Complete reducibility and labelling of irreps

1. Formula
   Observation

   The slide writes $\Gamma$ : $G \to$ GL(V) and later $\Gamma(g)$ for the matrix of the representation.

$\Gamma$

A completely reducible group representation of G on V.

Representation $\Gamma$ : $G \to$ GL(V).

1. Formula
   Observation

   The displayed statement begins with $\Gamma$ : $G \to$ GL(V).

G

The group being represented.

Group.

1. Formula
   Observation

   The slide states that one can find a basis of V such that the matrix notation becomes block diagonal.

V

The vector space carrying the representation.

Vector space.

1. Formula
   Observation

   The matrix is written as $\Gamma(g)$, with blocks $\Gamma^1(g)$, $\Gamma^2(g)$, $\Gamma^3(g)$, etc.

g

An element of the group G.

$g \in G$.

1. Formula
   Observation

   The slide says 'where the $\Gamma^j$ are irreducible ($d_j \times d_j$ blocks with $d_j = \dim \Gamma^j$)'.

$\Gamma^j$

An irreducible representation appearing as a diagonal block in the chosen basis.

Index j labels an irreducible summand.

1. Formula
   Observation

   The text explicitly gives $d_j = \dim \Gamma^j$.

$d_j$

The dimension of the irreducible representation $\Gamma^j$, equivalently the size of its square block.

Positive integer.

1. Formula
   Observation

   The relabeled decomposition is written as $\bigoplus_j a_j \Gamma^j$, followed by 'i.e. in $\Gamma$ the irreducible representation $\Gamma^j$ is contained $a_j$ times.'

$a_j$

Multiplicity number: how many times the irreducible representation $\Gamma^j$ occurs in $\Gamma$.

Nonnegative integer; in the spoken example, $a_j > 1$ when the same irrep appears several times.

1. Audio
   Observation

   The lecturer says: 'So first we have a completely reducible representation. This means by a suitable choice of basis we can write the representation matrices in this block diagonal form.'

2. Formula
   Observation

   The slide displays $\Gamma(g)$ as a block diagonal matrix with $\Gamma^1(g)$, $\Gamma^2(g)$, $\Gamma^3(g)$, $\ldots$ on the diagonal and zeros off the diagonal.

In this clip, a completely reducible representation is characterized operationally: after choosing a suitable basis of V, every representation matrix $\Gamma(g)$ can be written in block diagonal form, with irreducible blocks along the diagonal.

1. $\Gamma$ : $G \to$ GL(V) is completely reducible.

2. One chooses a suitable basis of V.

1. $\Gamma$
2. G
3. V
4. g

1. Audio
   Observation

   The lecturer says there are 'some square blocks here, maybe different sizes, where you have these irreducible representations sitting.'

2. Formula
   Observation

   The slide states: 'where the $\Gamma^j$ are irreducible ($d_j \times d_j$ blocks with $d_j = \dim \Gamma^j$).'

Each diagonal block $\Gamma^j(g)$ is an irreducible representation of the group on its own subspace. The block is square, and its size is the dimension of that irreducible representation.

1. The representation has already been put into the block diagonal form described above.

1. Completely reducible representation in block diagonal form
2. $\Gamma^j$
3. $d_j$

1. Audio
   Observation

   The lecturer says: 'And now it may make sense to just use these labels 1, 2, 3 and so on as labels for inequivalent irreducible representations.'

2. Formula
   Observation

   The slide introduces '(relabel)' and rewrites the decomposition as $\Gamma = \Gamma^1 \oplus \cdots \oplus \Gamma^1 \oplus \Gamma^2 \oplus \cdots \oplus \Gamma^2 \oplus \cdots = \bigoplus_j a_j \Gamma^j$.

Instead of numbering blocks merely by their order of appearance, one regroups equal or equivalent irreducible summands under a single label. After relabeling, each index j denotes one inequivalent irreducible representation.

1. The original block diagonal decomposition is available.

2. Equivalent irreducible summands are grouped under the same label.

1. Diagonal blocks are irreducible representations
2. $\Gamma^j$

1. Audio
   Observation

   The lecturer says: 'and then we have a multiplicity number here if a representation appears several times.'

