This article grew out of a sophisticated direction taught by way of the writer on the Fourier Institute (Grenoble, France). It serves as an advent to the combinatorics of symmetric capabilities, extra accurately to Schur and Schubert polynomials. additionally studied is the geometry of Grassmannians, flag types, and particularly, their Schubert types. This ebook examines profound connections that unite those topics. The booklet is split into 3 chapters. the 1st is dedicated to symmetric services and particularly to Schur polynomials. those are polynomials with optimistic integer coefficients during which all 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 way of A. Lascoux and M. P. Schutzenberger who deeply probed their combinatorial properties.It is proven, for instance, that those polynomials aid the delicate connections among difficulties of enumeration of diminished 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's dedicated to Schubert types, subvarieties of Grassmannians, and flag forms outlined by means of definite prevalence stipulations with mounted subspaces. This quantity makes available a couple of effects, making a reliable stepping stone for scaling extra bold heights within the zone. The author's purpose was once to stay simple: the 1st chapters require no earlier wisdom, the 3rd bankruptcy makes use of a few rudimentary notions of topology and algebraic geometry. consequently, a complete appendix at the topology of algebraic forms is equipped. This e-book is the English translation of a textual content formerly released in French.
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