Symmetric Functions, Schubert Polynomials, and Degeneracy Loci

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American Mathematical Soc., 2001 - Mathematics - 167 pages
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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."
  

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Contents

Introduction
1
The Ring of Symmetric Functions
7
Schubert Polynomials
57
Schubert Varieties
101
A Brief Introduction to Singular Homology
153
Bibliography
161
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Page 162 - Standard monomial theory. In Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), pages 279-322, Madras, 1991.
Page 163 - New symmetric plane partition identities from invariant theory work of De Concini and Procesi.
Page 163 - Pragacz, Symmetric polynomials and divided differences in formulas of intersection theory, in: Parameter Spaces, Banach Center Publications, 36 (1996), 125-177 [PR] P.
Page 162 - Jacobi. De functionibus alternantibus earumque divisione per productum e differentiis elementorum conflatum. J.
Page 162 - G. Kempf and D. Laksov, The determinantal formula of Schubert calculus, Acta Math. 132 (1974), 153-162.

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Lie Groups
Daniel Bump
Limited preview - 2004
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