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Questions tagged [characters]

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

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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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638 views

Character theory of $2$-Frobenius groups.

Edit Summary: I've posted this on MO and received a partial answer there. Can anybody help me expand on this? Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\...
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Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
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Reading the centralizer off of the character table

Assume that I am given the table of irreducible characters of a finite group $G$. I realize that we can see the order of the centralizer of any element $g \in G$ by summing the squares in the ...
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Irreducible 2-Brauer characters of $S_5$

Beginning with the ordinary character table of the symmetric group $S_5$, one immediately gets the following Brauer characters in characteristic two: $\begin{array}{c|c|c|c} S_5 & () & (...
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Differentiating a $p$-adic character

Let $L$ be a finite extension of $\mathbb Q_p$ with ring of integers $\mathcal{O}=\mathcal{O}_L$ and let $B_1(L):=\{z \in L \colon \vert z-1 \vert <1 \}$. Let $\widehat{\mathcal{O}}(L)_{\mathbb ...
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On the existence of semisimple (irreducible) characters of simply connected algebraic groups

I'm learning Deligne-Lusztig theory of complex characters of finite groups of Lie type and there are some difficulties for me in understanding the theory.Let $\mathcal{G}$ be a simple simply-connected ...
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67 views

Example of projective rep being used in Clifford theory

I'm trying to understand the use of projective representations in Clifford theory, and I'd like a small example where projective representations really help, and the ingredients are actually ...
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206 views

On Applications of the Murnaghan-Nakayama rule

The question is located below. In short, I am looking for an accessible explanation of the Murnaghan-Nakayama rule in relation to the following problem. Pardon the long setup. Let $Y$ be a standard ...
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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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Why chemists are interested in character tables?

I'm a french PhD student and I'm working on representation theory (of non-compact Lie groups). By the way I try to give a "concrete" sense to character tables of finite groups and I have a lot of ...
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64 views

Showing $\mathbb{F}_q^{\times}$ is cyclic using Character theory

I was wondering if there is a way to prove that the multiplicative of a finite field is cyclic by looking at the character table of such a group. In particular, I was wondering if there is a way to ...
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Character table of a group determines the set of commutators of the group.

We write $[x,y]$ for the commutator $x^{-1}y^{-1}xy$ of $x$ and $y$ in a group $G$. (A) Let $g \in G$ and fix $x \in G$. Show that $g$ is conjugate to $[x,y]$ for some $y \in G$ iff $$\sum_{\chi \...
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Linear independence of primitive Dirichlet characters and convolution

This is not an exercise but merely a question I have. Fix $N \in \mathbb{N}$ and suppose there exist some values $a_k \in \mathbb{C}$, for $k \in \mathbb{Z}_N$, such that $$ \sum_{k \in \mathbb{Z}_N} ...
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Every irreducible character of $G$ is an irreducible character of $H$?

Let $H$ be a proper subgroup of $G$ such that for all $\chi\in Irr(G)$, $\chi_H\in Irr(H)$. That is, the restriction of every irreducible character of $G$ to $H$ is an irreducible character of $H$. ...
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Characters of a finite group

Recently, I have been studying about Character Theory of Finite Groups, mostly from "Groups and Representations" by J. Alperin & R. Bell. In the aforementioned textbook, the characters of a finite ...
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Bounds for multi-dimensional Kloosterman Sums

I'm looking for a general bound (in terms of $p$) for the Kloosterman sum, working in $\mathbb{F}_{p}$, $$\sum\limits_{x_{1} \dots \ x_{n} = a} \psi(x_{1} + \dots + x_{n})$$ for $\psi$ a nontrivial ...
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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'...
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Character formula for $S_n$ and $GL(V)$

In a set of lecture notes I'm reading, we consider representations of the symmetric group $S_n$ via treatment of Young tableaux, partitions of $n$ etc. (in what I believe is the standard approach) - ...
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On a $\mathbb C$-linear map from $M(p-1,\mathbb C)$ to $\mathbb C^\hat G$, where $p$ is an odd prime and $G=\mathbb Z/(p) ^\times$

Let $p$ be an odd prime and $G=(\mathbb Z/(p))^\times=\{1,2,...,p-1\}$ i.e. $G$ is a cyclic group of order $p-1$. Let $\hat G:=\{\chi:G \to \mathbb C^\times : \chi $ is a group homomorphism $\}$. For ...
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Irreducible Characters & Representations of a Cube

