Symmetric Functions, Schubert Polynomials and Degeneracy Loci

ISBN-13: 9780821821541 / Angielski / Miękka / 2001 / 176 str.

Saint Martin D'her Laurent (University Of Grenoble Manivel

Opis

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. Schutzenberger 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."