- #1

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i.e. suppose I knew all the irreducible real representations of G, (how) could I then get all the irreducible complex and quaternionic representations?

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- A
- Thread starter hideelo
- Start date

- #1

- 91

- 14

i.e. suppose I knew all the irreducible real representations of G, (how) could I then get all the irreducible complex and quaternionic representations?

- #2

fresh_42

Mentor

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$$

\varphi_\mathbb{C} \, : \,G \longrightarrow GL(V_\mathbb{C}) \quad \text{ by } \quad \varphi_\mathbb{C}(g)(\lambda\cdot v) = \varphi_\mathbb{R}(g)(v) \otimes \lambda

$$

I'm not sure, however, whether they automatically will be irreducible again, as there are simply more eigenvalues available, so I doubt it.

- #3

Infrared

Science Advisor

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suppose I knew all the irreducible real representations of G, (how) could I then get all the irreducible complex and quaternionic representations?

Unfortunately, you can't. Consider the cyclic group of order 3. Its only irreducible real representation is the trivial one, but it has three irreducible complex representations.

Indeed, let the cyclic group of order 3 act on [itex]\mathbb{R}^2[/itex] with a generator corresponding to rotation by [itex]2\pi/3[/itex]. This representation is irreducible but its complexification isn't.I'm not sure, however, whether they automatically will be irreducible again, as there are simply more eigenvalues available, so I doubt it.

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