## Symmetric Functions, Schubert Polynomials, and Degeneracy LociThis 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. It is devoted to Schubert varieties, subvarieties of Grassmannians, and flag varieties defined by certain incidence conditions with fixed subspaces. This volume makes accessible a number of results, creating a solid stepping stone for scaling more ambitious heights in the area. The author's intent was to remain elementary: The first two chapters require no prior knowledge, the third chapter uses some rudimentary notions of topology and algebraic geometry. For this reason, a comprehensive appendix on the topology of algebraic varieties is provided. This book is the English translation of a text previously published in French. |

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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 |

161 | |

### Common terms and phrases

algebraic subvariety associated Bruhat order Chern class codimension coefficients cohomology ring columns commutative complete symmetric functions configuration consider contains coordinates COROLLARY deduce defined degeneracy loci degree denote desingularization determine diagram dimension divided differences element elementary entries equal example EXERCISE exists finite flag varieties following fashion fundamental class given Gm,n graph Grassmannian hence homology Hq(X identity implies induction integer intersection irreducible representations isomorphism Knuth correspondence Kostka-Foulkes polynomials lemma length line bundle Littlewood-Richardson rule locus matrix monomials Moreover morphism nonzero obtain particular permutation Pieri's formulas plane partitions preceding proposition projective space PROOF quotient bundles rank reduced decomposition reduced words REMARK Schubert cells Schubert class Schubert polynomials Schubert varieties Schur functions Schur polynomials semistandard tableaux sequence similarly singular skew tableau standard tableaux subspaces Suppose symmetric group tableaux with shape theorem transverse variables vector bundles verify vexillary weight zero

### Popular passages

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.