The aim of this book is to give an introduction to what has come to be known as Standard Monomial Theory (SMT). SMT 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 to these groups. Besides its intrinsic interest, SMT has applications to the study of the geometry of Schubert varieties. SMT 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, SMT was developed by the author in a series of papers written in collaboration with V. Lakshmibai and C. Musili.

This book is a reproduction of a course of lectures given 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 SMT due to the work of Peter Littelmann. Seshadri's course of lectures (reproduced in this book) remains an excellent introduction to SMT.

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Readership

Graduate students and research mathematicians interested in algebra and algebraic geometry.

Table of Contents

• Introduction
• Standard monomial theory on $SL_n (k)/Q$
• Applications
• Schubert varieties in $G/Q$
• Appendix A
• Appendix B
• Bibliography
• Notation
• Index
• Index of Symbols