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2 votes
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Does the formal character determine the representation?

Yes, it's true. The cancellation property does hold for finite-dimensional representations of $\mathfrak{sl}_2$, and more generally for finite-dimensional representations of any semisimple Lie algebra,...
Qiaochu Yuan's user avatar
2 votes
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On Frobenius–Schur indicator of real/complex representations

It means both in different contexts, as far as I know. It's unfortunate but they can be disambiguated based on whether the author is discussing real or complex representations. The Frobenius-Schur ...
Qiaochu Yuan's user avatar
1 vote

Can the sum of a nonlinear irreducible character's values on $Z(\chi)$ be zero?

This sum of values is not equal to zero if and only if $\chi$ is constant over $Z(\chi)$. First, write the restriction of $\chi$ to $Z(\chi)$ as $\chi_{Z(\chi)}=\chi(1)\lambda$, where $\lambda$ is a ...
Deif's user avatar
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0 votes

How to determine $\Gamma_{15}$?

The Chinese remainder theorem states that the ring homomorphism $\mathbb{Z}/mn\mathbb{Z} \to \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$ that maps $x + mn\mathbb{Z}$ to $(x + m\mathbb{Z},x + ...
arkeet's user avatar
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2 votes
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Exercise 2.19 in Isaacs's book on character theory

You have already done the hard part, which is establishing the hint. You can now finish the problem in one line, using a previous problem from chapter 2. (As mentioned in the comments below, this is 2....
Steve D's user avatar
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2 votes
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Problem 2.14 from Isaacs's Character Theory of Finite Group

I will show that $n\leq\chi(1)$. Since $n$ is a $p$-power for some prime $p$, if $H\cap\ker\chi\not=1$ for every $\chi\in\text{Irr}(G)$ then it would follow that $K\subseteq H\cap\ker\chi$ for the ...
Deif's user avatar
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