Seminar Sophus Lie 3 (1993) 239–253

Tables for an effective enumeration of real representations of quasi-simple Lie groups

Richard Bödi, Michael Joswig

The purpose of this paper is to provide the data which are necessary to calculate the (real) dimension, the centralizer and the kernel of a finitedimensional irreducible real representation of a quasi-simple Lie group.

The authors have implemented a software package based on the LiE system described in [7] which allows the user to access these data easily. For further information on this software package see the last section of this paper.

The main source for our tables in this paper is J. Tits’ Springer Lecture Note 40 [11]. This book contains all of the data we shall need. Unfortunately, much of this information is given in a rather implicit way and one has to perform additional calculations to obtain the final results which can be found in the tables below. Part of this data can also be found in [9]. Note that in this paper we use the (more standard) labelling of the Dynkin diagrams by Bourbaki [1], pp. 250– 275. This differs from Tits’ notation [11] in the case of the exceptional types E_{6} , E_{7} , E_{8} , and F_{4} . The theory of irreducible real representations of semi-simple or reductive Lie groups can be found in [5], [9], [4], [12], e. g. A summary of the basic facts of this theory is included in [11]. Extensive tables covering the complex representations of complex quasi-simple Lie groups are contained in [8] and [3].

For the sake of completeness and to fix notation we shall start with some basics on the complex representation theory and then proceed with the real case.

We would like to thank Theo Grundhöfer for giving us many helpful suggestions and comments.

Complex representations of quasi-simple Lie groups

Let G be a quasi-simple complex Lie group with 𝔤 as its (simple) Lie algebra. Every (complex) finite-dimensional continuous representation P : G → GL(V) yields a representation ρ : 𝔤 → 𝔤𝔩(V) by differentiation.

The adjoint representation ad is the (right) regular representation of 𝔤 defined by ad : 𝔤 → 𝔤𝔩(V) : x ↦ (z ↦ [z, x]), where V is 𝔤 considered just as a vector space. A Cartan subalgebra 𝔥 of 𝔤 is a nilpotent self-normalizing subalgebra of 𝔤. All Cartan subalgebras of 𝔤 are in fact conjugate and thus have the same dimension r. The integer r is called the rank of 𝔤. A weight λ of the representation ρ is an element of the dual 𝔥^{∗} with the property that there is a v ∈ V \{0} such that h^{ρ}(v) = λ(h)·v holds for every h ∈ 𝔥. A root of 𝔤 is a nonzero weight of the representation ad. Since 𝔥 is abelian, 𝔥 = {x ∈ 𝔤 | x^{ad(𝔥)} = 0} by Engel’s theorem. Thus the set Φ of all roots of 𝔤 generates the dual 𝔥^{∗} because 𝔤 is simple. Moreover, there is an ℝ−basis Δ ⊆ Φ of 𝔥^{∗} such that every root λ is a linear combination of elements of Δ having all coefficients either ≥ 0 (then λ is called a positive root) or ≤ 0 (and λ is called a negative root). The elements of Δ are called fundamental roots.

Since semisimple Lie algebras are reductive, we may restrict our attention to irreducible representations. For every finite-dimensional irreducible representation ρ there is a unique weight λ_{ρ} such that the difference λ_{ρ} − λ is a positive linear combination of fundamental roots for every weight λ of ρ. This weight λ_{ρ} is called the highest weight of ρ and the representation ρ is uniquely determined by λ_{ρ} up to equivalence.

The Killing form κ on 𝔤 is the symmetric bilinear form defined by κ(x, y) = trace(ad(x)ad(y)). Since κ is nondegenerate on 𝔥 × 𝔥, for any φ ∈ 𝔥^{∗} there is a unique element h_{φ} ∈ 𝔥 such that φ(h) = κ(h_{φ}, h) holds for every h ∈ 𝔥. Setting (φ, ψ) := κ(h_{φ}, h_{ψ}), the mapping (𝔥^{∗}, (. , .)) → (𝔥, κ) : φ ↦ h_{φ} hence is an isometry.

Now assume that Δ = {α_{1}, …, α_{r}}. For λ ∈ 𝔥^{∗} and 1 ≤ i ≤ r set f_{i}(λ) := 2(α_{i}, λ)(α_{i}, α_{i})^{−1} . Using the linear forms f_{i} we can decide whether or not an element λ ∈ 𝔥^{∗} is a weight or a highest weight of an irreducible representation ρ. Namely, λ is a (highest) weight of ρ iff f_{1}(λ), …, f_{r}(λ) ∈ ℤ (∈ ℕ_{0} ). If we define λ_{i} ∈ ⟨Φ⟩_{ℝ} by f_{j}(λ_{i}) = δ_{ij}, the highest weight λ_{ρ} of ρ can be written as

λ_{ρ} = ∑_{i=1}^{r} f_{i}(λ_{ρ}) · λ_{i}.

In particular, the elements λ_{i} are highest weights of irreducible representations ρ_{i} which are called the fundamental representations corresponding to the fundamental roots α_{i}. The free semigroup Λ_{+}(𝔤, 𝔥) := ⨁_{i=1}^{r} ℕ_{0}λ_{i} is called the weight space of 𝔤 with respect to 𝔥. Since all Cartan subalgebras of 𝔤 are conjugate, we may abbreviate Λ_{+}(𝔤) := Λ_{+}(𝔤, 𝔥). Summing up, we obtain (see [10], 3.2, [13], Thm. 4.7.1, [6], VII.3, or [4], 44.1)

Theorem 1. The elements of the weight space Λ_{+}(𝔤) and the equivalence classes of irreducible finite-dimensional (complex) representations of 𝔤 are in a one-to-one correspondence.

