I'm reading a book on representation theory (in Japanese). I'm stuck trying to understand the definition of the canonical decomposition that is given. The below is my best attempt at a translation.
Let $G$ be a finite group and $(V, \rho)$ a representation. Suppose there exist irreducible invariant (presumably under $G$) subspaces $W_1, \cdots, W_r$ such that
$$ V = W_1 \oplus \cdots \oplus W_r $$
Now, consider distinct irreducible representations $V_1, \cdots, V_k$ of $(V, \rho)$. We choose these in a way such that each $W_i$ from the direct sum decomposition is equal (isomorphic??) to some $V_j$. We let the sum of all $W_j$ equal to $V_i$ be $V^{(i)}$. Then,
$$V = \bigoplus_{i = 1}^{k}V^{(i)}$$
is known as the canonical decomposition.
I don't understand this definition at all. I tried looking for other books/notes on this topic but couldn't anything like this anywhere.
There is a chance that I have misinterpreted the Japanese and my translation is slightly wrong. If so I'd love to know what can be done to correct the definiton.
Otherwise, could someone perhaps give me an example or an explanation for this definition?
Edit:
The closest thing I could find was this on the wikipedia article for representation theory:
(The canonical decomposition)
To achieve a unique decomposition, one has to combine all the irreducible subrepresentations that are isomorphic to each other. That means, the representation space is decomposed into a direct sum of its isotypes. This decomposition is uniquely determined. It is called the canonical decomposition.
However, I don't know what an isotype is. My knowledge of rep. theory is only up to Schur's Lemma and Maschke's Theorem.
Best Answer
Let's do a nice example first. Let $G = C_2$ be the cyclic group of order $2$, with generator $g$, and let $V$ be a representation of it (not necessarily finite-dimensional, but over a field of characteristic $\neq 2$). $G$ has two irreducible representations, namely the trivial representation $gv = v$ and the sign representation $gv = -v$. The corresponding isotypic components (for some reason the Wikipedia article only considers the case of reductive Lie algebras but the definition applies to finite groups too) of $V$ are the "trivial subspace" or "positive subspace"
$$V_1 = \{ v \in V : gv = v \}$$
and what you might call the "sign subspace" or the "negative subspace"
$$V_{-1} = \{ v \in V : gv = -v \}.$$
Note that $g^2 = 1$ implies that $g$ has eigenvalues $\pm 1$ so these are also the two eigenspaces of $g$. $V_1$ can alternatively be defined as the sum of all subspaces of $V$ isomorphic to the trivial representation, and similarly $V_{-1}$ can be defined as the sum of all subspaces of $V$ isomorphic to the sign representation. Then the claim is that $V$ is canonically a direct sum $V_1 \oplus V_{-1}$ of the two isotypic components. What this says, explicitly, is that every vector can be written in a unique way as the sum of a vector fixed by $g$ and the sum of a vector negated by $g$. This can be written down more explicitly, using a special case of the discrete Fourier transform, as
$$v = \frac{v + gv}{2} + \frac{v - gv}{2}.$$
Already this $C_2$ case is a nice general pattern that you can see in more elementary parts of mathematics without knowing anything about groups or representations. For example if we consider the action $f(x) \mapsto f(-x)$ of $C_2$ on vector spaces of functions (say $\mathbb{R} \to \mathbb{R}$ or $\mathbb{C} \to \mathbb{C}$), we see that this gives a unique decomposition of any function as the sum of an even function and an odd function. A famous example is Euler's formula $e^{ix} = \cos x + i \sin x$, where the even part is $\cos x = \frac{e^{ix} + e^{-ix}}{2}$ and the odd part is $i \sin x = \frac{e^{ix} - e^{-ix}}{2}$.
It is a pleasant exercise to work out how to generalize this to the case of $G = C_n$ a cyclic group of order $n$, or more generally to any finite abelian group $G = A$ (here we need to work over a field $K$ of characteristic not dividing $|G|$). With the cyclic group of order $n$ there are $n$ isotypic components given by the $n^{th}$ roots of unity, which can equivalently be thought of as the $n$ eigenspaces $V_{\zeta} = \{ v \in V : gv = \zeta v \}$ of the generator $g$ (here $\zeta^n = 1$), and explicitly writing down the corresponding direct sum decomposition recovers the theory of the discrete Fourier transform. In the case of a finite abelian group $A$ there are $|A|$ isotypic components labeled by the characters $\chi : A \to K^{\times}$, namely $V_{\chi} = \{ v \in V : \forall a \in A, av = \chi(a) v \}$.
In the nonabelian case the isotypic components are a little more difficult to write down explicitly, but they are labeled by the irreducible representations $\chi$ of $G$ and can be defined as the sum (not a direct sum) $V_{\chi}$ of all subspaces isomorphic to a given irreducible. Then the claim is again that $V$ is canonically the direct sum $\bigoplus_{\chi} V_{\chi}$ of these subspaces, and what this again means is that every $v \in V$ is uniquely the sum of one vector $v_{\chi} \in V_{\chi}$ from each isotypic component. This follows from Maschke's theorem (this is also a good exercise). It is also possible to write down more explicitly a formula for these $v_{\chi}$ (it involves the character, which I will also denote $\chi$) which generalizes the discrete Fourier transform, which I can give if you're interested.
The point of doing all this is that the usual decomposition $V \cong \bigoplus_i n_i V_i$ of a representation into a direct sum of irreducibles is not canonical if there are multiplicities; if any $n_i > 1$ we have to choose specific subrepresentations isomorphic to the irreducible $V_i$. The isotypic decomposition is canonical; we replace the term $n_i V_i$ of the above decomposition with the sum (again, not direct) of all subrepresentations isomorphic to $V_i$, and that's the isotypic component. You can think of it as being given, canonically, by a tensor product $\text{Hom}(V_i, V) \otimes V_i$ where $\text{Hom}(V_i, V)$ is called the multiplicity space; it has dimension $n_i$. From here, getting a more ordinary-looking decomposition amounts to choosing a basis of the multiplicity space, but again, that isn't canonical and sometimes we don't want to do it.