Singular loci of Schubert varieties
Schubert varieties appear in a natural way in many parts of mathematics and its applications. They are on the crossroad of representation theory, combinatorics and algebraic, analytic and differential geometry. The structure of the singularities of the Schubert varieties is extremely complicated, but still manageable. This makes Schubert varieties extremely interesting objects and explain why the work on these varieties has contributed so much to mathematics. There is a vast literature on their properties, in particular on their singularities and their cohomology. The literature list of the present book contains nearly 160 items, most of them on singularities, and most of them are written after 1984. The book started out as a survey article. This is reflected in the text which consists mostly of explanation of concepts and statements of results. There are very few proofs, and the background material is limited to a minimum. In spite of this the book is more than 200 pages long. This illustrates well the size of the task that the authors have undertaken. That they have succeeded in writing a readable text is partly due to the fact that they both are experts in the area. About one fifth of the articles in the reference list are written by the authors. The book is closer to a survey of recent work on the singularities of Schubert varieties, than a textbook. What distinguishes it from a survey is that it contains a wealth of examples and calculations. This makes the book extremely valuable to those that want to learn the area. The contents of the book is: Chapter 2: This chapter contains background material on Schubert varieties, Bruhat-Chevalley orders and parabolic groups, weights, algebraic groups, Weyl groups, and representations of semi-simple algebraic groups. It would take several courses, and books, to cover properly this material alone. -- Chapter 3: To compensate for the rapid introduction of concepts in chapter 2 this chapter contains concrete computations of the Bruhat-Chevalley order in the classical groups, the grassmannians, the flag manifolds and their Schubert varieties. -- Chapter 4: This chapter is devoted to the study of the tangent space to a Schubert variety. -- Chapter 5: In this chapter results by P. Polo on the singular locus of a Schubert variety is recalled, and Polo's root system description of the tangent space is described. This is used to obtain a very detailed treatment of the classical groups. -- Chapter 6: Here Kazhdan-Lusztig polynomials and their use for characterizing rational singularities is explained. -- Chapter 7: In this chapter Shrawan Kumars criteria for smoothness, and generic smoothness are discussed. The relation between nil-Hecke rings and characters on the tangent cone is explained. This chapter also contains some proofs. -- Chapter 8: In this chapter combinatorial algorithms for testing smoothness and rational smoothness of Schubert varieties for the classical groups are given. The last four chapters are devoted to the singularities of special varieties. -- Chapter 9: In this chapter the minuscule and cominuscule cases are discussed. The major part is devoted to the small resolutions of A. Zelevinsky and of P. Sankaran and P. Vanchinatan. Also the description of their tangent space made by M. Brion and Polo is given. -- Chapter 10: This chapter is devoted to the rank \(2\) case. -- Chapter 11: In this chapter some results on the relation between smoothness and factorization of the Poincaré polynomial are described. -- Chapter 12: In this chapter determinantal varieties are treated. The book contains a wealth of results, and a large number of very useful and illustrating examples. It treats many central topics. The authors seem to concentrate on the work treating singularities of Schubert varieties only. There is a large body of work on the cohomology of manifold where the singularities of the Schubert varieties enter in an essential way, and where many of the techniques for treating singularities originated. It would have been very interesting to have a short description of this work and to get an explanation of the relations to the work on the singularities of Schubert varieties. Also it would have made the reference list more complete. The book is well written, and the background material clearly presented. It is valuable to both experts and beginners. Beginners will find a way to get an idea about the area before indulging into the large and technically complicated background material. Together with the experts they will find useful detailed examples, together with lists of minimal bad patterns and the singular loci of the special groups. The experts will find a wealth of interesting material, much of it new.
scientific article; zbMATH DE number 1520629