The complex dimension dim_{ℂ} ρ can be calculated by using Weyl’s formula (cp. [10], 3.8, [13], Thm. 4.14.6, [6], p. 257, or [4], 47.8):

dim_{ℂ} ρ = ∏_{φ∈Φ_{+}} \frac{(λ_{ρ} + δ, φ)}{(δ, φ)} with δ = ∑_{i=1}^{r} λ_{i},

where Φ_{+} ⊆ Φ denotes the set of all positive roots in Φ. Expressing dim_{ℂ} ρ in terms of f_{i}(λ_{ρ}) and (λ_{i}, φ) we have

dim_{ℂ} ρ = ∏_{φ∈Φ_{+}} \frac{∑_{i=1}^{r}(f_{i}(λ_{ρ}) + 1)(λ_{i}, φ)}{∑_{i=1}^{r}(λ_{i}, φ)}

and if φ = ∑_{j=1}^{r} k_{j}α_{j} ∈ Φ_{+} we moreover have

(λ_{i}, φ) = ∑_{j=1}^{r} k_{j}(λ_{i}, α_{j}) = k_{i}(λ_{i}, α_{i}) = \frac{1}{2}k_{i}(α_{i}, α_{i}).

The sets Φ_{+} of positive roots are listed in [11]. An algorithm for computing Φ_{+} is presented in [6], Chapt. IV, Thm. XVI. The values (α_{i}, α_{i}) can be found in [6], Chapt. IV, §5, or in [10], (2.14), e. g.

Real representations of quasi-simple Lie groups

By integration, every Lie algebra homomorphism ψ : 𝔤 → 𝔤′ gives rise to a homomorphism Ψ : G̃ → G̃′ between the associated simply connected groups G̃ and G̃′. We shall always write the corresponding uppercase letters for the integrated homomorphisms. In particular, by P_{i} we denote the representation of G̃ which corresponds to the fundamental representation ρ_{i} of 𝔤.

If 𝔤 is a simple real Lie algebra, then either

I) its complexification 𝔤_{ℂ} := 𝔤 ⊗_{ℝ} ℂ is a simple complex Lie algebra (and

𝔤 is called a real form of 𝔤_{ℂ}) or II) it is the realification of a simple complex Lie algebra 𝔤_{ℂ}, i. e. 𝔤 is the

algebra 𝔤_{ℂ} considered as an ℝ-algebra.

Now let ρ : 𝔤 → 𝔤𝔩(V) be an irreducible real representation. Then ρ_{ℂ} := ρ ⊗ id_{ℂ} is a complex representation of 𝔤_{ℂ} on V_{ℂ} := V ⊗_{ℝ} ℂ which may not be irreducible, see the cases (2), (3) of the next section. Via the natural embedding v ↦ v ⊗ 1 of V into V_{ℂ} we consider 𝔤𝔩(V) as a subspace of 𝔤𝔩(V_{ℂ}) and so GL(V) is a subset of GL(V_{ℂ}).

The following commutative diagram displays the situation.

𝔤  \overline{ρ} 𝔤𝔩(V)

[figure: ι; ⊆]

𝔤_{ℂ}  \overline{ρ_{ℂ}} 𝔤𝔩(V_{ℂ})

[figure: G̃; exp; P; GL(V); exp]

[figure: I; ⊆]

[figure: G̃_{ℂ}; exp; P_{ℂ}; GL(V_{ℂ}); exp]

Case I) 𝔤 is a real form of 𝔤_{ℂ}. The algebra 𝔤 is embedded in 𝔤_{ℂ} via the injection ι : g ↦ g ⊗ 1. The integrated map I : G̃ → G̃′ is not necessarily injective, but its kernel must lie in the center of G̃ since G̃ is quasi-simple. The kernel ker I, which is called the linearizer of G̃, turns out to be the intersection of the kernels of all representations IP_{i} of G̃. The quotient G̃/ker I is the maximal linearly representable group which is locally isomorphic to G̃. To determine the group G̃^{P} = G̃^{IP_{ℂ}} which is represented by P, we thus have to know how the center Z(G̃) is mapped via I to Z(G̃_{ℂ}). To identify the elements of Z(G̃), table 1 shows how Z(G̃) is embedded in the group Z(K̃), where K̃ denotes the universal covering of a maximal compact subgroup K of G̃/Z(G̃) . Table 2 lists the image of Z(G̃) under I as well as the linearizer. We shall use the symbol ⟨x⟩_{n} to express that the element x generates a cyclic group of order n, where n = ∞ means that ⟨x⟩_{∞} ≅ ℤ.

Table 1. The embedding of Z(G̃) in Z(K̃).

type of K̃  Z(K̃)  Z(G̃)
type A_{n}^{ℝ}
n = 1  ℝ  ℝ  ⟨z⟩_{∞} for some z ∈ ℝ
n = 2l  B_{l}^{ℝ,0}  ⟨y⟩_{2}  ⟨z⟩_{2} , z = y
n = 4l+1 D_{2l+1}^{ℝ,0}  ⟨y⟩_{4}  ⟨z⟩_{4} , z = y
n = 4l+3 D_{2l+2}^{ℝ,0}  ⟨y⟩_{2} × ⟨y′⟩_{2}  ⟨z⟩_{2} × ⟨z′⟩_{2} ,

z = y, z′ = y′

type A_{n}^{ℂ,i}
i = 0  A_{n}^{ℂ,0}  ⟨y⟩_{n+1}  ⟨z⟩_{n+1} , z = y
i > 0  A_{n−i}^{ℂ,0}×  ⟨y⟩_{r} × ⟨y′⟩_{i} × ℝ ⟨z⟩_{g} × ⟨z′⟩_{∞}
A_{i−1}^{ℂ,0}× with r = n−i+1 z = y^{r/g}y^{′i/g}
ℝ  z′ = xy^{a+b}y^{′−a}, x ∈ ℝ

g = gcd(n+1, i) = (n+1)a+ib

type A_{n}^{ℍ}
n = 2l+1  C_{l+1}^{ℍ,0}  ⟨y⟩_{2}  ⟨z⟩_{2} , z = y
type B_{n}^{ℝ,i}
i = 0  B_{n}^{ℝ,0}  ⟨y⟩_{2}  ⟨z⟩_{2} , z = y
i = 1, 2|n  D_{n}^{ℝ,0}  ⟨y⟩_{2} × ⟨y′⟩_{2}  ⟨z⟩_{2} , z = yy′
i = 1, 2|̸n D_{n}^{ℝ,0}  ⟨y⟩_{4}  ⟨z⟩_{2} , z = y^{2}
i = 2  B_{n−1}^{ℝ,0} × ℝ  ⟨y⟩_{2} × ℝ  ⟨z⟩_{2} × ⟨z′⟩_{∞},

z = y, z′ = x ∈ ℝ

Bödi, Joswig 243

Table 1. The embedding of Z(G̃) in Z(K̃), continued.

