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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quantum cohomology and the $ k$-Schur basis


Authors: Luc Lapointe and Jennifer Morse
Journal: Trans. Amer. Math. Soc. 360 (2008), 2021-2040
MSC (2000): Primary 05E05; Secondary 14N35
Published electronically: October 5, 2007
MathSciNet review: 2366973
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Abstract: We prove that structure constants related to Hecke algebras at roots of unity are special cases of $ k$-Littlewood-Richardson coefficients associated to a product of $ k$-Schur functions. As a consequence, both the 3-point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to $ \widehat{su}(\ell)$ are shown to be $ k$-Littlewood-Richardson coefficients. From this, Mark Shimozono conjectured that the $ k$-Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas $ k$-Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual $ k$-Schur functions defined on weights of $ k$-tableaux that, given Shimozono's conjecture, form the Schubert basis for the cohomology of the loop Grassmannian. We derive several properties of these functions that extend those of skew Schur functions.


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Additional Information

Luc Lapointe
Affiliation: Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile
Email: lapointe@inst-mat.utalca.cl

Jennifer Morse
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124
Address at time of publication: Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104
Email: morsej@math.miami.edu, morsej@math.drexel.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04287-0
PII: S 0002-9947(07)04287-0
Received by editor(s): September 2, 2005
Received by editor(s) in revised form: December 20, 2005
Published electronically: October 5, 2007
Additional Notes: Research of the first author was supported in part by FONDECYT (Chile) grant #1030114, the Anillo Ecuaciones Asociadas a Reticulados financed by the World Bank through the Programa Bicentenario de Ciencia y Tecnologia, and the Programa Reticulados y Ecuaciones of the Universidad de Talca
Research of the second author was supported in part by NSF grant #DMS-0400628
Article copyright: © Copyright 2007 American Mathematical Society



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