Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux. (French) Zbl 0542.14030

Lascoux, Alain; Schützenberger, Marcel-Paul

C. R. Acad. Sci., Paris, Sér. I 295, 629-633 (1982).

In a long series of interesting and sometimes deep articles the authors have exploited the properties of the cohomology ring and Grothendieck ring of flag manifolds. The present article extends formulas for Schur functions, used to prove that the representation ring of the symmetric group is a Hopf algebra, to formulas for (what the authors call) Schubert and Grothendieck polynomials, that are generalizations of Schur functions. As a result of their formulas they obtain a beautiful formula concerning reduced representations of the symmetric group.
Unfortunately their work is now so far developed in terminology and notation that it is hard for non-experts to read it. In addition, their presentation is extremely condensed and computational. Perhaps time has come to collect their contributions in a leisurely written monograph? Why not a successor to Macdonald’s book on symmetric polynomials [I. G. Macdonald, ”Symmetric functions and Hall polynomials” (1979; Zbl 0487.20007)]?

Reviewer: D.Laksov

MSC:

14M15	Grassmannians, Schubert varieties, flag manifolds
14F25	Classical real and complex (co)homology in algebraic geometry
20C30	Representations of finite symmetric groups
14C17	Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry

Keywords:

Schubert polynomial; Grothendieck polynomial; cohomology ring of flag manifold; Grothendieck ring of flag manifolds; Hopf algebra; representations of the symmetric group