# Questions tagged [characters]

For questions about characters (traces of representations of a group on a vector space).

60 questions
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### Why unitary characters for the dual group in Pontryagin duality if $G$ is not compact?

In harmonic analysis, for any locally compact abelian group, one constructs the dual group as the group of homomorphisms into the unit circle with the compact open topology. In other words, unitary ...
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### Easy way to get real irreducible characters (reps) from complex irreducible characters?

For plenty of groups, the real irreducible characters/representations aren't the same as the complex irreducible representations. I really enjoy James Montaldi's summary of real representations, for ...
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### Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$, where $G$ is abelian and $\mathbb{T}$ is the circle group.

Suppose $G$ is an abelian group and $a\in G$ and $$f:\left<a \right>\to\Bbb T$$ is a homomorphism. Can $f$ be extended to a homomorphism on $G$: $$g:G\to \Bbb T$$ ? $\Bbb T$ is the circle group....
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### Show that the characters of the representations $\phi_{n}$ of $SU(2)$ constitute a complete orthogonal set.

The question is given below: And the other questions mentioned are (I know the solutions of all of them): Sorry for the bad formulation of the my question at the first time I have ...
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### Why does the tensor product of an irreducible representation with the sign representation yield another irreducible representation?

I was writing this question, and I came up with an answer, so I thought I would answer it myself: In considering representations of $S_n$, among others, we have the "sign representation", that is the ...
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### A direct proof that for finite $G$ we have “$G$ abelian iff all irreducible characters are linear”

A finite group is abelian iff all its irreducible characters have dimension one (hence are linear). A common proof uses that the number of irreducible representations equals the number of conjugacy ...
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### Frobenius Reciprocity and a character theory problem

How Frobenius Reciprocity can help us to solve these two problems: Let $H$ be a subgroup with index $m$ in the finite group $G$. Let $F$ be an algebraic closed field of characteristic $0$. ...
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### Do “$K/k$ twisted” representations exist?

Given $k$-representations $V,W$ of a group $G$, where $k$ is a field, $K/k$ a field extension, if we have $V\otimes_k K\cong W\otimes_k K$ as $K$-representations, do we have that $V\cong W$? Being ...
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### Generalizing Artin's theorem on independence of characters

Artin's theorem says that for any field $K$ and any (semi) group $G$, the set of homomorphisms from $G$ into the multiplicative group $K^*$ is linearly independent over $K$. Can this theorem be ...
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### If Brauer characters are $\bar{\mathbb{Q}}$-linearly independent, why are they $\mathbb{C}$-linearly independent?

If Brauer characters are $\bar{\mathbb{Q}}$-linearly independent, why are they $\mathbb{C}$-linearly independent? I think this is a linear algebra fact showing up when proving the irreducible Brauer ...
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### Trying to understand why the zeta function is a rational function under certain conditions. Questions about some equations.

Information: I linked the pages below, which relate to my questions. I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th ...
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### Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.
I posted a question a short while ago on this but got no response. I have worked on this more and so now have a more specific question. To start with we work with the $\mathbb{Q}$ version of Hamilton'...