type of K̃ Z(K̃) Z(G̃)

type B_{n}^{ℝ,i}

i = 4k  D_{i/2}^{ℝ,0}×B_{n−i/2}^{ℝ,0} ⟨y⟩_{2} × ⟨y′⟩_{2} × ⟨y′′⟩_{2} ⟨z⟩_{2} × ⟨z′⟩_{2} ,

z = yy′, z′ = y′′

i = 4k+2 D_{i/2}^{ℝ,0}×B_{n−i/2}^{ℝ,0} ⟨y⟩_{4} × ⟨y′⟩_{2} ⟨z⟩_{2} × ⟨z′⟩_{2} ,

z = y^{2} , z′ = y′

i = 2k+1

2|n−k  D_{n−k}^{ℝ,0}×B_{k}^{ℝ,0} ⟨y⟩_{2} × ⟨y′⟩_{2} × ⟨y′′⟩_{2} ⟨z⟩_{2} × ⟨z′⟩_{2} ,

z = yy′, z′ = y′′

2|̸n−k D_{n−k}^{ℝ,0}×B_{k}^{ℝ,0} ⟨y⟩_{4} × ⟨y′⟩_{2} ⟨z⟩_{2} × ⟨z′⟩_{2} ,

z = y^{2} , z′ = y′

type C_{n}^{ℝ}

n = 2l A_{n−1}^{ℂ,0} × ℝ ⟨y⟩_{n} × ℝ ⟨z⟩_{2} × ⟨z′⟩_{∞},

z = y^{l}, z′ = xy, x ∈ ℝ

n = 2l+1 A_{n−1}^{ℂ,0} × ℝ ⟨y⟩_{n} × ℝ ⟨z⟩_{∞}, z = xy, x ∈ ℝ

type C_{n}^{ℍ,i}

i = 0  C_{n}^{ℍ,0}  ⟨y⟩_{2}  ⟨z⟩_{2} , z = y
i > 0  C_{i}^{ℍ,0}×C_{n−i}^{ℍ,0}  ⟨y⟩_{2} × ⟨y′⟩_{2}  ⟨z⟩_{2} , z = yy′

type D_{n}^{ℝ,i} i = 0

2|n D_{n}^{ℝ,0} ⟨y⟩_{2} × ⟨y′⟩_{2} ⟨z⟩_{2} × ⟨z′⟩_{2} ,

z = y, z′ = y′

2|̸n  D_{n}^{ℝ,0}  ⟨y⟩_{4}  ⟨z⟩_{4} , z = y
i = 1  B_{n−1}^{ℝ,0}  ⟨y⟩_{2}  ⟨z⟩_{2} , z = y

i = 2

2|n D_{n−1}^{ℝ,0} × ℝ ⟨y⟩_{4} × ℝ ⟨z⟩_{2} × ⟨z′⟩_{∞},

z = y^{2} , z′ = xy, x ∈ ℝ

2|̸n D_{n−1}^{ℝ,0} × ℝ ⟨y⟩_{2} × ⟨y′⟩_{2} × ℝ ⟨z⟩_{2} × ⟨z′⟩_{∞},

z = yy′, z′ = xy, x ∈ ℝ

i = 4k

2|n D_{2k}^{ℝ,0}×D_{n−2k}^{ℝ,0} ⟨y⟩_{2} × ⟨y′⟩_{2}× ⟨z⟩_{2} × ⟨z′⟩_{2} × ⟨z′′⟩_{2} ,
⟨y′′′⟩_{2} × ⟨y′′′′⟩_{2}  z = yy′, z′ = y′′′y′′′′,

z′′ = yy′′′

2|̸n  D_{2k}^{ℝ,0}×D_{n−2k}^{ℝ,0} ⟨y⟩_{2} × ⟨y′⟩_{2} × ⟨y′′⟩_{4} ⟨z⟩_{2} × ⟨z′⟩_{4} ,

z = yy′, z′ = yy′′ 244 Bödi, Joswig

Table 1. The embedding of Z(G̃) in Z(K̃), continued.

type of K̃ Z(K̃) Z(G̃)
type D_{n}^{ℝ,i}

i = 4k+2

2|n  D_{2k+1}^{ℝ,0}×  ⟨y⟩_{4}×  ⟨z⟩_{2} × ⟨z′⟩_{4} ,
D_{n−2k−1}^{ℝ,0}  ⟨y′⟩_{4}  z = y′^{2} , z′ = yy′
2|̸n  D_{2k+1}^{ℝ,0}×  ⟨y⟩_{4}×  ⟨z⟩_{2} × ⟨z′⟩_{4} ,
D_{n−2k−1}^{ℝ,0}  ⟨y′⟩_{2} × ⟨y′′⟩_{2}  z = y′y′′, z′ = yy′
i = 2k+1 B_{k}^{ℝ,0}×B_{n−k−1}^{ℝ,0}  ⟨y⟩_{2} × ⟨y′⟩_{2}  ⟨z⟩_{2} × ⟨z′⟩_{2} ,
z = y, z′ = y′

type D_{n}^{ℍ}

n = 2l A_{n−1}^{ℂ,0} × ℝ ⟨y⟩_{n} × ℝ ⟨z⟩_{2} × ⟨z′⟩_{∞},
z = y^{l}, z′ = xy,
x ∈ ℝ
n = 2l+1 A_{n−1}^{ℂ,0} × ℝ ⟨y⟩_{n} × ℝ ⟨z⟩_{∞}, z = xy, x ∈ ℝ

