This article grew out of a complicated direction taught through the writer on the Fourier Institute (Grenoble, France). It serves as an creation to the combinatorics of symmetric capabilities, extra accurately to Schur and Schubert polynomials. additionally studied is the geometry of Grassmannians, flag forms, and particularly, their Schubert types. This ebook examines profound connections that unite those topics. The ebook is split into 3 chapters. the 1st is dedicated to symmetric features and particularly to Schur polynomials. those are polynomials with optimistic integer coefficients during which all of the monomials correspond to a tender tableau with the valuables of being 'semistandard'. the second one bankruptcy is dedicated to Schubert polynomials, that have been chanced on by means of A. Lascoux and M. P. Schutzenberger who deeply probed their combinatorial properties.It is proven, for instance, that those polynomials aid the sophisticated connections among difficulties of enumeration of lowered decompositions of diversifications and the Littlewood-Richardson rule, a very efficacious model of that could be derived from those connections. the ultimate bankruptcy is geometric. it really is dedicated to Schubert kinds, subvarieties of Grassmannians, and flag types outlined by way of convinced prevalence stipulations with mounted subspaces. This quantity makes available a couple of effects, making a stable stepping stone for scaling extra formidable heights within the quarter. The author's motive was once to stay basic: the 1st chapters require no earlier wisdom, the 3rd bankruptcy makes use of a few rudimentary notions of topology and algebraic geometry. accordingly, a complete appendix at the topology of algebraic kinds is supplied. This e-book is the English translation of a textual content formerly released in French.
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