1.5. Lecture 5

1.5.1. Direct sums and reducibility

We start by defining sums of representations:

Definition 1.5.1.

Let (π,V)(\pi,V) and (ρ,W)(\rho,W) be representations of a group GG. The direct sum of the two representations is (πρ,VW)(\pi\oplus\rho,V\oplus W), where

(πρ)(g)(𝐯,𝐰):=(π(g)𝐯,ρ(g)𝐰)(\pi\oplus\rho)(g)({\bf v},{\bf w}):=(\pi(g){\bf v},\rho(g){\bf w})

for all gGg\in G, 𝐯V{\bf v}\in V, 𝐰W{\bf w}\in W.

We also write (π,V)(ρ,W)(\pi,V)\oplus(\rho,W) for the direct sum of the representations.

Example 1.5.2.

Let (χ1,)(\chi_{1},{\mathbb{C}}) and (χ2,)(\chi_{2},{\mathbb{C}}) be representations of Cn=a|an=eC_{n}=\langle a\,|\,a^{n}=e\rangle given by χ1(aj)=e2πij/n\chi_{1}(a^{j})=e^{2\pi ij/n} and χ2(aj)=e4πij/n\chi_{2}(a^{j})=e^{4\pi ij/n}. Then (χ1χ2,2)(\chi_{1}\oplus\chi_{2},{\mathbb{C}}^{2}) is the representation of CnC_{n} on 2{\mathbb{C}}^{2} given by χ1χ2(aj)=(e2πij/n,e4πij/n)\chi_{1}\oplus\chi_{2}(a^{j})=(e^{2\pi ij/n},e^{4\pi ij/n}).

Conversely, we also have the following definition.

Definition 1.5.3.

If (π,V)(\pi,V) is a representation with non-trivial subrepresentations (π,U)(\pi,U) and (π,W)(\pi,W) such that V=UWV=U\oplus W, then we say that (π,V)(\pi,V) is decomposable or reducible.

Note that if (π,V)(\pi,V) is decomposable , then

(π,V)(π,U)(π,W),(\pi,V)\cong(\pi,U)\oplus(\pi,W),

with a GG-isomorphism from UWU\oplus W to VV given by

(𝐮,𝐰)𝐮+𝐰({\bf u},{\bf w})\mapsto{\bf u}+{\bf w}

for all 𝐮U,𝐰W{\bf u}\in U,{\bf w}\in W (see Problem 1.5.4 below). We say that the representation (π,V)(\pi,V) has been decomposed into a direct sum of subrepresentations.

Example 1.5.4.

The permutation representation (π,n)(\pi,{\mathbb{C}}^{n}) decomposes as (π,W1)(π,W0)(\pi,W_{1})\oplus(\pi,W_{0}), where W0={(x1,,xn)n|x1++xn=0}W_{0}=\{(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}\,|\,x_{1}+\ldots+x_{n}=0\} and W1W_{1} is the linear span of (1,,1)(1,\ldots,1).

The following example shows that there are representations that are neither decomposable or irreducible:

Example 1.5.5.

Let (ρ,2)(\rho,{\mathbb{C}}^{2}) be the representation of (,+)({\mathbb{Z}},+) given by

ρ(n)=(1n01).\rho(n)=\left(\begin{matrix}1&n\\ 0&1\end{matrix}\right).

This representation is not irreducible, as (10){\mathbb{C}}\left(\begin{matrix}1\\ 0\end{matrix}\right) is a non-trivial subrepresentation.

The representation is not decomposable , as then there would be two non-zero vectors 𝐯1{\bf v}_{1} and 𝐯2{\bf v}_{2} such that 2=𝐯1𝐯2{\mathbb{C}}^{2}={\mathbb{C}}{\bf v}_{1}\oplus{\mathbb{C}}{\bf v}_{2}, and (ρ,𝐯1)(\rho,{\mathbb{C}}{\bf v}_{1}), (ρ,𝐯2)(\rho,{\mathbb{C}}{\bf v}_{2}) are subrepresentations. The vectors 𝐯1{\bf v}_{1} and 𝐯2{\bf v}_{2} would then be two linearly independent eigenvectors of all the matrices (1n01)\left(\begin{smallmatrix}1&n\\ 0&1\end{smallmatrix}\right), which is not possible - these matrices are not diagonalisable.

However, for finite groups, we have the following:

Theorem 1.5.6 (Maschke’s Theorem, 1899).

Let (π,W)(\pi,W) be a non-trivial subrepresentation of a finite-dimensional complex representation of a finite group GG. Then there exists a subrepresentation (π,W)(\pi,W^{\prime}) of (π,V)(\pi,V) such that

(π,V)=(π,W)(π,W).(\pi,V)=(\pi,W)\oplus(\pi,W^{\prime}).

Before proving this, we introduce another important notion:

1.5.2. Unitary representations

Definition 1.5.7.

