In mathematics, the Dynkin index
x
λ
of a representation with highest weight
|
λ
|
of a compact simple Lie algebra
g
that has a highest weight
λ
is defined by
t
r
(
t
a
t
b
)
=
2
x
λ
g
a
b
evaluated in the representation
|
λ
|
. Here
t
a
are the matrices representing the generators, and
g
a
b
is given by
t
r
(
t
a
t
b
)
=
2
g
a
b
evaluated in the defining representation.
By taking traces, we find that
x
λ
=
dim
|
λ
|
2
dim
g
(
λ
,
λ
+
2
ρ
)
where the Weyl vector
ρ
=
1
2
∑
α
∈
Δ
+
α
is equal to half of the sum of all the positive roots of
g
. The expression
(
λ
,
λ
+
2
ρ
)
is the value quadratic Casimir in the representation
|
λ
|
. The index
x
λ
is always a positive integer.
In the particular case where
λ
is the highest root, meaning that
|
λ
|
is the adjoint representation,
x
λ
is equal to the dual Coxeter number.
