Encyclopaedia of Mathematical Sciences

Standard Monomial Theory

Invariant Theoretic Approach

Authors: Lakshmibai, V., Raghavan, K. N.

  • Excellent overview of the theory of Schubert varieties

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eBook $119.00
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  • ISBN 978-3-540-76757-2
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Hardcover $159.00
price for USA
  • ISBN 978-3-540-76756-5
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  • Usually dispatched within 3 to 5 business days.
Softcover $159.00
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  • ISBN 978-3-642-09543-6
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About this book

Schubert varieties lie at the cross roads of algebraic geometry, combinatorics, commutative algebra, and representation theory. They are an important class of subvarieties of flag varieties, interesting in their own right, and providing an inductive tool for studying flag varieties. The literature on them is vast, for they are ubiquitous—they have been intensively studied over the last fifty years, from many different points of view and by many different authors.

This book is mainly a detailed account of a particularly interesting instance of their occurrence: namely, in relation to classical invariant theory. More precisely, it is about the connection between the first and second fundamental theorems of classical invariant theory on the one hand and standard monomial theory for Schubert varieties in certain special flag varieties - the ordinary, orthogonal, and symplectic Grassmannians - on the other. Historically, this connection was the prime motivation for the development of standard monomial theory. Determinantal varieties and basic concepts of geometric invariant theory arise naturally in establishing the connection.

The book also treats, in the last chapter, some other applications of standard monomial theory, e.g., to the study of certain naturally occurring affine algebraic varieties that, like determinantal varieties, can be realized as open parts of Schubert varieties.

Reviews

From the reviews:

"The goal of the book is to present the results of Classical Invariant Theory (CIT) and Standard Monomial Theory (SMT) and the connection between the two theories. … The book is written for a broad audience including prospective graduate students and young researchers. The exposition is self-contained. It may be used for a year long course on Invariant Theory and Schubert varieties." (Dmitrii A. Timashëv, Mathematical Reviews, Issue 2008 m)

"The book aims to describe the beautiful connection between Schubert varieties and their Standard Monomial Theory (SMT) on the one hand and Classical Invariant Theory (CIT) on the other. … make the presentation self-contained keeping in mind the needs of prospective graduate students and young researchers. … The book may be recommended as a nice introduction to SMT and related active research areas. It may be used for a year long course on Invariant Theory and Schubert varieties." (Ivan V. Arzhantsev, Zentralblatt MATH, Vol. 1137 (15), 2008)


Table of contents (13 chapters)

Buy this book

eBook $119.00
price for USA (gross)
  • ISBN 978-3-540-76757-2
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $159.00
price for USA
  • ISBN 978-3-540-76756-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $159.00
price for USA
  • ISBN 978-3-642-09543-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Standard Monomial Theory
Book Subtitle
Invariant Theoretic Approach
Authors
Series Title
Encyclopaedia of Mathematical Sciences
Series Volume
137
Copyright
2008
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-540-76757-2
DOI
10.1007/978-3-540-76757-2
Hardcover ISBN
978-3-540-76756-5
Softcover ISBN
978-3-642-09543-6
Series ISSN
0938-0396
Edition Number
1
Number of Pages
XIV, 266
Topics