Let (π,V)(\pi,V) be a representation of a group GG on an inner product space VV, with inner product ,\langle\cdot,\cdot\rangle. The representation is said to be unitary if

π(g)𝐯,π(g)𝐮=𝐯,𝐮\langle\pi(g){\bf v},\pi(g){\bf u}\rangle=\langle{\bf v},{\bf u}\rangle (1.5.8)

for all 𝐯,𝐮V{\bf v},{\bf u}\in V and gGg\in G.

We say that the inner product ,\langle\cdot,\cdot\rangle is GG-invariant if (1.5.8) holds for all 𝐯,𝐮V{\bf v},{\bf u}\in V and gGg\in G. Other ways of writing (1.5.8) include

π(g)𝐯,𝐮=𝐯,π(g)𝐮\langle\pi(g){\bf v},{\bf u}\rangle=\langle{\bf v},\pi(g)^{*}{\bf u}\rangle

and π(g)=π(g1)=π(g)1\pi(g)^{*}=\pi(g^{-1})=\pi(g)^{-1}.

Proposition 1.5.9.

Let (π,W)(\pi,W) be a subrepresentation of a unitary representation (π,V)(\pi,V). Then (π,W)(\pi,W^{\perp}) is also a subrepresentation, and hence

(π,V)=(π,W)(π,W).(\pi,V)=(\pi,W)\oplus(\pi,W^{\perp}).

Consequently, all unitary representations with a nontrivial subrepresentation are decomposable.

Proof.

We simply need to verify that WW^{\perp} is GG-invariant. Recall that

W={𝐯V|𝐯,𝐰=0 for all 𝐰W}.W^{\perp}=\{{\bf v}\in V\,|\,\langle{\bf v},{\bf w}\rangle=0\text{ for all }{% \bf w}\in W\}.

Given gGg\in G, 𝐯W{\bf v}\in W^{\perp} and any 𝐰W{\bf w}\in W, we have π(g)𝐯,𝐰=𝐯,π(g1)𝐰\langle\pi(g){\bf v},{\bf w}\rangle=\langle{\bf v},\pi(g^{-1}){\bf w}\rangle. However, (π,W)(\pi,W) is a subrepresentation, so π(g1)𝐰W\pi(g^{-1}){\bf w}\in W. Since 𝐯W{\bf v}\in W^{\perp}, we thus have 𝐯,π(g1)𝐰=0\langle{\bf v},\pi(g^{-1}){\bf w}\rangle=0, hence π(g)𝐯W\pi(g){\bf v}\in W^{\perp}. ∎

The following shows that every finite-dimensional representation of a finite group may be “made” unitary:

Proposition 1.5.10.

Let (π,V)(\pi,V) be a finite-dimensional complex representation of a finite group GG. Then there exists a GG-invariant inner product on VV.

Proof.

Since VV is finite-dimensional, it has an inner product ,\langle\cdot,\cdot\rangle (see Problem 1.5.4 below). We now define ,G:V×V\langle\cdot,\cdot\rangle_{G}:V\times V\rightarrow{\mathbb{C}} by the formula

𝐯,𝐮G=1|G|hGπ(h)𝐯,π(h)𝐮,\langle{\bf v},{\bf u}\rangle_{G}=\frac{1}{|G|}\sum_{h\in G}\langle\pi(h){\bf v% },\pi(h){\bf u}\rangle,

and claim that this is a GG-invariant inner product. We leave it as an exercise for the reader to verify that this is an inner product (see Problem 1.5.4). The GG-invariance is shown as follows:

π(g)𝐯,π(g)𝐮G\displaystyle\langle\pi(g){\bf v},\pi(g){\bf u}\rangle_{G} =1|G|hGπ(h)π(g)𝐯,π(h)π(g)𝐮\displaystyle=\frac{1}{|G|}\sum_{h\in G}\langle\pi(h)\pi(g){\bf v},\pi(h)\pi(g% ){\bf u}\rangle
=1|G|hGπ(hg)𝐯,π(hg)𝐮\displaystyle=\frac{1}{|G|}\sum_{h\in G}\langle\pi(hg){\bf v},\pi(hg){\bf u}\rangle
=1|G|h~Gπ(h~)𝐯,π(h~)𝐮=𝐯,𝐮G.\displaystyle=\frac{1}{|G|}\sum_{\widetilde{h}\in G}\langle\pi(\widetilde{h}){% \bf v},\pi(\widetilde{h}){\bf u}\rangle=\langle{\bf v},{\bf u}\rangle_{G}.

We note that as a consequence of the spectral theorem for unitary matrices, we have

Proposition 1.5.11.

Let (π,V)(\pi,V) be a finite-dimensional unitary representation of a group GG. Then for any gGg\in G, π(g)\pi(g) is diagonalisable, i.e. has an eigenbasis.

Proof.

See Exercise 1.5.4 below ∎

1.5.3. Decompositions into irreducibles

Note that Theorem 1.5.6 now follows directly from Propositions 1.5.9 and 1.5.10. In fact, we can give a stronger statement:

Theorem 1.5.12.

Let (π,V)(\pi,V) be a representation of GG. Then there exist irreducible subrepresentations (π,Wi)(\pi,W_{i}) such that

(π,V)=(π,W1)(π,W2)(π,Wn).(\pi,V)=(\pi,W_{1})\oplus(\pi,W_{2})\oplus\ldots\oplus(\pi,W_{n}).

