A graphical calculus for 2-block Spaltenstein varieties
type: Item
instance of: scholarly article
zbMATH Open document ID: 1243.14042
DOI: 10.1017/S0017089512000110
author: Gisa Schäfer
published in: Glasgow Mathematical Journal
publication date: 2012-05-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.6247
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14F25
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 18D10
zbMATH DE Number: 6031024
zbMATH Keywords: Springer fiber
zbMATH Keywords: Spaltenstein varieties
zbMATH Keywords: Young tableaux
zbMATH Keywords: dependence graphs
zbMATH Keywords: cup diagrams
zbMATH Keywords: circle diagrams
zbMATH Keywords: iterated fiber bundle
zbMATH Keywords: cohomology
zbMATH Keywords: colored cobordisms
MaRDI profile type: Publication
arXiv ID: 1202.6247
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OpenAlex ID: W2964027456
title: A graphical calculus for 2-block Spaltenstein varieties
review text: Let \(V = {\mathbb C}^n\) be an \(n\)-dimensional complex vector space. For a sequence of integers \(\{i_1, i_2, \dots, i_m\}\) with \(0 < i_1 < i_2 < \cdots < i_m = n\), a partial flag of \(V\) of type \((i_1,i_2,\dots, i_m)\) is a sequence of subspaces \(F_{i_1} \subset F_{i_2} \subset \cdots \subset F_{i_m}\) with \(\text{dim}F_{i_j} = i_j\). A complete (or full) flag is one of type \((1,2,\dots, n)\). Consider a nilpotent endomorphism \(N : V \to V\). The variety of all complete flags that are invariant under \(N\) is known as the Springer fiber of \(N\). More generally, the variety of \(N\)-invariant partial flags of some type \((i_1,i_2,\dots,i_m)\) is known as a Spaltenstein variety of type \((i_1,i_2,\dots,i_m)\). The nilpotent endomorphism \(N\) is determined by the sizes of its Jordan blocks and hence can be associated with a partition \(\lambda\) of \(n\). Spaltenstein showed that the irreducible components of the Springer fiber are in one-to-one correspondence with standard Young tableaux of shape \(\lambda\). More generally, the irreducible components of a Spaltenstein variety (of a given type) can also be (bijectively) associated with a set of tableaux determined by the type.NEWLINENEWLINEThe goal of this paper is to study the geometry of the irreducible (and so-called ``generalized irreducible'') components of a Spaltenstein variety in the case when \(N\) has two Jordan blocks by generalizing known results for the special case of the Springer fiber. The author first gives an explicit description of these irreducible components by generalizing the description given for Springer fibers by \textit{F. Fung} [Adv. Math. 178, No. 2, 244--276 (2003; Zbl 1035.20004)]. Following \textit{C. Stroppel} and \textit{B. Webster} [Comment. Math. Helv. 87, No. 2, 477--520 (2012; Zbl 1241.14009)], associated to a row-strict tableaux of a given type, the author defines the notion of a ``generalized irreducible component'' of a Spaltenstein variety (of that given type). …
reviewed by: Christopher P. Bendel
description: scientific article; zbMATH DE number 6031024