Let $A_4$ act on the four long diagonals (labeled $1,2, 3, 4$) inscribed in a cube (which is $S_4$). Then $A_4$ acts on the faces, the edges, and vertices of the cube. This gives rise to three ...
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stabilizer of an induced character

Suppose $K\trianglelefteq N\trianglelefteq G$ are finite groups (where $K$ isn't necessarily normal in $G$), and suppose $\chi\in\mathrm{Irr}(N)$ is induced from $\mu\in\mathrm{Irr}(K)$. Let $V=\...
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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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157 views

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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Relation between the characters of subgroups of a finite group

Let $ H $ and $ K $ be subgroups of a finite group $ G $. Let $ \chi_1(H) $ and $ \chi_1(K) $ denote the trivial characters of $ H $ and $ K $ over an algebraically closed field of characteristic $ 0 ...
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Character group of $\mathbb{Z}$

I am trying to compute the character group of $\mathbb{Z}$ which contains homomorphisms that map into $\mathbb{C}^\times$. I have determined that each homomorphism $\phi \in \hat{\mathbb{Z}}$ may be ...
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How much we know about the Group from its Complex character table?

Suppose $G$ is a finite group and suppose that complex character table of $G$ is given.It is well known that from character table we cannot determine the Group uniquely (For example $Q_8$ and $D_8$ ...
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a finite group of order $360$ never has an irreducible character of degree $15$

The following problem is an exercise in character theory: "Prove that a finite group of order $360$ never has an irreducible character of degree $15$." Could you help me for it?
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Question on irreducible character of regular representation of the symmetric group

Consider the symmetric group $S_n$ acting on $A=\{1,..,n\}$, for any nonnegative integer $k\leq n/2$, denote $A_k$ to be the collection of all $k$-element subsets of $A$. Let $\chi_k$ be the character ...
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Characters of the fundamental representations of $SU(3)$

Let us denote $3$ and $\bar{3}$ the fundamental representations of $SU(3)$. According to my lecture notes, the characters read as follows: $\chi_{[3]} = e^{\omega_1} + e^{\omega_1 - \alpha_1} + e^{\...
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$\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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Proof check that the product of the periods of a generating set of a finite abelian group $G$ is $|G|$.

A friend recently showed me some nice machinery from Representation Theory that allows you to show relatively simply that if $G$ is abelian, then it has $|G|$ distinct characters (homomorphisms into $\...
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Spectrum of the Cayley Graph

From wikipedia: "Every group character $\chi$ of the group $G$ induces an eigenvector of the adjacency matrix of $\mathcal{C}(G,S)$. The associated eigenvalue is $\sum_{s \in S} \chi (s)$." It seems ...
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is this paper wrong? — “irreducible characters of finite algebra groups” by andre

Hoping someone familiar with the subject will find this. In Thm 4.1 of this paper (p. 14), it says that the irreducible characters of a finite algebra group (like the unipotent uppertriangular matrix ...
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(No) difference between irreducible $\bar{\mathbb{Q}}$-valued characters and irreducible $\mathbb{C}$-valued characters of a finite group?

I would be most grateful if someone would check my logic below... Many thanks! Let $G$ be a finite group and let $\bar{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$ inside $\mathbb{C}.$ ...
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Role of projective representations in representation theory of semidirect products?

I am interested in the representations of a finite group $G=N\rtimes H$. There is an article by A. Reyes that might be helpful, but I can't find it for free anywhere. This is what I know so far. If $...
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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 ...
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A question about irreducible characters of defect one

Let $G$ be a finite group and $p$ a prime divisor of $|G|$. Also let $\mathrm{Irr}(G)$ and $\mathrm{IBr}(G)$ denote the set of irreducible complex characters and irreducible $p$-Brauer characters of $...
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Legendre symbols as multiplicative homomorphisms in number fields

$\newcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}$ Suppose we have a finite set of rational primes $B=\{p_1,\ldots,p_k\}$, and $V=\{ x\in\mathbb{Q}^*~|~x\text{ contains only primes in B} \}$. So ...
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What is number of irreducible characters in modular representation?

In ordinary representation, I know that the number of irreducible characters is the number of conjugacy classes, what about the modular representation? Can I find a good and simple book on modular ...