Symmetric Functions, Schubert Polynomials and Degeneracy Loci

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 fur Mathematik This text serves as an introduction to the combinatorics of symmetric functions, more precisely to Schur and Schubert polynomials. Simultaneously, it studies the geometry of Grassmannians, flag varieties and especially their Schubert varieties, and examines the profound connections that unite these subjects. The first chapter of the book 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 the study of Schubert polynomials. 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.