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I am going through Hamilton's book Mathematical Gauge Theory, in which he says

If the representation $\rho$ of $G$ is irreducible, it may happen that the representation $\rho|_H$ of $H$ is reducible and decomposes as a direct sum. The actual form of the decomposition of a representation $\rho$ under restriction to a subgroup $H \subset G$ is called the branching rule.

I do not fully understand this definition and would like to see an example of a branching rule through the following exercise from his book. Let $V = \mathbb{C}^{2n}$ be the complex fundamental representation of $SO(2n)$, he asks the reader to determine the branching rule of the representation $V$ under restriction to the subgroup $U(n) \subset SO(2n)$.

Does finding the branching rule mean we determine how the representation decomposes into a direct sum when restricted to $U(n)$? If so, how would one do this?

CBBAM
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  • When restricted to $U(n)$ you mean? Yes, exactly. In general the construction of branching rules can be quite complicated (see here) so if the book hasn't given you any more tools for this case then I suspect it wants you to work it out by hand. In this case, it's not too hard to see, as $n$ increases, there are only $3$ representations of $U(n)$ smaller than $\mathbb{C}^{2n}$ which are $\mathbb{C}^{n}$, $(\mathbb{C}^{n})^*$ and the trivial representation so it isn't too hard to guess the form of the decomposition and then prove it. – Callum Jun 14 '23 at 13:35
  • @Callum Yes that was a typo, thank you for catching that. How would one work out or notice that those are the only three representations smaller than $\mathbb{C}^{2n}$? – CBBAM Jun 14 '23 at 18:08
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    Just by familiarity with the representations of $U(n)$. The fundamental representations are the exterior powers $\bigwedge^k\mathbb{C}^n$ which have dimension $n$ choose $k$ (quickly exceeding $2n$ except for $k=1,n$). The other smallest representations are going to be the symmetric square $S^2\mathbb{C}$ and its dual which have dimension $n+1$ choose $2$ and the adjoint representation with dimension $n^2 -1$. Each of these have dimensions growing quadratically with $n$ so clearly exceed $2n$ eventually but actually this happens fairly early if you crunch the numbers. – Callum Jun 14 '23 at 18:33
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    Assuming there is a consistent pattern, we now only have a few choices, of which the most likely to me would seem to be $\mathbb{C}^n \oplus (\mathbb{C}^n)^*$ (as it is nicely symmetric). So now you can simply ask: does the subgroup $U(n) \subset SO(2n)$ fix two complementary subspaces of dimension $n$ and does it act on them the same or dual to each other – Callum Jun 14 '23 at 18:38
  • @Callum Thank you for your comment. I am familiar with exterior powers but I am fairly new to representation theory (including representations of specific groups such as $U(n)$, so it looks like I will have to do some more reading before attempting this problem. – CBBAM Jun 14 '23 at 18:40
  • Something like Fulton and Harris's Representation Theory might be a good shout for that. The second half is all about Lie groups. Note the representations of $\operatorname{U}(n)$ are the same as those of $\operatorname{SL}(n,\mathbb{C})$ and $\operatorname{sl}(n,\mathbb{C})$ and these are usually the first examples discussed – Callum Jun 15 '23 at 15:58

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