Real and Complex Representations

This note extends Schur’s Lemma to real representations of a compact Lie group, expanding on some of the material in §5 of Chapter II in Bröcker–tom Dieck.

Throughout, let G be a compact Lie group. Our object is to compute the endomorphism ring End_{ℝ}(W) of an irreducible real representation W by appealing to the complex version of Schur’s Lemma.

Two constructions We need two constructions that relate real and complex representations. Given a complex representation V of G, we may regard V as a real vector space (of twice the dimension) and treat it as a real representation of G, the realification rV of V. Trivially, r\overline{V} = rV.

In the other direction, given a real representation W of G, we have the complexification cW = ℂ ⊗_{ℝ} W, equipped with the obvious actions of ℂ and G. Then t ⊗ W: \overline{cW} ≅ cW, where t: ℂ → ℂ denotes the conjugation in ℂ. (The identity morphism does not work!)

In this note, V will always denote a complex representation and W a real representation. We recall that rcW ≅ W ⊕ W and crV ≅ V ⊕ \overline{V}.

The first half of Schur’s Lemma carries over without change (by the same proof).

Lemma 1 If W_{1} and W_{2} are real irreducible representations of G, any morphism f: W_{1} → W_{2} is either an isomorphism or zero.

Corollary 2 For any irreducible real representation W of G, the endomorphism ring End_{ℝ}(W) is a finite-dimensional skew field over ℝ.

Complex vector spaces with conjugation The above space cW is more than just a complex vector space.

Definition 3 A complex vector space X has a real structure or conjugation if we are given an involution x ↦ \overline{x} that is additive and antilinear: \overline{zx} = \overline{z} \overline{x} for z ∈ ℂ. Its real part ℜX is the real vector subspace consisting of those x ∈ X that satisfy \overline{x} = x.

If X is a complex representation of G, we require the conjugation to preserve the G-action, \overline{gx} = g\overline{x}; then ℜX is a real representation of G.

In the case of cW above, the conjugation is inherited from ℂ and does indeed preserve the G-action; also, ℜcW = 1 ⊗ W ≅ W.

Lemma 4 If X is a complex vector space with conjugation, it is canonically isomorphic, as a complex vector space, to ℂ ⊗_{ℝ} ℜX. Explicitly, it is the direct sum of the real subspaces ℜX and iℜX.

Proof For any x, x + \overline{x} and i\overline{x} − ix lie in ℜX, so x = (x + \overline{x})/2 + i(i\overline{x} − ix)/2 ∈ ℜX + iℜX. On the other hand, if x ∈ (ℜX) ∩ (iℜX), we have x = iy with y ∈ ℜX, and \overline{x} = −iy = −x. Since \overline{x} = x, this implies x = 0.

Lemma 5 Let W be any real representation of G. Then:

(a) The complexification cW is a complex representation with conjugation;

(b) The complex endomorphism ring End_{ℂ}(cW) is a complex vector space with conjugation;

(c) We may identify the real part ℜ End_{ℂ}(cW) of End_{ℂ}(cW) with the real endomorphism ring End_{ℝ}(W) of W itself.

Proof In (a), given a complex endomorphism f of cW, we define its conjugate \overline{f} by

\overline{f}v = \overline{f\overline{v}}.  (6)

It is easy to check that \overline{f} is again ℂ-linear and that this conjugation is antilinear.

In (b), suppose that f: cW → cW is an endomorphism. We decompose cW = (1⊗_{ℝ}W) ⊕ (i⊗_{ℝ}W) and consider the restriction f|1⊗W. We may write f(1⊗w) = 1⊗f_{1}w + i⊗f_{2}w, where f_{1} and f_{2} are real endomorphisms of W. Then by ℂ-linearity of f, f(i⊗w) = −1⊗f_{2}w + i⊗f_{1}w, which shows that the real endomorphisms f_{1} and f_{2} determine f and may be chosen arbitrarily. The conjugation simply changes the sign of f_{2}, so that ℜf is essentially just f_{1}.

Remark It will be useful to rewrite the definition and identify the real endomorphisms of W with those complex endomorphisms of cW that commute with the conjugation on cW.

Lemma 7 Given a real representation W of G, suppose that V is a self-conjugate invariant complex subspace of cW. Then V has the form ℂ ⊗_{ℝ}W_{1}, where W_{1} is some invariant real subspace of W.

