Quotients of Representation Rings

TL;DR

This paper demonstrates how the representation rings of orthogonal and symplectic groups can be derived as quotients of a universal ring using fusion rings and q-deformations, providing insights into branching rules and algebra structures.

Contribution

It introduces a new approach to obtain representation rings as quotients via fusion rings and q-deformations, linking to reflection groups and branching rules.

Findings

Representation rings of orthogonal and symplectic groups are quotients of a universal ring.

Detailed description of the quotient map in terms of a reflection group.

Analysis of the structure of q-Brauer algebras in nonsemisimple cases.

Abstract

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(\infinity)). This is obtained here as a limiting case for analogous quotient maps for fusion categories, with the level going to \infinity. 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 of Gl(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.