exceptional types

E_{6(−78)}  E_{6(−78)}  ⟨y⟩_{3}  ⟨z⟩_{3} , z = y
E_{6(−26)}  F_{4(−52)}  1l  1l
E_{6(−14)}  D_{5}^{ℝ,0} × ℝ  ⟨y⟩_{4} × ℝ  ⟨z⟩_{∞}, z = xy, x ∈ ℝ
E_{6(2)}  A_{5}^{ℂ,0}× A_{1}^{ℂ,0}  ⟨y⟩_{6} × ⟨y′⟩_{2}  ⟨z⟩_{6} , z = yy′
E_{6(6)}  C_{4}^{ℍ,0}  ⟨y⟩_{2}  ⟨z⟩_{2} , z = y
E_{7(−133)}  E_{7(−133)}  ⟨y⟩_{2}  ⟨z⟩_{2} , z = y
E_{7(−25)}  E_{6(−78)} × ℝ  ⟨y⟩_{3} × ℝ  ⟨z⟩_{∞}, z = xy, x ∈ ℝ
E_{7(−5)}  D_{6}^{ℝ,0}× A_{1}^{ℂ,0}  ⟨y⟩_{2} × ⟨y′⟩_{2} × ⟨y′′⟩_{2} ⟨z⟩_{2} × ⟨z′⟩_{2} ,
z = yy′, z′ = y′′
E_{7(7)}  A_{7}^{ℂ,0}  ⟨y⟩_{8}  ⟨z⟩_{4} , z = y^{2}
E_{8(−248)}  E_{8(−248)}  1l  1l
E_{8(−24)}  E_{7(−133)}× A_{1}^{ℂ,0}  ⟨y⟩_{2} × ⟨y′⟩_{2}  ⟨z⟩_{2} , z = yy′
E_{8(8)}  D_{8}^{ℝ,0}  ⟨y⟩_{2} × ⟨y′⟩_{2}  ⟨z⟩_{2} , z = y
F_{4(−52)}  F_{2(−52)}  1l  1l
F_{4(−20)}  B_{4}^{ℝ,0}  ⟨y⟩_{2}  1l
F_{4(4)}  C_{3}^{ℍ,0}× A_{1}^{ℂ,0}  ⟨y⟩_{2} × ⟨y′⟩_{2}  ⟨z⟩_{2} , z = yy′
G_{2(−14)}  G_{2(−14)}  1l  1l
G_{2(2)}  A_{1}^{ℂ,0}× A_{1}^{ℂ,0}  ⟨y⟩_{2} × ⟨y′⟩_{2}  ⟨z⟩_{2} , z = yy′

Bödi, Joswig 245

Table 2. Description of I Z(G̃) and linearizer.

I Z(G̃)  linearizer

type A_{n}^{ℝ} Z(G̃_{ℂ}) = ⟨x⟩_{n+1}

n = 1  z ↦ x  ⟨z^{2}⟩
n = 2l  z ↦ 1  Z
n = 4l+1 z ↦ x^{2l+1}  ⟨z^{2}⟩
n = 4l+3 z ↦ x^{2l+2} , z′ ↦ x^{2l+2}  ⟨zz′⟩

type A_{n}^{ℂ,i} Z(G̃_{ℂ}) = ⟨x⟩_{n+1}

i = 0  z ↦ x  1l
i > 0  z ↦ x^{(n+1)/g}, z′ ↦ x^{b}  ⟨z^{g−b}z^{′(n+1)/g}⟩
g = gcd(n+1, i) = (n+1)a+ib

type A_{n}^{ℍ} Z(G̃_{ℂ}) = ⟨x⟩_{n+1}

n = 2l+1 z ↦ x^{l+1}  1l

type B_{n}^{ℝ,i} Z(G̃_{ℂ}) = ⟨x⟩_{2}

i ≤ 1  z ↦ x  1l
i ≥ 2  z ↦ x, z′ ↦ x  ⟨zz′⟩

type C_{n}^{ℝ} Z(G̃_{ℂ}) = ⟨x⟩_{2}

2|n  z ↦ x, z′ ↦ 1  ⟨z′⟩
2|̸n  z ↦ x  ⟨z^{2}⟩

type C_{n}^{ℍ,i} Z(G̃_{ℂ}) = ⟨x⟩_{2}

z ↦ x  1l

type D_{n}^{ℝ,i} Z(G̃_{ℂ}) = ⟨x⟩_{2} × ⟨x′⟩_{2}, if 2|n

Z(G̃_{ℂ}) = ⟨x⟩_{4}, if 2|̸n

i = 0

2|n  z ↦ x, z′ ↦ x′  1l
2|̸n  z ↦ x  1l

i = 1

2|n  z ↦ xx′  1l
2|̸n  z ↦ x^{2}  1l

i = 2

2|n  z ↦ xx′, z′ ↦ x  ⟨z′^{2}⟩
2|̸n  z ↦ x^{2} , z′ ↦ x  ⟨zz′^{2}⟩

i = 4k

2|n  z ↦ xx′, z′ ↦ xx′, z′′ ↦ x  ⟨zz′⟩
2|̸n  z ↦ x^{2} , z′ ↦ x  ⟨zz′^{2}⟩

i = 4k+2

2|n  z ↦ xx′, z′ ↦ x  ⟨z′^{2}⟩
2|̸n  z ↦ x^{2} , z′ ↦ x  ⟨zz′^{2}⟩

246 Bödi, Joswig

Table 2. Description of I Z(G̃) and linearizer, continued.

I Z(G̃)  linearizer

type D_{n}^{ℝ,i} Z(G̃_{ℂ}) = ⟨x⟩_{2} × ⟨x′⟩_{2}, if 2|n

Z(G̃_{ℂ}) = ⟨x⟩_{4}, if 2|̸n

i = 2k+1

2|n  z ↦ xx′, z′ ↦ xx′  ⟨zz′⟩
2|̸n  z ↦ x^{2} , z′ ↦ x^{2}  ⟨zz′⟩

type D_{n}^{ℍ} Z(G̃_{ℂ}) = ⟨x⟩_{2} × ⟨x′⟩_{2}, if 2|n

Z(G̃_{ℂ}) = ⟨x⟩_{4}, if 2|̸n

2|n  z ↦ x, z′ ↦ xx′  ⟨z′^{2}⟩
2|̸n  z ↦ x  ⟨z^{4}⟩

type E_{6} Z(G̃_{ℂ}) = ⟨x⟩_{3}

E_{6(−78)}  z ↦ x  1l
E_{6(−26)}  —  1l
E_{6(−14)}  z ↦ x  ⟨z^{3}⟩
E_{6(2)}  z ↦ x  ⟨z^{3}⟩
E_{6(6)}  z ↦ 1  Z

