This article grew out of a complicated direction taught by way of the writer on the Fourier Institute (Grenoble, France). It serves as an creation to the combinatorics of symmetric services, extra accurately to Schur and Schubert polynomials. additionally studied is the geometry of Grassmannians, flag kinds, and particularly, their Schubert types. This e-book examines profound connections that unite those topics. The e-book is split into 3 chapters. the 1st is dedicated to symmetric services and particularly to Schur polynomials. those are polynomials with optimistic integer coefficients within which all the monomials correspond to a tender tableau with the valuables of being 'semistandard'. the second one bankruptcy is dedicated to Schubert polynomials, which have been came across via 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 diminished decompositions of diversifications and the Littlewood-Richardson rule, a very efficacious model of that may be derived from those connections. the ultimate bankruptcy is geometric. it's dedicated to Schubert kinds, subvarieties of Grassmannians, and flag kinds outlined via definite prevalence stipulations with fastened subspaces. This quantity makes available a few effects, making a strong stepping stone for scaling extra formidable heights within the sector. The author's motive was once to stay uncomplicated: the 1st chapters require no past wisdom, the 3rd bankruptcy makes use of a few rudimentary notions of topology and algebraic geometry. hence, a entire appendix at the topology of algebraic types is equipped. This e-book is the English translation of a textual content formerly released in French.
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