2. Formula
   Observation

   The slide states: 'i.e. in $\Gamma$ the irreducible representation $\Gamma^j$ is contained $a_j$ times.'

After relabeling, the coefficient $a_j$ records how many copies of the irreducible representation $\Gamma^j$ occur inside $\Gamma$. If $a_j > 1$, the same irreducible type appears several times on different subspaces.

1. The decomposition has been relabeled so that each j corresponds to one inequivalent irreducible representation.

1. Relabeling by inequivalent irreducible representations
2. $a_j$
3. $\Gamma^j$

1. Audio
   Observation

   'This means by a suitable choice of basis we can write the representation matrices in this block diagonal form.'

2. Formula
   Observation

   The displayed remark begins: 'Given a completely reducible representation $\Gamma$ : $G \to$ GL(V), we can find a basis of V such that in matrix notation ...'

If $\Gamma$ : $G \to$ GL(V) is completely reducible, then there exists a basis of V in which every matrix $\Gamma(g)$ is block diagonal with irreducible diagonal blocks.

1. $\Gamma$ : $G \to$ GL(V) is completely reducible.

For all $g \in G$, after one suitable choice of basis of V, $\Gamma(g)$ has the displayed block diagonal form.

1. Audio
   Observation

   'So here the sum is over inequivalent irreducible representations and then we have a multiplicity number here if a representation appears several times.'

2. Formula
   Observation

   The slide writes $\Gamma = \bigoplus_j a_j \Gamma^j$ and explains that $\Gamma^j$ is contained $a_j$ times.

After relabeling, a completely reducible representation can be written as a direct sum over inequivalent irreducible representations, with integer coefficients recording how often each one occurs.

1. $\Gamma$ is completely reducible.

2. Equivalent irreducible summands are grouped under the same label.

The outer sum runs over inequivalent irreducible representations $\Gamma^j$; each $a_j$ counts repeated occurrences of that label.

1. Audio
   Observation

   The lecturer first says the blocks had been numbered 'as they appeared, gamma 1, gamma 2, gamma 3 and so on', then proposes using those labels for inequivalent irreducible representations, and finally explains that repeated equivalent blocks are collected with a multiplicity number.

2. Formula
   Observation

   The slide moves from the block matrix display to the relabeled expression $\Gamma = \bigoplus_j a_j \Gamma^j$.

1. Expression

   $$\Gamma(g)=\begin{pmatrix}\Gamma^1(g)&&&&0\\&\Gamma^2(g)&&&\\&&\Gamma^3(g)&&\\&&&\ddots&\\0&&&&\end{pmatrix}$$
   Explanation

   Start from a block diagonal matrix form obtained after a suitable basis change.

   Justification

   Stated directly on the slide and in the audio as the meaning of complete reducibility.

   Shown in the video
2. Expression

   $$\Gamma^j \text{ are irreducible},\quad d_j=\dim \Gamma^j$$
   Explanation

   Identify each diagonal block as an irreducible representation of some dimension $d_j$.

   Justification

   Explicit text on the slide.

   Shown in the video
3. Expression

   $$\text{initially: labels }1,2,3,\ldots\text{ by order of appearance}$$
   Explanation

   The lecturer notes that the original numbering was just sequential according to how the blocks appeared.

   Justification

   Audio: 'I just numbered them as they appeared'.

   Shown in the video
4. Expression

   $$\text{relabel so that } j \text{ labels inequivalent irreducible representations}$$
   Explanation

   Replace appearance-order indexing by indexing equivalence classes of irreducible summands.

   Justification

   Audio: 'use these labels ... as labels for inequivalent irreducible representations'.

   Shown in the video
5. Expression

   $$\Gamma = \underbrace{\Gamma^1 \oplus \cdots \oplus \Gamma^1}_{a_1\ \text{times}} \oplus \underbrace{\Gamma^2 \oplus \cdots \oplus \Gamma^2}_{a_2\ \text{times}} \oplus \cdots = \bigoplus_j a_j \Gamma^j$$
   Explanation

   Collect all copies of the same irreducible type together and encode their number by $a_j$.