Moreover, the number of times each isomorphism class of an irreducible representation shows up in the above decomposition is independent of the exact choice of decomposition.

The existence of such a decomposition follows as a corollary from repeated application of Maschke’s Theorem (Theorem 1.5.6). To complete the proof of the second part of the theorem we need the following two lemmas:

Lemma 1.5.13.

Let (π1,U)(\pi_{1},U), (π2,V)(\pi_{2},V), and (π3,W)(\pi_{3},W) be representations of a group GG. Then

  1. (i)

    HomG(UV,W)HomG(U,W)HomG(V,W)\mathrm{Hom}_{G}(U\oplus V,W)\cong\mathrm{Hom}_{G}(U,W)\oplus\mathrm{Hom}_{G}(% V,W)

  2. (ii)

    HomG(W,UV)HomG(W,U)HomG(W,V)\mathrm{Hom}_{G}(W,U\oplus V)\cong\mathrm{Hom}_{G}(W,U)\oplus\mathrm{Hom}_{G}(% W,V)

Proof.

Exercise: Problem 1.5.4 below. ∎

Lemma 1.5.14.

Let (π,V)=i=1n(π,Wi)(\pi,V)=\bigoplus_{i=1}^{n}(\pi,W_{i}) be a decomposition of (π,V)(\pi,V) into irreducible subrepresentations. For any irreducible representation (ρ,U)(\rho,U), the number of (π,Wi)(\pi,W_{i}) that are isomorphic to (ρ,U)(\rho,U) is dimHomG(V,U)\operatorname{dim}\operatorname{Hom}_{G}(V,U).

Proof.

By Lemma 1.5.13, dimHomG(V,U)=idimHomG(Wi,U)\operatorname{dim}\operatorname{Hom}_{G}(V,U)=\sum_{i}\operatorname{dim}% \operatorname{Hom}_{G}(W_{i},U). By Corollary 1.4.2 to Schur’s lemma, dimHomG(Wi,U)=1\operatorname{dim}\operatorname{Hom}_{G}(W_{i},U)=1 if (π,Wi)(ρ,U)(\pi,W_{i})\cong(\rho,U), and zero otherwise. ∎

Proof of Theorem 1.5.12.

The existence of such a decomposition follows from repeated application of Theorem 1.5.6. The second part of the statement follows from Lemma 1.5.14. ∎

1.5.4. Exercises

.

Problem 31. Let (π,U)(\pi,U) and (π,W)(\pi,W) be subrepresentations of a representation (π,V)(\pi,V) such that V=UWV=U\oplus W. Verify that the map

(𝐮,𝐰)𝐮+𝐰({\bf u},{\bf w})\mapsto{\bf u}+{\bf w}

is a GG-isomorphism from (π,U)(π,W)(\pi,U)\oplus(\pi,W) to (π,V)(\pi,V).

Problem 32. Verify that

π(g)𝐯,π(g)𝐮=𝐯,𝐮𝐯,𝐮V,gG\langle\pi(g){\bf v},\pi(g){\bf u}\rangle=\langle{\bf v},{\bf u}\rangle\qquad% \forall{\bf v},{\bf u}\in V,\;g\in G

is equivalent to

π(g)𝐯,𝐮=𝐯,π(g1)𝐮𝐯,𝐮V,gG.\langle\pi(g){\bf v},{\bf u}\rangle=\langle{\bf v},\pi(g^{-1}){\bf u}\rangle% \qquad\forall{\bf v},{\bf u}\in V,\;g\in G.

Problem 33. Let VV be a finite-dimensional complex vector space with a basis {𝐛i}i=1,,n\{{\bf b}_{i}\}_{i=1,\ldots,n}, where n=dimVn=\operatorname{dim}V. Show that

i=1nxi𝐛i,j=1nyj𝐛j:=i=1nxiyi¯,\left\langle\sum_{i=1}^{n}x_{i}{\bf b}_{i},\sum_{j=1}^{n}y_{j}{\bf b}_{j}% \right\rangle:=\sum_{i=1}^{n}x_{i}\overline{y_{i}},

for xi,yjx_{i},\,y_{j}\in{\mathbb{C}}, defines an inner product on VV.

Problem 34. Fill in the remaining details of the proof of Proposition 1.5.10, i.e. show that ,G\langle\cdot,\cdot\rangle_{G} is linear in the first argument, conjugate-symmetric, and positive-definite.

Problem 35. Let (π,V)(\pi,V) be a finite-dimensional unitary representation of a group GG, with GG-invariant inner product ,\langle\cdot,\cdot\rangle. Show that the matrix of any π(g)\pi(g) with respect to an orthonormal basis for ,\langle\cdot,\cdot\rangle is unitary (recall that a matrix is unitary if M(=M¯)t=M1M^{*}(=\overline{M}{\,{}^{\mathrm{t}}\!}\,)=M^{-1}).

Problem 36. Prove Lemma 1.5.13.