Symmetric functions, Schubert polynomials and degeneracy loci. Transl. from the French by John R. Swallow.

*(English)*Zbl 0998.14023
SMF/AMS Texts and Monographs 6; Cours Spécialisés (Paris) 3. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2154-7/pbk). vii, 167 p. (2001).

There is a surprising number of fascinating connections between combinatorics, algebra and geometry. Among the most beautiful illustrations of such ties are the relations between symmetric functions, and in particular Schur polynomials, on the one hand, the representation theory of the symmetric- and general linear groups, on the other-, and the theory of Schubert polynomials, and the geometry and cohomology theory of flag varieties and Schubert varieties, on the third hand. These connections have intrigued mathematicians for more than hundred years, and have proved extremely fruitful for the development of the fields. The area is still active and expanding. New surprising ties appear amazingly often, and there remain many interesting problems and conjectures. This book contains an introduction to the basic material of the fields and their interrelations. It offers a nice complement to the books “Symmetric functions and Hall polynomials” (1998; Zbl 0899.05068), and the notes on “Schubert polynomials”, Lond. Math. Soc. Lect. Note Ser. 166, 73-99 (1991; Zbl 0784.05061) by I. G. Macdonald, and to W. Fulton’s book “Young tableaux. With applications to representation theory and geometry” [Lond. Math. Soc. Student Texts. 35 (1997; Zbl 0878.14034)]. At many places the book gives a different point of view and chooses different techniques. Is is written in a clear and quick style. Sometimes so quick that more examples would have been useful. The book consists of three separate parts:

Symmetric functions and Schur polynomials.

Schubert polynomials.

The geometry of Schubert varieties and the cohomology of flags varieties.

The contents is roughly as follows:

In the first part the Schur functions are introduced in the original way of C. Jacobi and interpreted in terms of Young tableaux. The Pieri formula, the Jacobi-Trudi formulas, and Giambelli’s formula are proved in the usual way. Combinatorial correspondences like those of Knuth, Schensted and Robinson are explained, and the Plactic monoid is defined. Using these tools the Littlewood-Richardson rule for multiplication of Schur polynomials is given. The presentation is an alternative to that used by Macdonald in the book on symmetric functions and Hall polynomials mentioned above.

Several applications of the theory are mentioned and the important Kostka-Foulkes polynomials are defined and their main properties are proved. In a separate section the classical connection between Schur polynomials and the irreducible characters of the symmetric group is given.

The second part of the book is devoted to Schubert polynomials. There are several possible approaches to Schubert polynomials. Here the author chooses the one proposed by S. Fomin and A. N. Kirillov via the Yang-Baxter equation and the Hecke algebras of the symmetric groups. It is shown how Schubert polynomials can be computed, and the main properties of the Schubert polynomials, like symmetries, the Cauchy formula, bases, interpolation, and specialization are proved. The lattice path method of I. Gessel and G. Viennot is explained and used. An interesting problem is to determine how the Schubert polynomials are multiplied. The partial formulas of Monk and the Pieri Formula for Schubert polynomials are proved.

In the third part of the book Grassmann varieties and their Schubert varieties are introduced and studied. Their coordinate rings are described, and the fundamental properties of the singularities of the Schubert varieties are given. Also the cohomology ring of the Grassmann variety is determined. The well known correspondence between Chern classes and the classes of the special Schubert varieties is described, and the well known Thom-Porteous formula is proved. In order to study degeneracy loci the more general theory of flag varieties and their Schubert varieties is studied, and the cohomology rings of the flag varieties are described. One of the highlights of the book is the study of degeneracy loci of maps between vector bundles, and a proof of the beautiful result of Fulton that gives the relation between the classes of certain degeneracy loci and the Schubert polynomials.

Symmetric functions and Schur polynomials.

Schubert polynomials.

The geometry of Schubert varieties and the cohomology of flags varieties.

The contents is roughly as follows:

In the first part the Schur functions are introduced in the original way of C. Jacobi and interpreted in terms of Young tableaux. The Pieri formula, the Jacobi-Trudi formulas, and Giambelli’s formula are proved in the usual way. Combinatorial correspondences like those of Knuth, Schensted and Robinson are explained, and the Plactic monoid is defined. Using these tools the Littlewood-Richardson rule for multiplication of Schur polynomials is given. The presentation is an alternative to that used by Macdonald in the book on symmetric functions and Hall polynomials mentioned above.

Several applications of the theory are mentioned and the important Kostka-Foulkes polynomials are defined and their main properties are proved. In a separate section the classical connection between Schur polynomials and the irreducible characters of the symmetric group is given.

The second part of the book is devoted to Schubert polynomials. There are several possible approaches to Schubert polynomials. Here the author chooses the one proposed by S. Fomin and A. N. Kirillov via the Yang-Baxter equation and the Hecke algebras of the symmetric groups. It is shown how Schubert polynomials can be computed, and the main properties of the Schubert polynomials, like symmetries, the Cauchy formula, bases, interpolation, and specialization are proved. The lattice path method of I. Gessel and G. Viennot is explained and used. An interesting problem is to determine how the Schubert polynomials are multiplied. The partial formulas of Monk and the Pieri Formula for Schubert polynomials are proved.

In the third part of the book Grassmann varieties and their Schubert varieties are introduced and studied. Their coordinate rings are described, and the fundamental properties of the singularities of the Schubert varieties are given. Also the cohomology ring of the Grassmann variety is determined. The well known correspondence between Chern classes and the classes of the special Schubert varieties is described, and the well known Thom-Porteous formula is proved. In order to study degeneracy loci the more general theory of flag varieties and their Schubert varieties is studied, and the cohomology rings of the flag varieties are described. One of the highlights of the book is the study of degeneracy loci of maps between vector bundles, and a proof of the beautiful result of Fulton that gives the relation between the classes of certain degeneracy loci and the Schubert polynomials.

Reviewer: Dan Laksov (Stockholm)

##### MSC:

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

05E10 | Combinatorial aspects of representation theory |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

05E05 | Symmetric functions and generalizations |

20C30 | Representations of finite symmetric groups |