Introduction to the Theory of Standard Monomials, Second Edition

Authors: C. S. Seshadri



Discusses about the full linear group and standard monomial theory and its applications

Reproduces the lectures the author delivered and appeared in the Brandeis Lectures Notes

Is authored by the winner of the Padma Bhushan

The book is a reproduction of a course of lectures delivered by the author in 1983-84 which appeared in the Brandeis Lecture Notes series. The aim of this course was to give an introduction to the series of papers by concentrating on the case of the full linear group. In recent years, there has been great progress in standard monomial theory due to the work of Peter Littelmann. The author’s lectures (reproduced in this book) remain an excellent introduction to standard monomial theory.

Standard monomial theory deals with the construction of nice bases of finite dimensional irreducible representations of semi-simple algebraic groups or, in geometric terms, nice bases of coordinate rings of flag varieties (and their Schubert subvarieties) associated with these groups. Besides its intrinsic interest, standard monomial theory has applications to the study of the geometry of Schubert varieties. Standard monomial theory has its origin in the work of Hodge, giving bases of the coordinate rings of the Grassmannian and its Schubert subvarieties by “standard monomials”. In its modern form, standard monomial theory was developed by the author in a series of papers written in collaboration with V. Lakshmibai and C. Musili. In the second edition of the book, conjectures of a standard monomial theory for a general semi-simple (simply-connected) algebraic group, due to Lakshmibai, have been added as an appendix, and the bibliography has been revised.

Table of contents

Front Matter
Pages i-xvi

Schubert Varieties in the Grassmannian
Pages 1-53

Standard monomial theory on SLn(k)/Q
Pages 55-80

Applications
Pages 81-105

Schubert varieties in G/Q
Pages 107-137

Back Matter
Pages 139-224