Does representation irreducibility ensure non-zero determinant?
Does representation irreducibility ensure non-zero determinant?
If a set of matrix representation $M(g)$ for a group $G$ is irreducible, what can we say about their determinant for every $gin G$? Are they all of non-zero determinant?
Thank you very much!
Cheers,
Collin
P.S.: I'm a physics graduate student. So please use as little math terminology as possible, I would really appreciate that!
1 Answer
1
The inverse of $M(g)$ is $M(g^-1)$.
And of course, invertible matrices have non-zero determinant.
Note that this is true for all representations, not just irreducible ones.
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Oh, Yes! You're so right sir. I can't believe this. I just read this sentence yesterday. Thank you! ^. ^ @DanielMroz
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– Collin
Sep 15 '18 at 22:03