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Littlewood-Richardson rules for Grassmannians. (English) Zbl 1053.05121
One of the equivalent forms of the classical Littlewood-Richardson rule describes the structure constants obtained when the cup product of two Schubert classes in the cohomology ring of a complex Grassmannian is written as a linear combination of Schubert classes. In the paper under review the authors give a short and self-contained argument which shows that this rule is a direct consequence of the Pieri formula [M. Pieri, Lomb. Ist. Rend. (2) XXVI. 534–546 (1893), XXVII. 258–273 (1894; JFM 25.1038.02)] for the product of a Schubert class with a special Schubert class.
The Littlewood-Richardson rule has a generalization [J. R. Stembridge, Adv. Math. 74, No. 1, 87–134 (1989; Zbl 0677.20012)] for the Grassmannians which parametrize maximal isotropic subspaces of \({\mathbb C}^n\) equipped with a symplectic or orthogonal form. The same arguments work equally well in more difficult cases and give a simple derivation of the rule of Stembridge from the analogues of the Pieri formula in [H. Hiller and B. Boe, Adv. Math. 62, 49–67 (1986; Zbl 0611.14036)].

MSC:
05E10 Combinatorial aspects of representation theory
05E05 Symmetric functions and generalizations
14M15 Grassmannians, Schubert varieties, flag manifolds
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References:
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