ON THE FACES OF THE TENSOR CONE OF SYMMETRIZABLE KAC-MOODY LIE ALGEBRAS
Sur les faces du cône tensoriel d'une algèbre de Kac-Moody symétrisable
Résumé
In this paper, we are interested in the decomposition of the tensor product of two representations of a symmetrizable Kac-Moody Lie algebra g, and more precisely in the tensor cone of g. Let P + be the set of dominant integral weights. For λ ∈ P + , L(λ) denotes the (irreducible) integrable, highest weight representation of g with highest weight λ. Let P +,Q be the rational convex cone generated by P +. Consider the tensor cone Γ(g) := {(λ 1 , λ 2 , µ) ∈ P 3 +,Q : ∃N ≥ 1 such that L(Nµ) ⊂ L(Nλ 1)⊗L(Nλ 2)}. If g is finite dimensional, Γ(g) is a polyhedral convex cone described in [BK06] by an explicit finite list of inequalities. In [Res10] this list of inequalities is proved to be irredundant: each inequality corresponds to a codimension one face. In general, Γ(g) is neither polyhedral, nor closed. Brown-Kumar [BK14] obtained a list of inequalities that describe Γ(g) conjecturally. Here, we prove that each of Brown-Kumar's inequalities corresponds to a codimension one face of Γ(g).
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