The support of the tensor product decomposition of integrable irreducible highest weight representations of a symmetrizable Kac-Moody Lie algebra
g
\mathfrak {g}
defines a semigroup of triples of weights. Namely, given
λ
\lambda
in the set
P
+
P_+
of dominant integral weights,
V
(
λ
)
V(\lambda )
denotes the irreducible representation of
g
\mathfrak {g}
with highest weight
λ
\lambda
. We are interested in the tensor semigroup
Γ
N
(
g
)
≔
{
(
λ
1
,
λ
2
,
μ
)
∈
P
+
3
|
V
(
μ
)
⊂
V
(
λ
1
)
⊗
V
(
λ
2
)
}
,
\begin{equation*} \Gamma _{\mathbb {N}}(\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+}^3\,|\, V(\mu )\subset V(\lambda _1)\otimes V(\lambda _2)\}, \end{equation*}
and in the tensor cone
Γ
(
g
)
\Gamma (\mathfrak {g})
it generates:
Γ
(
g
)
≔
{
(
λ
1
,
λ
2
,
μ
)
∈
P
+
,
Q
3
|
∃
N
≥
1
V
(
N
μ
)
⊂
V
(
N
λ
1
)
⊗
V
(
N
λ
2
)
}
.
\begin{equation*} \Gamma (\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+,{\mathbb {Q}}}^3\,|\,\exists N\geq 1 \quad V(N\mu )\subset V(N\lambda _1)\otimes V(N\lambda _2)\}. \end{equation*}
Here,
P
+
,
Q
P_{+,{\mathbb {Q}}}
denotes the rational convex cone generated by
P
+
P_+
.
In the special case when
g
\mathfrak {g}
is a finite-dimensional semisimple Lie algebra, the tensor semigroup is known to be finitely generated and hence the tensor cone to be convex polyhedral. Moreover, the cone
Γ
(
g
)
\Gamma (\mathfrak {g})
is described in Belkale and Kumar [Invent. Math. 166 (2006), pp. 185–228] by an explicit finite list of inequalities.
In general,
Γ
(
g
)
\Gamma (\mathfrak {g})
is neither polyhedral, nor closed. In this article we describe the closure of
Γ
(
g
)
\Gamma (\mathfrak {g})
by an explicit countable family of linear inequalities for any untwisted affine Lie algebra, which is the most important class of infinite-dimensional Kac-Moody algebra. This solves a Brown-Kumar’s conjecture in this case (see Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936]).
The difference between the tensor cone and the tensor semigroup is measured by the saturation factors. Namely, a positive integer
d
d
is called a saturation factor, if
V
(
N
λ
1
)
⊗
V
(
N
λ
2
)
V(N\lambda _1)\otimes V(N\lambda _2)
contains
V
(
N
μ
)
V(N\mu )
for some positive integer
N
N
then
V
(
d
λ
1
)
⊗
V
(
d
λ
2
)
V(d\lambda _1)\otimes V(d\lambda _2)
contains
V
(
d
μ
)
V(d\mu )
, assuming that
μ
−
λ
1
−
λ
2
\mu -\lambda _1-\lambda _2
belongs to the root lattice. For
g
=
s
l
n
\mathfrak {g}={\mathfrak {sl}}_n
, the famous Knutson-Tao theorem asserts that
d
=
1
d=1
is a saturation factor (see Knutson and Tao [J. Amer. Math. Soc. 12 (1999), pp. 1055–1090]). More generally, for any simple Lie algebra, explicit saturation factors are known. In the Kac-Moody case,
Γ
N
(
g
)
\Gamma _{\mathbb {N}}(\mathfrak {g})
is not necessarily finitely generated and hence the existence of such a factor is unclear a priori. Here, we obtain explicit saturation factors for any affine Kac-Moody Lie algebra. For example, in type
A
~
n
\tilde A_n
, we prove that any integer
d
≥
2
d\geq 2
is a saturation factor, generalizing the case
A
~
1
\tilde A_1
shown in Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936].