# Questions tagged [characters]

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

59 questions
886 views

### 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 ...
560 views

### 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 ...
971 views

### 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....
92 views

### 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 ...
1k views

2k views

### 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 ...
209 views

### 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 ...
508 views

1k views

### Order of group elements from a character table

Most questions that I can find on here (or anywhere else on the internet) deal with constructing a character table given a description of the group. I'm trying to answer a question which goes the ...
1k views

### Computing Brauer characters of a finite group

I am studying character theory from the book "Character Theory of Finite Groups" by Martin Isaac. (I am not too familiar with valuations and algebraic number theory.) In the last chapter on modular ...
351 views

108 views

### 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 ...
400 views

123 views

### 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 ...
143 views

### 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 ...
62 views