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# Nicolas Ressayre, Universite Claude Bernard Lyon 1 – Geometric Methods in Represenation Theory Seminar

## April 12, 2018 @ 4:00 pm - 5:00 pm

**Title: ***On the tensor semigroup of affine Kac-Moody Lie algebras.*

**Abstract:** In this talk, we are interested in the decomposition of the tensor product of two representations of a symmetrizable Kac-Moody Lie algebra 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. Consider the tensor cone

Γ(g):={(λ1,λ2,μ)∈P+3 |∃N >1 L(Nμ)⊂L(Nλ1)⊗L(Nλ2)}.

If g is finite dimensional, Γ(g) is a polyhedral convex cone described by Belkale-Kumar by an explicit finite list of inequalities. In general, Γ(g) is nor polyhedral, nor closed. We will describe the closure of Γ(g) by an explicit countable family of linear inequalities, when g is untwisted affine. This solves a Brown-Kumar’s conjecture in this case.