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This is a proceedings article reviewing a recent combinatorial constructionof the su(n) WZNW fusion ring by C. Stroppel and the author. It contains onenovel aspect: the explicit derivation of an algorithm for the computation offusion coefficients different from the Kac-Walton formula. The discussion ispresented from the point of view of a vertex model in statistical mechanicswhose partition function generates the fusion coefficients. The statisticalmodel can be shown to be integrable by linking its transfer matrix to aparticular solution of the Yang-Baxter equation. This transfer matrix can beidentified with the generating function of an (infinite) set of polynomials ina noncommutative alphabet: the generators of the local affine plactic algebra.The latter is a generalisation of the plactic algebra occurring in the contextof the Robinson-Schensted correspondence. One can define analogues of Schurpolynomials in this noncommutative alphabet which become identical to thefusion matrices when represented as endomorphisms over the state space of theintegrable model. Crucial is the construction of an eigenbasis, the Bethevectors, which are the idempotents of the fusion algebra.

The su(n) WZNW fusion ring as integrable model: a new algorithm to compute fusion coefficients