G with Lie algebra b. Given a B-stable abelian subalgebra a of the nilradical of b, we…

Let $X, Y$ be complete, simply connected Riemannian surfaces with pinched negative curvature $-b^2 \leq K \leq -1$. We show that if $f : \partial X \to \partial Y$ is a Moebius homeomorphism between the boundaries at infinity of $X, Y$,…

Let G be a simple and simply-connected complex algebraic group, P \subset G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH^*(G/P) of a flag variety is, up to localization, a…

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…

The prime spectra of two families of algebras, $S^w$ and $\check{S}^w$, $w\in W,$ indexed by the Weyl group $W$ of a semisimple finitely dimensional are studied. The algebras $S^w$ have been introduced by A.~Joseph; they are $q$-analogues…

A permutation is called smooth if the corresponding Schubert variety is smooth. Gilboa and Lapid prove that in the symmetric group, multiplying the reflections below a smooth element $w$ in Bruhat order in a compatible order yields back the…

This paper studies the M\"obius function and related questions about the finiteness of the poset of submodules of semisimple and general modules. We show how to calculate the M\"obius function for semisimple modules based on endomorphism…

An affine variety $X$ of dimension $\ge 2$ is called {\em flexible} if its special automorphism group SAut$(X)$ acts transitively on the smooth locus $X_{reg}$ \cite{AKZ}. Recall that the special automorphism group SAut$(X)$ is the subgroup…

The degenerate affine and affine BMW algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic Lie algebras and quantum groups, respectively. Cyclotomic BMW algebras, affine Hecke algebras, cyclotomic Hecke…

The higher Bruhat orders are partial orders that generalize the weak order on the symmetric group $S_n$, and the second higher Bruhat order is a poset on commutation classes of reduced words for the longest element in $S_n$, where covering…

The Demazure product gives a natural monoid structure on any Coxeter group. Such structure occurs naturally in many different areas in Lie Theory. This paper studies the Demazure product of an extended affine Weyl group $\tilde W$. The main…

Crystal base of the level 0 part of the modified quantum affine algebra $\widetilde U_q(\widehat{sl_2})_0$ is given by path. Weyl group actions, extremal vectors and crystal structure of all irreducible components are described explicitly.

We construct a homomorphism from the affine Yangian associated with $\widehat{\mathfrak{sl}}(q_u-q_{u+1})$ to the universal enveloping algebra of a $W$-algebra associated with $\mathfrak{gl}(\sum_{s=1}^lq_s)$ and a nilpotent element of type…