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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The moduli space of $SU(3)$-flat connections
and the fusion rules


Author: Masato Hayashi
Journal: Proc. Amer. Math. Soc. 127 (1999), 1545-1555
MSC (1991): Primary 58D29, 57M05; Secondary 17B81, 17B10
Published electronically: January 29, 1999
MathSciNet review: 1476136
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Abstract | References | Similar Articles | Additional Information

Abstract: The aim of this paper is to determine the existence condition of the moduli space of $SU(3)$-flat connections on $3$-holed $2$-sphere $D$, the so-called pair of pants, and to study its relationship to the $\widehat {\mathfrak{sl}}(3;\mathbb{C})$ fusion rules. The existence condition can be expressed by a system of inequalities with the entries of highest weights with respect to the fundamental weights. This gives a necessary condition for the fusion coefficents to be nontrivial. We also find that the fusion coefficient of a triplet of extremal highest weights equals one. This can be considered a quantum counterpart of the PRV-conjecture.


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Additional Information

Masato Hayashi
Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo
Email: hayashim@ms318sun.ms.u-tokyo.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04674-2
PII: S 0002-9939(99)04674-2
Keywords: Representations of the fundamental group of a surface, Bruhat decomposition, fusion rules, PRV-conjecture
Received by editor(s): May 13, 1996
Received by editor(s) in revised form: September 3, 1997
Published electronically: January 29, 1999
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 1999 American Mathematical Society