type E_{7} Z(G̃_{ℂ}) = ⟨x⟩_{2}

E_{7(−133)}  z ↦ x  1l
E_{7(−25)}  z ↦ x  ⟨z^{2}⟩
E_{7(−5)}  z ↦ 1, z′ ↦ x  ⟨z⟩
E_{7(7)}  z ↦ x  ⟨z^{2}⟩

type E_{8} Z(G̃_{ℂ}) = 1l

E_{8(−248)}  —  1l
E_{8(−24)}  z ↦ 1  Z
E_{8(8)}  z ↦ 1  Z

type F_{4} Z(G̃_{ℂ}) = 1l

F_{4(−52)}  —  1l
F_{4(−20)}  —  1l
F_{4(4)}  z ↦ 1  Z

type G_{2} Z(G̃_{ℂ}) = 1l

G_{2(−14)}  z ↦ 1  Z
G_{2(2)}  —  1l

Case II) 𝔤 is the realification of a simple complex Lie algebra 𝔤_{ℂ}. The complexification 𝔤 ⊗_{ℝ} ℂ of 𝔤 is isomorphic to 𝔤_{ℂ} ⊕ \overline{𝔤_{ℂ}} via the mapping g ⊗ c ↦ (g, 0), g ⊗ \overline{c} ↦ (0, g), where c = \frac{1}{2}(1 + i). The embedding ι of 𝔤 into 𝔤 ⊗_{ℝ} ℂ is now given by ι : g ↦ g ⊗ 1 = g ⊗ c + g ⊗ \overline{c} ↦ (g, g) : 𝔤 → 𝔤_{ℂ} ⊕ \overline{𝔤_{ℂ}}. Every real representation ρ : 𝔤 → 𝔤𝔩(V) is induced by a pair of irreducible complex representations ρ_{1}, ρ_{2} : 𝔤_{ℂ} → 𝔤𝔩(V_{ℂ}). The representation (ρ_{1}, ρ_{2}) of 𝔤_{ℂ} ⊕ \overline{𝔤_{ℂ}} is equivalent to the representation (ρ_{1}, ρ_{2}^{∗}) of 𝔤_{ℂ} ⊕ 𝔤_{ℂ}, where ρ_{2}^{∗} is the contragradient representation of ρ_{2} , cp. [5], §11 and §12. Every complex representation τ : 𝔤 → 𝔤𝔩(V) arises as the pair (τ, 0), which is equivalent to (0, τ^{∗}). The integrated map I : G̃ → G̃_{ℂ} × G̃_{ℂ} is injective and the image of I is diagonal in G̃_{ℂ} × G̃_{ℂ}. Thus the I-image of Z(G̃) is obvious and therefore we have omitted tables for this case.

The contragradient representation of the fundamental representation ρ_{j} is always ρ_{j}, except for the following cases: If 𝔤_{ℂ} is of type A_{n}, then ρ_{j}^{∗} = ρ_{n+1−j}. If 𝔤_{ℂ} is of type D_{n}, then ρ_{n−1}^{∗} = ρ_{n} and ρ_{n}^{∗} = ρ_{n−1} for n odd. If 𝔤_{ℂ} is of type E_{6} , then ρ_{1}^{∗} = ρ_{6} , ρ_{6}^{∗} = ρ_{1} , ρ_{3}^{∗} = ρ_{5} , ρ_{5}^{∗} = ρ_{3} .

Extend one (of the mutually conjugate [2], 20.9(ii)) maximal ℝ-split toral subalgebras of 𝔤 to a Cartan subalgebra 𝔱 of 𝔤. Then 𝔱_{ℂ} := 𝔱 ⊗ ℂ is a Cartan subalgebra of 𝔤_{ℂ}. Now complex conjugation acts on 𝔤_{ℂ} via σ : g ⊗ c ↦ g ⊗ \overline{c} and thus also on Λ_{+}(𝔤_{ℂ}, 𝔱_{ℂ}), see [12], 3.1. This action is explicitly given in Table 3 for case I. Note that the action of ⟨σ⟩ on Λ_{+}(𝔤_{ℂ}, 𝔥_{ℂ}) may be different for 𝔥 not conjugate to 𝔱 in 𝔤. For case II the action is simply the exchange of the two components. We have ([5], [9], p. 290/291, [12], 7.2, 8.2, or [4], section 55)

Theorem 2. The orbits of ⟨σ⟩ on Λ_{+}(𝔤_{ℂ}, 𝔱_{ℂ}) and the equivalence classes of finite-dimensional irreducible real representations of 𝔤 are in a one-to-one correspondence.

Centralizers and real dimensions

Let 𝔤 be a simple real Lie algebra and ρ : 𝔤 → 𝔤𝔩(V) an irreducible real representation. Then the centralizer Cs(ρ) of the image 𝔤^{ρ} in 𝔤𝔩(V) is a skew field by Schur’s lemma. Moreover, Cs(ρ) obviously contains ℝ = ℝ · id_{V} in its center. Thus Cs(ρ) ≅ ℝ, ℂ, ℍ by a result of Frobenius. Let λ be the highest weight of ρ_{ℂ}. We consider the subsemigroup Σ_{+}(𝔤) of Λ_{+}(𝔤_{ℂ}) of those weights that are fixed by σ. By §4 and §9 of [5] there exists an additive mapping α_{𝔤} : Σ_{+}(𝔤) → ℤ/2ℤ, the index of 𝔤, such that either

(1) λ ∈ Σ_{+}(𝔤), α_{𝔤}(λ) = 0, Cs(ρ) ≅ ℝ, dim_{ℝ}(ρ) = dim_{ℂ}(ρ_{ℂ}),

and ρ_{ℂ} is irreducible, or (2) λ ∈ Σ_{+}(𝔤), α_{𝔤}(λ) = 1, Cs(ρ) ≅ ℍ, dim_{ℝ}(ρ) = 2 dim_{ℂ}(ρ_{ℂ}),

and ρ_{ℂ} is the direct sum of two equivalent irreducible complex representations corresponding to λ,

or (3) λ ∉ Σ_{+}(𝔤), Cs(ρ) ≅ ℂ and dim_{ℝ}(ρ) = 2 dim_{ℂ}(ρ_{ℂ}), and ρ_{ℂ} is the direct sum of two non-equivalent irreducible complex representations corresponding to λ and λ^{σ}.

