The Grassmannian Variety
Geometric and Representation-Theoretic Aspects
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Beschreibung
This book gives a comprehensive treatment of the Grassmannian varieties and their Schubert subvarieties, focusing on the geometric and representation-theoretic aspects of Grassmannian varieties. Research of Grassmannian varieties is centered at the crossroads of commutative algebra, algebraic geometry, representation theory, and combinatorics. Therefore, this text uniquely presents an exciting playing field for graduate students and researchers in mathematics, physics, and computer science, to expand their knowledge in the field of algebraic geometry. The standard monomial theory (SMT) for the Grassmannian varieties and their Schubert subvarieties are introduced and the text presents some important applications of SMT including the Cohen–Macaulay property, normality, unique factoriality, Gorenstein property, singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory.
This textwould serve well as a reference book for a graduate work on Grassmannian varieties and would be an excellent supplementary text for several courses including those in geometry of spherical varieties, Schubert varieties, advanced topics in geometric and differential topology, representation theory of compact and reductive groups, Lie theory, toric varieties, geometric representation theory, and singularity theory. The reader should have some familiarity with commutative algebra and algebraic geometry.
This book gives a comprehensive treatment of the Grassmannian varieties and their Schubert subvarieties, focusing on the geometric and representation-theoretic aspects of Grassmannian varieties. Research of Grassmannian varieties is centered at the crossroads of commutative algebra, algebraic geometry, representation theory, and combinatorics. Therefore, this text uniquely presents an exciting playing field for graduate students and researchers in mathematics, physics, and computer science, to expand their knowledge in the field of algebraic geometry. The standard monomial theory (SMT) for the Grassmannian varieties and their Schubert subvarieties are introduced and the text presents some important applications of SMT including the Cohen–Macaulay property, normality, unique factoriality, Gorenstein property, singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory.
This text would serve well as a reference book for a graduate work on Grassmannian varieties and would be an excellent supplementary text for several courses including those in geometry of spherical varieties, Schubert varieties, advanced topics in geometric and differential topology, representation theory of compact and reductive groups, Lie theory, toric varieties, geometric representation theory, and singularity theory. The reader should have some familiarity with commutative algebra and algebraic geometry.
Presents an exciting playing field for graduate students and researchers in mathematics, physics, and computer science, to expand their knowledge in the field of algebraic geometry Comprehensive treatment brings graduate students and new researchers to the forefront of the field Serves as excellent supplementary material for a graduate course on Grassmannian varieties
Autor*in
V. Lakshmibai
Themen in »The Grassmannian Variety«
Bott-Samelson varieties Cohen-Macaulay property Flag varieties Geometry of spherical varieties Gorenstein property Groebner bases Schubert varieties Singular locus Standard monomial theory The Grassmannian variety algebraic geometry classical invariant theory combinatorics
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“The present book gives a detailed treatment of the standard monomial theory (SMT) for the Grassmannians and their Schubert subvarieties along with several applications of SMT. It can be used as a reference book by experts and graduate students who study varieties with a reductive group action such as flag and toric varieties.” (Valentina Kiritchenko, zbMATH 1343.14001, 2016)
“The book under review is more elementary; it is exclusively devoted to Grassmannians and their Schubert subvarieties. The book is divided into three parts. … This is a nicely written book, one that may appeal to students and researchers in related areas.” (Felipe Zaldivar, MAA Reviews, maa.org, December, 2015)
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