Quotients of Representation Rings

We give a proof using so-called fusion rings and q-deformations of Brauer algebras that the representation ring of an orthogonal or symplectic group can be obtained as a quotient of a ring Gr(O(∞)). This is obtained here as a limiting case for analogous quotient maps for fusion categories, with the level going to ∞. This in turn allows a detailed description of the quotient map in terms of a reflection group. As an application, one obtains a general description of the branching rules for the restriction of representations ofGl(N) to O(N) and Sp(N) as well as detailed information about the structure of the q-Brauer algebras in the nonsemisimple case for certain specializations. It is well known that one can study the combinatorics of the finite dimensional representations of the general linear groups Gl(N) in a uniform way essentially independently of the dimension N , using the representation theory of the symmetric groups. A similar approach is possible for orthogonal and symplectic groups. In particular, it is possible to obtain their Grothendieck rings as quotients from a large ring, denoted here by Gr(O(∞)) (see e.g. [KT]). More recently, quotients Gr(G) of the Grothendieck ring Gr(G) of a semisimple Lie group G, depending on a positive integer , have been studied which have only finitely many simple objects up to isomorphism. They originally arose in mathematical physics and are usually called fusion rings. One of the observations in this paper is that there is more than just a formal similarity between these two quotients: We show that the quotient map from Gr(O(∞)) onto the Grothendieck ring Gr(Sp(N)) of a symplectic group can be obtained as a limit of the quotient map Gr(Sp(M)) to Gr(Sp(M)) (M) for M → ∞; here (M) depends on M in a simple linear way. A similar result also holds for orthogonal groups. This allows, among other things, for a simple explicit description of the quotient map in terms of a reflection group. One of the applications is an explicit formula for the restriction multiplicities for representations from Gl(N) to O(N) and to Sp(N), respectively; see below for more details and related results. As another application, we also obtain results about the structure of a q-deformation Ĉn(r, q) of Brauer’s centralizer algebras Cn(x) when r = q−N−1, with N > 0 even. In our approach, we get these results as a limiting case of rather deep results about tilting modules of quantum groups. This also makes the appearance of parabolic Kazhdan-Lusztig polynomials fairly natural. Here is the contents of our paper in more detail. The first section reviews results about so-called fusion rings, which are certain quotients of the representation Received by the editors December 11, 2006 and, in revised form, January 11, 2011. 2010 Mathematics Subject Classification. Primary 22E46. This work was partially supported by NSF grant DMS 0302437. c ©2011 American Mathematical Society Reverts to public domain 28 years from publication