Note that the semigroup Σ_{+}(𝔤) is generated by the set {λ_{i}, λ_{j} +λ_{j}^{σ} | 1 ≤ i, j ≤ rank 𝔤_{ℂ}, λ_{i}^{σ} = λ_{i}}. Since α_{𝔤}(μ + μ^{σ}) = 0, see [5], Lemma 11, we can extend the definition of α_{𝔤} to Λ_{+}(𝔤_{ℂ}) by setting α ≡ 0 on Λ_{+}(𝔤_{ℂ}) \ Σ_{+}(𝔤). Thus, the mapping α_{𝔤} is uniquely determined by the images of the fundamental weights λ_{i}. Table 3 lists the index for the real forms, whereas for case II we have α_{𝔤} ≡ 0 on the whole weightspace Λ_{+}(𝔤_{ℂ}). The canonical projection of ℤ to ℤ/2ℤ is denoted by par.

Table 3. The action of σ on Λ_{+}(𝔤_{ℂ}, 𝔱_{ℂ}) and the index α_{𝔤}.
type  action of σ on Λ_{+}(𝔤_{ℂ}, 𝔱_{ℂ})  α_{𝔤}
A_{n}^{ℝ}  λ_{j}^{σ} = λ_{j}  λ_{j} ↦ 0
A_{n}^{ℂ,i}  λ_{j}^{σ} = λ_{n+1−j}  λ_{j} ↦ \{ \begin{matrix}par(\frac{n+1}{2} −i) & if j = \frac{(n+1)}{2} \\ 0 & otherwise \end{matrix}
A_{n}^{ℍ}  λ_{j}^{σ} = λ_{j}  λ_{j} ↦ par(j)
B_{n}^{ℝ,i}  λ_{j}^{σ} = λ_{j}  λ_{j} ↦ \{ \begin{matrix}0 & if j <n or n−i ≢ 1,2 (4) \\ 1 & otherwise \end{matrix}
C_{n}^{ℝ}  λ_{j}^{σ} = λ_{j}  λ_{j} ↦ 0
C_{n}^{ℍ,i}  λ_{j}^{σ} = λ_{j}  λ_{j} ↦ par(j)
D_{n}^{ℝ,i}  \begin{matrix}λ_{j}^{σ} =λ_{j} (j <n−1 or 2|n−i), \\ λ_{n−1}^{σ} =λ_{n}, λ_{n}^{σ} =λ_{n−1}(2|̸n−i) \end{matrix} λ_{j} ↦ \{ \begin{matrix}0 & if j <n−1 or 4|n−i \\ 1 & otherwise \end{matrix}

D_{n}^{ℍ}  \begin{matrix}λ_{j}^{σ} = λ_{j} (j <n−1 or 2|n), \\ λ_{n−1}^{σ} =λ_{n}, λ_{n}^{σ} =λ_{n−1}(2|̸n−i) \end{matrix} λ_{j} ↦ \{ \begin{matrix}par(j) & if j <n−1 \\ 0 & if j =n−1 and 2|n \\ 1 & otherwise \end{matrix}

E_{6(−78)}  \begin{matrix}λ_{1}^{σ} =λ_{6}, λ_{2}^{σ} =λ_{2}, λ_{3}^{σ} =λ_{5} \\ λ_{4}^{σ} =λ_{4}, λ_{5}^{σ} =λ_{3}, λ_{6}^{σ} =λ_{1} \end{matrix}  λ_{j} ↦ \{ \begin{matrix}0 & if j =2, 4 \\ 1 & otherwise \end{matrix}
E_{6(−26)} λ_{j}^{σ} = λ_{j}  λ_{j} ↦ 0
E_{6(−14)} see E_{6,−78}  see E_{6,−78}
E_{6(2)}  see E_{6,−78}  see E_{6,−78}
E_{6(6)}  see E_{6,−26}  see E_{6,−26}
E_{7(−133)} λ_{j}^{σ} = λ_{j}  λ_{j} ↦ \{ \begin{matrix}0 & if j =1, 3, 4, 6 \\ 1 & otherwise \end{matrix}
E_{7(−25)} λ_{j}^{σ} = λ_{j}  λ_{j} ↦ 0
E_{7(−5)} see E_{7,−133}  see E_{7,−133}
E_{7(7)}  see E_{7,−25}  see E_{7,−25}
E_{8(i)}  λ_{j}^{σ} = λ_{j}  λ_{j} ↦ 0
F_{4(i)}  λ_{j}^{σ} = λ_{j}  λ_{j} ↦ 0
G_{2(i)}  λ_{j}^{σ} = λ_{j}  λ_{j} ↦ 0

Kernels of representations

Let 𝔤 be a simple real Lie algebra and let 𝔤_{ℂ} be its complexification, with corresponding simply connected groups G̃ and G̃_{ℂ}, resp. Consider a non-trivial irreducible representation ρ : 𝔤 → 𝔤𝔩(V) with highest weight λ.

As 𝔤 is simple the representation ρ is faithful and ker P = ker(IP_{ℂ}) is discrete and therefore central in the group G̃. Of course Z(G̃) is mapped to

Z(GL(V_{ℂ})) by P. Note that Z(G̃)^{P} is a cyclic group of roots of unity. Define a group homomorphism ε_{P} : Z(G̃) → ℂ^{×} by ε_{P}(z) · id_{V_{ℂ}} = z^{P} . Then ε_{P} can be written as

(∗)  ε_{P}(z) = ∏_{i=1}^{r} ε_{P_{i}}(z)^{f_{i}(λ)}.

