A Conjectural Presentation of Fusion Algebras

Let G be a connected, simply-connected, simple algebraic group over C. We fix a Borel subgroup B of G and a maximal torus T ⊂ B. We denote their Lie algebras by g, b, h respectively. Let P+ ⊂ h be the set of dominant integral weights. For any λ ∈ P+, let V (λ) be the finite dimensional irreducible g-module with highest weight λ. We fix a positive integer l and let Rl(g) be the free Z-module with basis {V (λ) : λ ∈ Pl}, where Pl := {λ ∈ P+ : λ(θ) ≤ l}, θ is the highest root of g and θ is the associated coroot. There is a product structure on Rl(g), called the fusion product (cf. Section 3), making it a commutative associative (unital) ring. In this paper, we consider its complexification Rl (g), called the fusion algebra, which is a finite dimensional (commutative and associative) algebra without nilpotents. Let R(g) be the Grothendieck ring of finite dimensional representations of g and let RC(g) be its complexification. As in subsection ??, there is a surjective ring homomorphism β : R(g) → Rl(g). Let β : RC(g) → Rl (g) be its complexification and let Il(g) denote the kernel of β. Since Rl (g) is an algebra without nilpotents, Il(g) is a radical ideal. The main aim of this note is to conjecturally describe this ideal Il(g). Before we describe our result and conjecture, we briefly describe the known results in this direction. Identify the complexified representation ring RC(g) with the polynomial ring C[χ1, . . . , χr], where r is the rank of g and χi denotes the character of V (ωi), the i th fundamental representation of g. It is generally believed (initiated by the physicists) that there exists an explicit potential function F = Fl(g) (depending upon g and l) in Z[χ1, . . . , χr], coming from representation theory of g, with the property that the ideal generated over the integers by the gradient of F , i.e., 〈∂F/∂χ1, . . . , ∂F/∂χr〉, is precisely the