# Questions tagged [characters]

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

229 questions
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### Subgroups as isotropy subgroups and regular orbits on tuples

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In some ...
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### If $g$ is commutator then so is $g^m$ for $(m,o(g))=1$

There are certain theorems in finite group theory whose proofs involve character theory and for which there are still no character-free proofs. Among such is Frobenius theorem on transitive ...
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### minimal polynomial of algebraic integer over $\mathbb{Z}$ and $\mathbb{Q}$

It was an exercise in Isaac's character theory: Let $\alpha$ be an algebraic integer (so $\alpha$ satisfies monic irreducible polynomial $g(x)$ over $\mathbb{Z}$). Let $\alpha$ satisfies a ...
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### If $T$ is an algebraic torus, is there a difference between $\operatorname{Irr}(T)$ and $X(T)$?

Suppose $T$ is a maximal torus in a linear algebraic group, for example, the diagonal matrices in $GL_n$. In this context, what does it most commonly mean when authors refer to the irreducible ...
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### Representation Theory and Character Table of the Monster

Does anybody know of a good reference for computing the character table of the Monster, with a clear demonstration of the theory behind the methods?
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### Show that over finite fields of characteristic two the finite group $C_3$ has no nontrivial representation

Let $G = \mathbb Z / 3\mathbb Z$ be the cyclic group of order three. I conjecture that every irreducible representation over a finite field of characteristic $2$ must be trivial. But how to prove this?...
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### Is there a stable probability distribution on the rational numbers?

Does there exist a (non-trivial) probability distribution on the rational numbers $$\sum_{r\in\mathbb{Q}}p_r=1$$ with $0\leq p_r$, which is stable, meaning that the sum of two i.i.d. random variables ...
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### Additive character sum over intersection of additive and multiplicative subgroups of finite fields

Let $H$ be a multiplicative subgroup of the finite field $\mathbb{F}_q$ with $q$ elements, say $H$ is the subgroup of $d$-th powers, $d \mid q-1$. Let $L$ be a subspace of $\mathbb{F}_q$ over some ...
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### $\chi$ varies over characters of $F$ of order dividing $m$, $\chi^{'}$ varies over characters of $F_s$ of order dividing $m$

A month ago I've asked two questions about rationality of the zeta function. The pages that belongs to my question are (linked here) Unfortunately I'm still clueless, but some steps are clear now. ...
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### What is a primitive character of a Galois groups of a finite cyclic extension of local fields?

Let $K$ be a local field and $F/K$ be a cyclic extension of degree $n$, meaning that $n = [F:K]$ and $\operatorname{Gal}(F/K) \simeq C_n$ is cyclic. In the proof of Lemma 2 of the paper "Euler ...
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### An exercise from Isaac's character theory (4.6)

Let $\chi$ denotes (complex) irreducible character of a finite group $G$. Define $\chi^{(n)}$ to be the function from $G$ to $\mathbb{C}$ by $\chi^{(n)}(g)=\chi(g^n)$. Problem: For fixed $n>0$, ...
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### Help with finishing a Character Table

Can anyone help me to finish the character table following the steps below. (I know how to do part (b) but i'm having trouble with the rest). Let $G$ be a finite group with conjugacy class ...
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### Finding all the characters of $\mathbb C^\times$

I am trying to determine all the characters on the group $\mathbb C^\times$ of non-zero complex numbers under multiplication. I believe that all the characters are given by: \chi : z=re^{ix} \to {r\...
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### Conductor as the volume of integers

I am working on Tate's thesis, and I have some problems with computations, yet the result seems to be a good natural motivation for introducing the arithmetic conductor of a character. Let $F$ be a ...
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### Two exercises on characters on Marcus (part 1)

I am trying to solve exercises 17 and 18 on Marcus book (page 210, chapter 7). Let's look at the first. This should be solved and correct now. 17) Let $m$ be even, $m\ge3$ and suppose $\chi$ is a ...