Obviously, ker P = ker ε_{P} . Together with the information on I|_{Z(_{G})} already given above this is enough to compute the kernel of P if one knows the kernels of all fundamental representations of G̃_{ℂ}. This is contained in Table 4 below, where ξ_{k} denotes a k th primitive root of unity. The group G̃/ker P is called the group which is represented by P.

Table 4. The images of the center Z(G̃_{ℂ}) under the fundamental representations P_{j} of G̃_{ℂ}

type  Z(G̃_{ℂ})  P_{j} Z(G̃_{ℂ})
A_{n}  ⟨x⟩_{n+1}  x ↦ ξ_{n+1}^{j}
B_{n}  ⟨x⟩_{2}  x ↦ \{ \begin{matrix}1 & if j < n \\ −1 & if j = n \end{matrix}
C_{n}  ⟨x⟩_{2}  x ↦ (−1)^{j}
D_{2l}  ⟨x⟩_{2} × ⟨x′⟩_{2} x ↦ \{ \begin{matrix}(−1)^{j} & if j < 2l − 1 \\ 1 & if j = 2l − 1 \\ −1 & if j = 2l \end{matrix}

x′ ↦ \{ \begin{matrix}(−1)^{j} & if j < 2l − 1 \\ −1 & if j = 2l − 1 \\ 1 & if j = 2l \end{matrix}

D_{2l+1} ⟨x⟩_{4}  x ↦ \{ \begin{matrix}(−1)^{j} & if j < 2l \\ \sqrt{−1} & if j = 2l \\ −\sqrt{−1} & if j = 2l + 1 \end{matrix}

E_{6}  ⟨x⟩_{3}  x ↦ \{ \begin{matrix}ξ_{3} & if j = 1, 5 \\ ξ_{3}^{2} & if j = 3, 6 \\ 1 & if j = 2, 4 \end{matrix}
E_{7}  ⟨x⟩_{2}  x ↦ \{ \begin{matrix}−1 & if j = 2, 5, 7 \\ 1 & if j = 1, 3, 4, 6 \end{matrix}
E_{8}  1l  —
F_{4}  1l  —
G_{2}  1l  —

About the implementation

Our initial motivation was to compute a list of the equivalence classes of all irreducible real representations of a given simple real Lie algebra 𝔤 up to a certain dimension. We found this to be not only tedious but also very fault prone to do by hand.

Obviously one can establish a (lexicographic) ordering on the elements of the weight space Λ_{+}(𝔤_{ℂ}). Moreover the dimension function (Weyl formula) is increasing if the multiplicities of all roots in the weight space but one are fixed. This ensures that there is only a finite number of equivalence classes of irreducible representations for 𝔤 up to a given dimension. Furthermore these representations can be enumerated in that way.

So what is left is to deal with a fixed representation ρ : 𝔤 → 𝔤𝔩(V). Again let λ be the highest weight of ρ_{ℂ} and let G̃ denote the simply connected group associated with 𝔤.

(1) Determine dim_{ℂ} ρ_{ℂ} via the Weyl formula.

(2) Derive ρ_{ℂ}^{σ} and the index α_{𝔤}(λ) from Table 3. This gives the centralizer Cs(ρ) and, together with (1), dim_{ℝ} ρ.

(3) Compute ker P = ker ε_{P} as follows: Table 4 gives the images of each element of the generating system S for Z(G̃) (as given in table 1) under all fundamental representations P_{i}. Substitute these data and λ in formula (∗) to obtain the images x^{P} for each x ∈ S. Using the Chinese-Remainder-Algorithm one obtains a generating system for ker ε_{P} . By some reduction strategy the result can be transformed into a canonical form such that the name for the represented group G̃/ker P can be looked up in a list.

The first implementation of the algorithm sketched above has been done in the LiE language [7]. To give the reader an idea of the complexity: this LiE based implementation takes 5 minutes, approximately, on a 486 IBM computer (50 MHz) to determine all irreducible representations of all simple real Lie algebras up to dimension 16. We have to mention that it was not possible to run our present LiE based program with the original LiE 2.0 MS–DOS version. This is mainly due to the fact that this particular LiE version used short integers for its memory management, which did not meet our requirements. However, since the sources were included we were able to do a recompilation of the (slightly modified) source code with the GNU C compiler. A significantly faster ANSI C based version of our representation package is in preparation. This implementation will also be more portable, since it will be independent from LiE. The output (of both versions) is an ASCII file that contains all information computed. Because this file uses some crude format, we have written a TEX back end that transforms the plain ASCII file into a TEX file which produces tables that look like the ones listed in the next section.

Example Tables

A{ℂ,1 \atop 3}  —  𝔰𝔲_{4}ℂ(1)  —  real dimension 15

Real representations from dimension 2 to dimension 300. Center Z of the universal covering : Z = ⟨z⟩ ≅ ℤ The linearizer is ⟨z^{4}⟩. By SαU_{3}ℍ we denote the antiunitary group on ℍ^{3} .