Proof We define the invariant subspace W_{1} by 1⊗W_{1} = V ∩(1⊗W); since 1⊗W_{1} ⊂ V, we have ℂ ⊗_{ℝ} W_{1} ⊂ V. Conversely, given an element v = 1⊗w_{1} + i⊗w_{2} ∈ V, its conjugate 1⊗w_{1} − i⊗w_{2} also lies in V, from which it follows that 1 ⊗ w_{1}, i ⊗ w_{2} and 1 ⊗ w_{2} all lie in V. Thus w_{1} and w_{2} lie in W_{1} and v ∈ ℂ ⊗_{ℝ} W_{1}.

We use Lemma 5 to compute the desired endomorphism rings directly. Let W be an irreducible real representation. We decompose the complex representation cW into irreducible complex representations; since rcW ≅ W ⊕ W, cW decomposes into either one or two complex representations. There are three cases.

Case 1: V = cW is irreducible. By Schur’s Lemma, End_{ℂ}(V) ≅ ℂ consists of the endomorphisms fv = λv, for λ ∈ ℂ. Then \overline{f}v = \overline{λ\overline{v}} = \overline{λ}v, so that \overline{f} corresponds to \overline{λ}. As a trivial example, we have the trivial representation ℝ of any group G.

Theorem 8 Let W be an irreducible real representation of G. If cW is irreducible as a complex representation, the endomorphism ring End_{ℝ}(W) ≅ ℝ consists of the endomorphisms fw = λw with λ real.

Otherwise, we have the decomposition cW = V_{1} ⊕ V_{2} into exactly two irreducible invariant complex subspaces. By Lemma 7, the only self-conjugate invariant complex subspaces of cW are 0 and cW. In particular, we deduce that V_{1} ∩ \overline{V}_{1} = 0 and V_{1} + \overline{V}_{1} = cW, so that cW = V_{1} ⊕ \overline{V}_{1}. This case in turn divides into two.

Case 2: cW = V ⊕ \overline{V}, where V is irreducible and not isomorphic to \overline{V}. By Schur’s Lemma, End_{ℂ}(cW) ≅ ℂ ⊕ ℂ; the endomorphism corresponding to the pair (λ, μ) ∈ ℂ ⊕ ℂ is simply f_{λ,μ}(v_{1}, \overline{v}_{2}) = (λv_{1}, μ\overline{v}_{2}). The conjugation in cW is obvious, \overline{(v_{1}, \overline{v}_{2})} = (v_{2}, \overline{v}_{1}), and (6) yields

\overline{f}_{λ,μ}(v_{1}, \overline{v}_{2}) = \overline{f_{λ,μ}(v_{2}, \overline{v}_{1})} = \overline{(λv_{2}, μ\overline{v}_{1})} = (\overline{μ}v_{1}, \overline{λ}\overline{v}_{2}) = f_{\overline{μ},\overline{λ}}(v_{1}, \overline{v}_{2}).

Thus f_{λ,μ} is real if and only if μ = \overline{λ}, and then λ ∈ ℂ determines f_{λ,μ}.

Theorem 9 Let W be an irreducible real representation of G. If cW = V ⊕ \overline{V}, where V is not isomorphic to \overline{V}, then End_{ℝ}(W) ≅ ℂ.

As a simple example, we have the standard representation of SO(2) on ℝ^{2}.

Case 3: cW = V ⊕\overline{V}, where V is irreducible and V ≅ \overline{V}. So cW ≅ V ⊕V, and by Schur’s Lemma, End_{ℂ}(cW) ≅ M_{2}(ℂ). The 2×2 complex matrix A with entries a_{j,k} acts on V ⊕ V in the usual way,

A(v_{1}, v_{2}) = (a_{1,1}v_{1} + a_{1,2}v_{2}, a_{2,1}v_{1} + a_{2,2}v_{2}).

We need to be specific. We choose some isomorphism ϕ: V ≅ \overline{V}, and use it to form the isomorphism Φ = id ⊕ ϕ: V ⊕ V ≅ V ⊕ \overline{V}. In order to work entirely in V, it will be convenient to write ϕv = \overline{θv}, where θ: V → V is an antilinear bijection, so that Φ(v_{1}, v_{2}) = (v_{1}, \overline{θv_{2}}).

Now θ^{∘}θ is ℂ-linear, and therefore has the form θθv = λv for some λ ∈ ℂ by Schur’s Lemma, where λ ≠ 0. Moreover, by writing θ^{∘}θ^{∘}θ two different ways, θθθv = (θθ)θv = λθv and θθθv = θ(θθv) = θ(λv) = \overline{λ}θv, we see that λ must be real. Further, if we replace ϕ by kϕ, where k is real, λ changes to k^{2}λ, which allows us to arrange λ = ±1 by a suitable choice of ϕ.

