# Standard Monomial Theory

## Invariant Theoretic Approach

Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 137)

Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 137)

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.

Classical invariant theory Combinatorics Determinantal varieties Grassmannians Representation theory Schubert varieties algebra algebraic varieties standard monomial theory

- DOI https://doi.org/10.1007/978-3-540-76757-2
- Copyright Information Springer-Verlag Berlin Heidelberg 2008
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-540-76756-5
- Online ISBN 978-3-540-76757-2
- Series Print ISSN 0938-0396
- About this book