Standard Monomial Theory: Invariant Theoretic Approach

Venkatramani Lakshmibai, Komaranapuram N. Raghavan (auth.)3540767568, 9783540767565

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.

Table of contents :
Front Matter….Pages i-xiv
Introduction….Pages 1-9
Generalities on algebraic varieties….Pages 11-16
Generalities on algebraic groups….Pages 17-28
Grassmannian….Pages 29-46
Determinantal varieties….Pages 47-54
Symplectic Grassmannian….Pages 55-69
Orthogonal Grassmannian….Pages 71-83
The standard monomial theoretic basis….Pages 85-93
Review of GIT….Pages 95-120
Invariant theory….Pages 121-136
SL n ( K )-action….Pages 137-158
SO n ( K )-action….Pages 159-185
Applications of standard monomial theory….Pages 187-217
Back Matter….Pages 219-265