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Quantum multiplication of Schur polynomials. (English) Zbl 0936.05086
This paper presents formulas for the quantum multiplication of classes of Schubert varieties: These formulas are “quantum analogues” of the classical Littlewood-Richardson rule and the Kostka numbers, which describe the expansion of products of Schur functions. The basic algorithm involves rim hooks of Young diagrams: Quantum Kostka numbers are shown to be the numbers of tableaux satisfying a certain condition, an efficient algorithm for computing the quantum Littlewood-Richardson numbers is derived. The natural isomorphism of the quantum cohomology rings of Grassmannians \(\text{QH}^*(\text{Gr}(l,l+k))\) and \(\text{QH}^*(\text{Gr}(k,l+k))\) leads to dual versions of these results. Finally, the relation between the quantum cohomology ring and the Verlinde (or fusion) algebra is described.

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