A combinatorial description of the sl(n) fusion ring

Korff, C. and Stroppel, C. (2009) A combinatorial description of the sl(n) fusion ring. Oberwolfach Reports, 6(1), pp. 846-850. (doi: 10.4171/OWR/2009/15)

Abstract

This talk presents a combinatorial construction of the fusion ring (or Verlinde algebra) associated with the bsl(n)k-Wess-Zumino-Novikov-Witten (WZNW) model in conformal field theory. The aim of the talk is to describe a precise relationship between this fusion ring and the (small) quantum cohomology of the Grassmannian. As a result we get explicit identities between the structure constants of the two rings. We describe the structure constants of the fusion ring in terms of cyclic non-commutative Schur functions acting on a space Hk. This allows a simplified proof of associativity of the fusion product and a simple derivation of the Verlinde formula. Moreover, we explicitly construct a common eigenbasis for our Schur functions using the so-called Bethe Ansatz (a standard tool in quantum integrable models). The eigenvalues are then given by certain Weyl characters expressed in terms of commutative Schur functions, and one can deduce that the non-commutative symmetric functions share many nice properties with the usual commutative symmetric function. This construction directly relates to results of Rietsch ([12]) on the quantum cohomology side and is motivated by the work of Postnikov ([10]). Finally these eigenvalues will also be used to show (via the Verlinde formula) that our combinatorially defined ring is in fact the fusion ring, the eigenvalues from above turn up as entries in the modular S-matrix defining the structure constants of the fusion ring. Details and proofs will appear in [9].

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Korff, Professor Christian and Stroppel, Dr Catharina
• Authors: Korff, C., and Stroppel, C.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Oberwolfach Reports
• ISSN: 1660-8933
• ISSN (Online): 1660-8941