1.5. Lecture 5
1.5.1. Direct sums and reducibility
We start by defining sums of representations:
Definition 1.5.1.
Let and be representations of a group . The direct sum of the two representations is , where
for all , , .
We also write for the direct sum of the representations.
Example 1.5.2.
Let and be representations of given by and . Then is the representation of on given by .
Conversely, we also have the following definition.
Definition 1.5.3.
If is a representation with non-trivial subrepresentations and such that , then we say that is decomposable or reducible.
Note that if is decomposable , then
with a -isomorphism from to given by
for all (see Problem 1.5.4 below). We say that the representation has been decomposed into a direct sum of subrepresentations.
Example 1.5.4.
The permutation representation decomposes as , where and is the linear span of .
The following example shows that there are representations that are neither decomposable or irreducible:
Example 1.5.5.
Let be the representation of given by
This representation is not irreducible, as is a non-trivial subrepresentation.
The representation is not decomposable , as then there would be two non-zero vectors and such that , and , are subrepresentations. The vectors and would then be two linearly independent eigenvectors of all the matrices , 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 be a non-trivial subrepresentation of a finite-dimensional complex representation of a finite group . Then there exists a subrepresentation of such that
Before proving this, we introduce another important notion:
1.5.2. Unitary representations
Definition 1.5.7.
Let be a representation of a group on an inner product space , with inner product . The representation is said to be unitary if
for all and .
We say that the inner product is -invariant if (1.5.8) holds for all and . Other ways of writing (1.5.8) include
and .
Proposition 1.5.9.
Let be a subrepresentation of a unitary representation . Then is also a subrepresentation, and hence
Consequently, all unitary representations with a nontrivial subrepresentation are decomposable.
Proof.
We simply need to verify that is -invariant. Recall that
Given , and any , we have . However, is a subrepresentation, so . Since , we thus have , hence . ∎
The following shows that every finite-dimensional representation of a finite group may be “made” unitary:
Proposition 1.5.10.
Let be a finite-dimensional complex representation of a finite group . Then there exists a -invariant inner product on .
Proof.
Since is finite-dimensional, it has an inner product (see Problem 1.5.4 below). We now define by the formula
and claim that this is a -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 -invariance is shown as follows:
∎
We note that as a consequence of the spectral theorem for unitary matrices, we have
Proposition 1.5.11.
Let be a finite-dimensional unitary representation of a group . Then for any , 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 be a representation of . Then there exist irreducible subrepresentations such that
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 , , and be representations of a group . Then
-
(i)
-
(ii)
Proof.
Exercise: Problem 1.5.4 below. ∎
Lemma 1.5.14.
Let be a decomposition of into irreducible subrepresentations. For any irreducible representation , the number of that are isomorphic to is .
1.5.4. Exercises
Problem 31. Let and be subrepresentations of a representation such that . Verify that the map
is a -isomorphism from to .
Problem 32. Verify that
is equivalent to
Problem 33. Let be a finite-dimensional complex vector space with a basis , where . Show that
for , defines an inner product on .
Problem 34. Fill in the remaining details of the proof of Proposition 1.5.10, i.e. show that is linear in the first argument, conjugate-symmetric, and positive-definite.
Problem 35. Let be a finite-dimensional unitary representation of a group , with -invariant inner product . Show that the matrix of any with respect to an orthonormal basis for is unitary (recall that a matrix is unitary if ).
Problem 36. Prove Lemma 1.5.13.