an:00691453
Zbl 0830.20064
Georgiev, Galin; Mathieu, Olivier
Fusion rings for modular representations of Chevalley groups
EN
Sally, Paul J. jun. (ed.) et al., Mathematical aspects of conformal and topological field theories and quantum groups. AMS-IMS-SIAM summer research conference, June 13-19, 1992, Mount Holyoke College, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 175, 89-100 (1994).
1994
a
20G05 20G45 20C20 17B10 20G15
characters of reduced tensor products; finite dimensional representations; semi-simple algebraic groups; dominant weights; bottom alcove; tilting modules; tilting summands; highest weights; explicit formula; fusion rule; quantum groups; affine Kac-Moody algebras
Let \(V\) and \(W\) be finite dimensional representations of a semi-simple algebraic group over a field of characteristic \(p > 0\). Suppose all dominant weights of \(V\) and \(W\) are in the bottom alcove so that \(V\) and \(W\) are (partial) tilting modules [cf. \textit{S. Donkin}'s paper Math. Z. 212, 39-60 (1993; Zbl 0798.20035)]. By the `reduced' tensor product of \(V\) and \(W\) we shall understand the summand of \(V \otimes W\) obtained by ignoring all tilting summands whose highest weights are outside the first alcove. In this paper the authors deduce an explicit formula (a fusion rule) for the character of this `reduced' tensor product.
Similar results covering also quantum groups at roots of unity have been obtained by the reviewer and \textit{J. Paradowski} [Commun. Math. Phys. 169, 563-588 (1995; Zbl 0827.17010)]. In his 1993 Harvard thesis \textit{M. Finkelberg} has presented analogous results for affine Kac-Moody algebras.
For the entire collection see [Zbl 0801.00049].
H.H.Andersen (Aarhus)
Zbl 0827.17010; Zbl 0798.20035