A notion of inversion number associated to certain quiver flag varieties
type: Item
instance of: scholarly article
title: A notion of inversion number associated to certain quiver flag varieties
zbMATH Open document ID: 1423.14276
author: Lucas Fresse
published in: The Electronic Journal of Combinatorics
publication date: 2018-09-07T00:00:00Z
full work available at URL: v25i3p41
review text: Summary: We define an algebraic variety \(X(d,A)\) consisting of matrices whose rows and columns are partial flags. This is a smooth, projective variety, and we describe it as an iterated bundle of Grassmannian varieties. Moreover, we show that \(X(d,A)\) has a cell decomposition, in which the cells are parametrized by certain matrices of sets and their dimensions are given by a notion of inversion number. On the other hand, we consider the Spaltenstein variety of partial flags which are stabilized by a given nilpotent endomorphism. We partition this variety into locally closed subvarieties which are affine bundles over certain varieties called \(Y_T\), parametrized by semistandard tableaux \(T\). We show that the varieties \(Y_T\) are in fact isomorphic to varieties of the form \(X(d,A)\). We deduce that each variety \(Y_T\) has a cell decomposition, in which the cells are parametrized by certain row-increasing tableaux obtained by permuting the entries in the columns of \(T\) and their dimensions are given by the inversion number recently defined by \textit{P. Drube} [Electron. J. Comb. 23, No. 1, Research Paper P1.43, 22 p. (2016; Zbl 1333.05035)] for such row-increasing tableaux.
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 6933723
zbMATH Keywords: flag varieties
zbMATH Keywords: cell decomposition
zbMATH Keywords: Spaltenstein variety
zbMATH Keywords: semistandard Young tableaux
zbMATH Keywords: inversion number
MaRDI profile type: Publication
cites work: Q3999894
cites work: Lectures on the Geometry of Flag Varieties
cites work: Cohomology of Spaltenstein varieties
cites work: Inversions of semistandard Young tableaux
cites work: Generating functions for inverted semistandard Young tableaux and generalized ballot numbers
cites work: Betti numbers of Springer fibers in type \(A\)
cites work: Existence of affine pavings for varieties of partial flags associated to nilpotent elements
cites work: Polarizations in the classical groups
cites work: Q4670625
cites work: Schubert polynomials and Bott-Samelson varieties
cites work: Homology of moduli spaces of instantons on ALE spaces. I
cites work: Q4112123
cites work: Classes unipotentes et sous-groupes de Borel
description: scientific article; zbMATH DE number 6933723