dimension centralizer dominant weight kernel represented group
8  ℂ  λ_{1}  ⟨z^{4}⟩  SU_{4}ℂ (1)
12  ℍ  λ_{2}  ⟨z^{2}⟩  SαU_{3}ℍ
15  ℝ  λ_{1} + λ_{3}  Z  PSU_{4}ℂ (1)
20  ℝ  2λ_{2}  Z  PSU_{4}ℂ (1)
20  ℂ  2λ_{1}  ⟨z^{2}⟩  SαU_{3}ℍ
40  ℂ  λ_{1} + λ_{2}  ⟨z^{4}⟩  SU_{4}ℂ (1)
40  ℂ  3λ_{1}  ⟨z^{4}⟩  SU_{4}ℂ (1)
70  ℂ  4λ_{1}  Z  PSU_{4}ℂ (1)
72  ℂ  2λ_{1} + λ_{3}  ⟨z^{4}⟩  SU_{4}ℂ (1)
84  ℝ  2λ_{1} + 2λ_{3}  Z  PSU_{4}ℂ (1)
90  ℂ  2λ_{1} + λ_{2}  Z  PSU_{4}ℂ (1)
100  ℍ  3λ_{2}  ⟨z^{2}⟩  SαU_{3}ℍ
105  ℝ  4λ_{2}  Z  PSU_{4}ℂ (1)
112  ℂ  5λ_{1}  ⟨z^{4}⟩  SU_{4}ℂ (1)
120  ℂ  λ_{1} + 2λ_{2}  ⟨z^{4}⟩  SU_{4}ℂ (1)
128  ℍ  λ_{1} + λ_{2} + λ_{3}  ⟨z^{2}⟩  SαU_{3}ℍ
140  ℂ  3λ_{1} + λ_{3}  ⟨z^{2}⟩  SαU_{3}ℍ
168  ℂ  3λ_{1} + λ_{2}  ⟨z^{4}⟩  SU_{4}ℂ (1)
168  ℂ  6λ_{1}  ⟨z^{2}⟩  SαU_{3}ℍ
175  ℝ  λ_{1} + 2λ_{2} + λ_{3}  Z  PSU_{4}ℂ (1)
240  ℂ  4λ_{1} + λ_{3}  ⟨z^{4}⟩  SU_{4}ℂ (1)
240  ℂ  7λ_{1}  ⟨z^{4}⟩  SU_{4}ℂ (1)
252  ℂ  2λ_{1} + 2λ_{2}  ⟨z^{2}⟩  SαU_{3}ℍ
280  ℂ  λ_{1} + 3λ_{2}  ⟨z^{4}⟩  SU_{4}ℂ (1)
280  ℂ  2λ_{1} + λ_{2} + λ_{3}  ⟨z^{4}⟩  SU_{4}ℂ (1)
280  ℂ  4λ_{1} + λ_{2}  ⟨z^{2}⟩  SαU_{3}ℍ
300  ℝ  3λ_{1} + 3λ_{3}  Z  PSU_{4}ℂ (1)

D4  —  𝔬_{8}ℂ  —  real dimension 56

Real representations from dimension 2 to dimension 100. Center Z of the universal covering : Z = ⟨z, z′⟩ ≅ ℤ_{2} × ℤ_{2} The linearizer is 1.

dimension centralizer dominant weight kernel represented group
16  ℂ  (λ_{4}, 0)  ⟨z⟩  SO_{8}ℂ
16  ℂ  (λ_{3}, 0)  ⟨z′⟩  SO_{8}ℂ
16  ℂ  (λ_{1}, 0)  ⟨zz′⟩  SO_{8}ℂ
56  ℂ  (λ_{2}, 0)  Z  PSO_{8}ℂ
64  ℝ  (λ_{4}, λ_{4}^{∗})  Z  PSO_{8}ℂ
64  ℝ  (λ_{3}, λ_{3}^{∗})  Z  PSO_{8}ℂ
64  ℝ  (λ_{1}, λ_{1}^{∗})  Z  PSO_{8}ℂ
70  ℂ  (2λ_{4}, 0)  Z  PSO_{8}ℂ
70  ℂ  (2λ_{3}, 0)  Z  PSO_{8}ℂ
70  ℂ  (2λ_{1}, 0)  Z  PSO_{8}ℂ

E{ℝ \atop 6(−78)}  —  real dimension 78

Real representations from dimension 2 to dimension 35000. Center Z of the universal covering : Z = ⟨z⟩ ≅ ℤ_{3} The linearizer is 1.

dimension centralizer dominant weight kernel
54  ℂ  λ_{1}  1
78  ℝ  λ_{2}  Z
650  ℝ  λ_{1} + λ_{6}  Z
702  ℂ  λ_{3}  1
702  ℂ  2λ_{1}  1
2430  ℝ  2λ_{2}  Z
2925  ℝ  λ_{4}  Z
3456  ℂ  λ_{1} + λ_{2}  1
6006  ℂ  3λ_{1}  Z
11648  ℂ  λ_{1} + λ_{3}  Z
14742  ℂ  λ_{1} + λ_{5}  1
15444  ℂ  2λ_{1} + λ_{6}  1
34749  ℝ  λ_{1} + λ_{2} + λ_{6}  Z

References

[1] Bourbaki, N., “Groupes et algèbres de Lie, Chapitres IV – VI”, Paris: Hermann, 1968.

[2] Borel, A., “Linear Algebraic Groups”, Berlin–Heidelberg–New York: Springer, 2nd ed., 1991.

[3] Bremner, M. R., R. V. Moody, J. Patera, “Tables of dominant weight multiplicities for representations of simple Lie algebras”, Monographs and Textbooks in Pure and Appl. Math. 40, New York: Dekker, 1985.

[4] Freudenthal, H., H. de Vries, “Linear Lie Groups”, New York–London: Academic Press, 1969.

[5] Iwahori, N., On real irreducible representation of Lie algebra, Nagoya Math. J. 14 (1959), 59–83.

[6]  Jacobson, N., “Lie algebras”, New York: Interscience, 1962.

[7] van Leeuwen, M. A. A., A. M. Cohen, B. Lisser, “LiE – A Package for Lie Group Computations”, Amsterdam: Computer Algebra Nederland, 1992.

[8] McKay, W. G., J. Patera, “Tables of dimensions, indices and branching rules for representations of simple Lie algebras”, Lect. Notes in Pure and Appl. Math. 69, New York: Dekker, 1981.

[9] Onishchik, A. L., E. B. Vinberg, “Lie Groups and Algebraic Groups”, Berlin–Heidelberg–New York: Springer, 1990.

[10] Samelson, H., “Notes on Lie Algebras”, Berlin – Heidelberg – New York: Springer, 1990.

[11] Tits, J., “Tabellen zu den einfachen Liegruppen und ihren Darstellungen”, Lecture Notes 40, Berlin–Heidelberg–New York: Springer, 1967.

[12] Tits, J., Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, J. Reine Angew. Math. 247 (1971), 196–220.

[13] Varadarajan, V. S., “Lie groups, Lie algebras, and their representations”, Englewood Cliffs (N.J.): Prentice-Hall; Berlin etc.: Springer, 1984.

Universität Tübingen Mathematisches Institut Auf der Morgenstelle 10 72076 Tübingen

Received June 13, 1993 and in final form November 2, 1993