Singular loci of Schubert varieties
type: Item
instance of: scholarly article
title: Singular loci of Schubert varieties
zbMATH Open document ID: 0959.14032
published in: Progress in Mathematics
publication date: 2000-10-23T00:00:00Z
review text: Schubert varieties appear in a natural way in many parts of mathematics and its applications. They are on the crossroad of representation theory, combinatorics and algebraic, analytic and differential geometry. The structure of the singularities of the Schubert varieties is extremely complicated, but still manageable. This makes Schubert varieties extremely interesting objects and explain why the work on these varieties has contributed so much to mathematics. There is a vast literature on their properties, in particular on their singularities and their cohomology. The literature list of the present book contains nearly 160 items, most of them on singularities, and most of them are written after 1984. The book started out as a survey article. This is reflected in the text which consists mostly of explanation of concepts and statements of results. There are very few proofs, and the background material is limited to a minimum. In spite of this the book is more than 200 pages long. This illustrates well the size of the task that the authors have undertaken. That they have succeeded in writing a readable text is partly due to the fact that they both are experts in the area. About one fifth of the articles in the reference list are written by the authors. The book is closer to a survey of recent work on the singularities of Schubert varieties, than a textbook. What distinguishes it from a survey is that it contains a wealth of examples and calculations. This makes the book extremely valuable to those that want to learn the area. The contents of the book is: Chapter 2: This chapter contains background material on Schubert varieties, Bruhat-Chevalley orders and parabolic groups, weights, algebraic groups, Weyl groups, and representations of semi-simple algebraic groups. It would take several courses, and books, to cover properly this material alone. -- Chapter 3: To compensate for the rapid introduction of concepts in chapter 2 this chapter contains concrete computations of…
reviewed by: Dan Laksov
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14M12
zbMATH DE Number: 1520629
zbMATH Keywords: Schubert varieties
zbMATH Keywords: singular loci
zbMATH Keywords: roots
zbMATH Keywords: Weyl groups
zbMATH Keywords: tangent cone
zbMATH Keywords: parabolic groups
zbMATH Keywords: Bruhat-Chevalley orders
zbMATH Keywords: grassmannian
zbMATH Keywords: Kazhdan-Lusztig polynomials
zbMATH Keywords: classical groups
zbMATH Keywords: nil-Hecke ring
zbMATH Keywords: minuscule groups
zbMATH Keywords: Poincaré polynomial
zbMATH Keywords: determinantal varieties
zbMATH Keywords: Dynkin diagrams
author: Sara Billey
author: V. Lakshmibai
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1520629