   Justification

   Displayed formula plus audio explanation of repeated equivalent representations and multiplicity.

   Shown in the video

The decomposition is reorganized from a list of blocks into a sum over inequivalent irreducible types with explicit multiplicities.

1. Animation
   Observation

   An orange oval is drawn around the word 'once' in the sentence 'Here an irreducible representation can appear more than once, (relabel)'.

1. the printed word 'once'

2. orange annotation oval

1. The word 'once' is circled to draw attention to repetition before relabeling.

1. The underlying slide text remains unchanged.

The visual cue marks the situation that motivates introducing multiplicity numbers: the same irreducible representation may occur more than once.

1. Animation
   Observation

   Orange boxes are drawn successively around the diagonal entries $\Gamma^1(g)$, $\Gamma^2(g)$, $\Gamma^3(g)$, and the lower-right continuation of the block diagonal matrix.

1. matrix $\Gamma(g)$

2. diagonal blocks $\Gamma^1(g)$, $\Gamma^2(g)$, $\Gamma^3(g)$, $\ldots$

3. orange rectangular annotations

1. Each diagonal block is highlighted in turn.

2. The off-diagonal zero structure remains visible.

1. The matrix stays in block diagonal form.

2. The labels of the blocks do not change during this highlighting.

The animation visually identifies the individual irreducible summands sitting on the diagonal of the representation matrix.

1. Animation
   Observation

   Orange circles are drawn around terms in the relabeled direct sum, including repeated $\Gamma^1$ terms, later $\Gamma^2$ terms, and the summation index j in $\bigoplus_j a_j \Gamma^j$; arrows point between corresponding repeated terms and toward the compact notation.

1. expanded direct sum $\Gamma^1 \oplus \cdots \oplus \Gamma^1 \oplus \Gamma^2 \oplus \cdots$

2. compact form $\bigoplus_j a_j \Gamma^j$

3. orange circles and arrows

1. Repeated identical labels are grouped visually.

2. Arrows connect the expanded sum to the condensed notation.

3. The index j and coefficient $a_j$ are singled out.

1. The mathematical content of the decomposition does not change; only the presentation is regrouped.

The drawing demonstrates how several copies of the same irreducible are collected under one label and counted by a multiplicity coefficient.

1. Audio
   Observation

   The lecturer contrasts the earlier numbering 'as they appeared' with the later use of labels for 'inequivalent irreducible representations'.

One might think that the indices 1,2,3,... always label distinct irreducible types simply because they mark different diagonal blocks.

In this clip, the lecturer explicitly reinterprets the labels after relabeling: the index j is meant to label inequivalent irreducible representations, while repetitions of the same type are counted separately by $a_j$.

1. Audio
   Observation

   'This means by a suitable choice of basis we can write the representation matrices in this block diagonal form.'

The notion of complete reducibility is presented through the existence of a basis in which the representation decomposes into irreducible diagonal blocks.

1. Audio
   Observation

   The lecturer moves from blocks numbered by appearance to labels for inequivalent irreducible representations.

Once the irreducible blocks are identified, one can regroup them by equivalence class rather than by position in the matrix.

1. Formula
   Observation

   The compact notation $\bigoplus_j a_j \Gamma^j$ is introduced immediately after the relabeling discussion.

The relabeled direct sum naturally carries integer coefficients $a_j$ that count how many copies of each irreducible type occur.

1. Audio
   Observation

   The lecturer defines complete reducibility via a suitable basis giving block diagonal matrices.

1. Completely reducible representation in block diagonal form
2. Complete reducibility implies block diagonal matrix form

1. Audio
   Observation

   The lecturer says it makes sense to use labels for inequivalent irreducible representations instead of mere appearance order.

1. Relabeling by inequivalent irreducible representations
2. Block position is not the same as irreducible type

1. Formula
   Observation

   The slide states that $\Gamma^j$ is contained $a_j$ times.

1. Multiplicity coefficient $a_j$
2. $a_j$