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Zbl 0878.14034
Fulton, William
Young tableaux. With applications to representation theory and geometry. (English)
[B] London Mathematical Society Student Texts. 35. Cambridge: Cambridge University Press. ix, 260 p. \sterling 14.95, \$ 19.95 pbk; \sterling 40.00, \$ 59.95 hbk (1997). ISBN 0-521-56724-6/pbk

Young tableaux are well studied in representation theory and combinatorics [see the classic books: {\it G. James} and {\it A. Kerber}, ``The representation theory of the symmetric group'' (1981; Zbl 0491.20010) or {\it G. E. Andrews}, ``The theory of partition'' (1976; Zbl 0371.10001)]. The well-known author gives the definitions and main theorems in these fields. But his main emphasis is on the application of Young tableaux in (algebraic) geometry. Defining equations for Grassmannians and flag varieties are given. We find Schubert varieties in flag manifolds, Chern classes are introduced, and the geometry of flag varieties is used to construct the Schubert polynomials of Lascoux and Schützenberger. The basic facts about intersection theory on Grassmannians are given, proofs at some times are delegated to the standard books of {\it R. Hartshorne} [``Algebraic geometry'' (3rd edition 1983; Zbl 0531.14001)] and {\it I. R. Shafarevich} [``Basic algebraic geometry. I and II'' (2nd edition 1994; Zbl 0797.14001 and Zbl 0797.14002)] on algebraic geometry. The wealth of presented material makes this excusable.\par There are numerous exercises with answers in each chapter and a lot of references to be found at the end of the book.
[B.Richter (Berlin)]

MSC 2000:: *14M15 Grassmannians, Schubert varieties
05E10 Tableaux, etc.
14-02 Research monographs (algebraic geometry)
20-02 Research monographs (group theory)
05-02 Research monographs (combinatorics)
20C30 Representations of finite symmetric groups
20C33 Representations of finite groups of Lie type
22E45 Analytic repres.of Lie and linear algebraic groups over real fields

Keywords: Young tableau; Grassmannian; Schubert polynomials

Citations: Zbl 0491.20010; Zbl 0371.10001; Zbl 0531.14001; Zbl 0797.14001; Zbl 0797.14002

Cited in: Zbl 1208.14002 Zbl 1229.17004 Zbl 1205.05239 Zbl 1069.81023 Zbl 0998.14023 Zbl 0991.13006 Zbl 1025.20032 Zbl 0933.14031

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