The conjugation in V ⊕ \overline{V} corresponds by Φ to the conjugation \overline{(v_{1}, v_{2})} = (θv_{2}, θ^{−1}v_{1}) in V ⊕V. By Lemma 5, the matrix A lies in the real part of End_{ℂ}(V ⊕V) if and only if it commutes with conjugation. If we conjugate first and then apply A, we find

(a_{1,1}θv_{2} + a_{1,2}θ^{−1}v_{1}, a_{2,1}θv_{2} + a_{2,2}θ^{−1}v_{1}).

If we apply A first and then conjugate, we find, using the antilinearity of θ and θ^{−1},

(\overline{a}_{2,1}θv_{1} + \overline{a}_{2,2}θv_{2}, \overline{a}_{1,1}θ^{−1}v_{1} + \overline{a}_{1,2}θ^{−1}v_{2}).

Comparison of the coefficients of θ^{−1}v_{1} in the second entries yields a_{2,2} = \overline{a}_{1,1}. To continue, we rewrite \overline{a}_{2,1}θv_{1} as \overline{a}_{2,1}λθ^{−1}v_{1}, to obtain a_{1,2} = λ\overline{a}_{2,1}. The terms with v_{2} give nothing new. Thus ℜ End_{ℂ}(V ⊕V) consists of those 2×2 complex matrices of the form

A = ( \begin{matrix}a & λ\overline{b} \\ b & \overline{a} \end{matrix} ) .

However, we observe that det(A) = |a|^{2} − λ|b|^{2}, which vanishes if λ = 1 and a = b; in view of Corollary 2, we conclude that we must have λ = −1. We have recovered a standard construction of the quaternions ℍ.

As an obvious example, the group S^{3} of unit quaternions acts on ℍ by left multiplication; the endomorphism corresponding to q ∈ ℍ is right multiplication by \overline{q}.

Theorem 10 Let W be an irreducible real representation of G. If cW = V ⊕ \overline{V}, where V ≅ \overline{V}, the endomorphism ring End_{ℝ}(W) is isomorphic to ℍ.

Note that we have constructed the skew field ℍ of quaternions out of thin air.

Characters One can define the character function χ_{W} (g) of a real representation W just as for complex representations; it is of course real-valued. Further, distinct irreducible real representations W_{1} and W_{2} yield orthogonal characters χ_{W_{1}} and χ_{W_{2}}, just as in the complex case. However, they need not be unit vectors in the L^{2}-norm ∥−∥_{2}. In fact, the norm gives an easy way to distinguish the three classes of real representations.

First, we need to record the behavior of the operations r and c on characters.

Proposition 11 If W is a real representation, χ_{cW} = χ_{W}. If V is a complex representation, χ_{rV} = 2ℜχ_{V}.

Proof The action of g ∈ G on W is described by a real matrix A(g). The action of g on cW is described by the same matrix A(g), reinterpreted as a complex matrix; the trace is unchanged.

Suppose V has the ℂ-basis (e_{1}, e_{2}, …, e_{n}), with the action of g given by the complex matrix B(g). Then V has the real basis (e_{1}, ie_{1}, e_{2}, ie_{2}, …, ie_{n}). The effect on B(g) is to replace each complex entry z = a + ib by the 2×2 matrix ({a \atop b} {−b \atop a}). In computing the trace, each z on the diagonal is replaced by two copies of a.

Theorem 12 Let W be an irreducible real representation.

(a) If End_{ℝ}(W) ≅ ℝ, then ∥χ_{W} ∥_{2} = 1.

(b) If End_{ℝ}(W) ≅ ℂ, then ∥χ_{W} ∥_{2} = \sqrt{2}.

(c) If End_{ℝ}(W) ≅ ℍ, then ∥χ_{W} ∥_{2} = 2.

Proof In (a), or Case 1, V = cW is irreducible and χ_{W} = χ_{V} is a unit vector.

In (b), or Case 2, cW = V ⊕ \overline{V}, so that χ_{W} = χ_{V} + χ_{\overline{V}}, where χ_{V} and χ_{\overline{V}} are orthogonal unit vectors.

In (c), or Case 3, cW = V ⊕ \overline{V} ≅ V ⊕ V, and χ_{W} = 2χ_{V}.