zbMATH — the first resource for mathematics

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.

05E10 Combinatorial aspects of representation theory
05E05 Symmetric functions and generalizations
14M15 Grassmannians, Schubert varieties, flag manifolds
PDF BibTeX Cite
Full Text: DOI arXiv
[1] S. Agnihotri, Quantum Cohomology and the Verlinde Algebra, Ph.D. thesis, University of Oxford, 1995.
[2] Beauville, A., The Verlinde formula for PGL_{p}, The mathematical beauty of physics (saclay, 1996), Advanced series in mathematical physics, 24, (1997), World Scientific River Edge, p. 141-151 · Zbl 1058.14500
[3] Bertram, A., Quantum Schubert calculus, Adv. math., 128, 289-305, (1997) · Zbl 0945.14031
[4] Ciocan-Fontanine, I., On quantum cohomology rings of partial flag varieties, Duke math. J., 98, 485-524, (1999) · Zbl 0969.14039
[5] Cummins, C.J., su(n) and sp(2n) WZW fusion rules, J. phys. A, 24, 391-400, (1991) · Zbl 0727.17016
[6] Fulton, W., Young tableaux with applications to representation theory and geometry, (1997), Cambridge Univ. Press Cambridge · Zbl 0878.14034
[7] Fulton, W.; Pandharipande, R., Notes on stable maps and quantum cohomology, Algebraic geometry santa cruz 1995, Proceedings of symposia in pure mathematics, 62, (1997), Amer. Math. Soc Providence, p. 45-96 · Zbl 0898.14018
[8] Fomin, S.; Gelfand, S.; Postnikov, A., Quantum Schubert polynomials, J. amer. math. soc., 10, 565-596, (1997) · Zbl 0912.14018
[9] Goodman, F.; Wenzl, H., Littlewood – richardson coefficients for Hecke algebras at roots of unity, Adv. math., 82, 244-265, (1990) · Zbl 0714.20004
[10] James, G.D.; Kerber, A., The representation theory of the symmetric group, Encyclopedia of mathematics and its applications, 16, (1981), Addison-Wesley Reading
[11] Kac, V., Infinite-dimensional Lie algebras, (1990), Cambridge Univ. Press Cambridge · Zbl 0716.17022
[12] Macdonald, I.G., Symmetric functions and Hall polynomials, (1995), Oxford Univ. Press London · Zbl 0487.20007
[13] Siebert, B.; Tian, G., On quantum cohomology rings of Fano manifolds and a formula of Vafa and intriligator, Asian J. math., 1, 679-695, (1997) · Zbl 0974.14040
[14] Walton, M., Fusion rules in wess – zumino – witten models, Nuclear phys. B, 340, 777-790, (1990)
[15] Witten, E., The Verlinde algebra and the cohomology of the Grassmannian, Geometry, topology, and physics, Conference Proceedings and lecture notes in geometric topology, IV, (1995), International Press Cambridge, p. 357-422 · Zbl 0863.53054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.