  In this paper, we are interested in the decomposition of the tensor product
of two representations ofa symmetrizable Kac-Moody Lie algebra ${\mathfrak g}$,
or more precisely in the tensor cone of~${\mathfrak g}$.As usual, we
parametrize the integrable, highest weight (irreducible) representations
of~${\mathfrak g}$ by their highest weights. Then, the triples of such
representations such that the last one is contained in the tensor product of
the first two is a semigroup.This semigroup generates a rational convex cone $
\Gamma({\mathfrak g})$ called tensor cone.If ${\mathfrak g}$~is
finite-dimensional, $\Gamma({\mathfrak g})$~is a polyhedral convex cone. In
2006, Belkale and the first author described this cone by an explicit finite
list of inequalities.In 2010, this list of inequalities was proved to be
irredundant by the second author:each such inequality corresponds to a
codimension one face.In general, $\Gamma({\mathfrak g})$~is neither polyhedral,
nor closed.Brown and the first author obtained a list of inequalities that
describe $\Gamma({\mathfrak g})$ conjecturally. Here, we prove that each of
these inequalities corresponds to a codimension one face of~$\Gamma({\mathfrak
g})$.
