Introduction to the Theory of Standard Monomials

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 basesof 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.

C.S. SESHADRI, FRS, an eminent Indian mathematician, is director-emeritus of the Chennai Mathematical Institute, India. He is known for his work in algebraic geometry. The well-known “Seshadri constant” is named after him. His work with M.S. Narasimhan on unitary vector bundles and the Narasimhan–Seshadri theorem has influenced the field. His work on geometric invariant theory and on Schubert varieties, in particular his introduction of standard monomial theory, is widely recognized. A recipient of the Padma Bhushan in 2009, the third highest civilian honor in India, he was elected Fellow of the Indian Academy of Sciences in 1971. Professor Seshadri worked in the School of Mathematics at the Tata Institute of Fundamental Research, Mumbai, during 1953–1984, starting as a research scholar and rising to a senior professor. From 1984 to 1989, he worked in Institute of Mathematical Sciences, Chennai, India. From 1989 to 2010, he worked as the founding director of the Chennai Mathematical Institute.