On the tensor semigroup of affine Kac-Moody lie algebras
Authors: Ressayre Nicolas 
Published: February 2022

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma Subscript double-struck upper N Baseline left-parenthesis German g right-parenthesis colon-equal StartSet left-parenthesis lamda 1 comma lamda 2 comma mu right-parenthesis element-of upper P Subscript plus Superscript 3 Baseline vertical-bar upper V left-parenthesis mu right-parenthesis subset-of upper V left-parenthesis lamda 1 right-parenthesis circled-times upper V left-parenthesis lamda 2 right-parenthesis EndSet comma">

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:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma left-parenthesis German g right-parenthesis colon-equal StartSet left-parenthesis lamda 1 comma lamda 2 comma mu right-parenthesis element-of upper P Subscript plus comma double-struck upper Q Superscript 3 Baseline vertical-bar there-exists upper N greater-than-or-equal-to 1 upper V left-parenthesis upper N mu right-parenthesis subset-of upper V left-parenthesis upper N lamda 1 right-parenthesis circled-times upper V left-parenthesis upper N lamda 2 right-parenthesis EndSet period">

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].