This text grew out of an advanced course taught by the author at the Fourier Institute (Grenoble, France). It serves as an introduction to the combinatorics of symmetric functions, more precisely to Schur and Schubert polynomials. Also studied is the geometry of Grassmannians, flag varieties, and especially, their Schubert varieties. This book examines profound connections that unite these two subjects.

The book is divided into three chapters. The first is devoted to symmetric functions and especially to Schur polynomials. These are polynomials with positive integer coefficients in which each of the monomials correspond to a Young tableau with the property of being "semistandard". The second chapter is devoted to Schubert polynomials, which were discovered by A. Lascoux and M.-P. Schützenberger who deeply probed their combinatorial properties. It is shown, for example, that these polynomials support the subtle connections between problems of enumeration of reduced decompositions of permutations and the Littlewood-Richardson rule, a particularly efficacious version of which may be derived from these connections. The final chapter is geometric. It is devoted to Schubert varieties, subvarieties of Grassmannians, and flag varieties defined by certain incidence conditions with fixed subspaces.

This volume makes accessible a number of results, creating a solid stepping stone for scaling more ambitious heights in the area. The author's intent was to remain elementary: The first two chapters require no prior knowledge, the third chapter uses some rudimentary notions of topology and algebraic geometry. For this reason, a comprehensive appendix on the topology of algebraic varieties is provided. This book is the English translation of a text previously published in French.

Titles in this series are co-published with Société Mathématique de France. SMF members are entitled to AMS member discounts.

Readership

Graduate students and research mathematicians interested in combinatorics, algebraic geometry, group theory and generalizations, and manifolds and cell complexes.

Reviews

"The book ... is written in a clear and quick style."

-- Zentralblatt MATH

From reviews of the French Edition:

"Well-written book ... all of the concepts are clearly defined and presented in an informal and pleasant way ... an attractive book which presents the interplay between many diverse topics of algebraic combinatorics and their geometric realizations in Schubert calculus. It will be of great use to anyone wishing a brief and well-organized treatment of this material and particularly good for graduate students."

-- Mathematical Reviews

"Excellent text ... with numerous further-leading exercises and remarks, and with a rich bibliography, which makes the study of it very profitable."

-- Zentralblatt MATH

Table of Contents

• Introduction
• The ring of symmetric functions
• Schubert polynomials
• Schubert varieties
• A brief introduction to singular homology
• Bibliography
• Index