# Symmetric Functions, Schubert Polynomials and Degeneracy Loci

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*Laurent Manivel*

A co-publication of the AMS and Société Mathématique de France

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 & Endorsements

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

-- Zentralblatt MATH

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