# Introduction to the Theory of Standard Monomials: Second Edition

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*C. S. Seshadri*

A publication of Hindustan Book Agency

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.

In this edition, Conjectures of a Standard Monomial Theory (SMT) for
a general semi-simple (simply-connected) algebraic group, due to
Lakshmibai, have been added as Appendix C. Many
typographical errors have been corrected, and the bibliography has
been revised.

A publication of Hindustan Book Agency; distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for all commercial channels.

#### Readership

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