1.5. Lecture 5
1.5.1. Direct sums and reducibility
We start by defining sums of representations:
Definition 1.5.1.
Let
for all
We also write
Example 1.5.2.
Let
Conversely, we also have the following definition.
Definition 1.5.3.
If
Note that if
with a
for all
Example 1.5.4.
The permutation representation
The following example shows that there are representations that are neither decomposable or irreducible:
Example 1.5.5.
Let
This representation is not irreducible, as
The representation is not decomposable , as then there would be two non-zero vectors
However, for finite groups, we have the following:
Theorem 1.5.6 (Maschke’s Theorem, 1899).
Let
Before proving this, we introduce another important notion:
1.5.2. Unitary representations
Definition 1.5.7.
Let
for all
We say that the inner product
and
Proposition 1.5.9.
Let
Consequently, all unitary representations with a nontrivial subrepresentation are decomposable.
Proof.
We simply need to verify that
Given
The following shows that every finite-dimensional representation of a finite group may be “made” unitary:
Proposition 1.5.10.
Let
Proof.
Since
and claim that this is a
∎
We note that as a consequence of the spectral theorem for unitary matrices, we have
Proposition 1.5.11.
Let
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
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
-
(i)
-
(ii)
Proof.
Exercise: Problem 1.5.4 below. ∎
Lemma 1.5.14.
Let
1.5.4. Exercises
Problem 31.
Let
is a
Problem 32. Verify that
is equivalent to
Problem 33.
Let
for
Problem 34.
Fill in the remaining details of the proof of Proposition 1.5.10, i.e. show that
Problem 35.
Let
Problem 36. Prove Lemma 1.5.13.