For questions about characters (homomorphisms from a group into the multiplicative group of a field).

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3
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63 views

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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1answer
37 views

Groups and Characters exercise hint

I am trying to make my way through Grove's Groups and Characters but am having some trouble with the following seemingly benign exercise: If a group $G$ has a normal $p$-complement $K$ show that ...
0
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1answer
103 views

Proving that $\pi_{X \times Y} \simeq \pi_X \otimes \pi_Y$

If $X$ and $Y$ are $G$-sets and $X \times Y$ is a G-set by $g \cdot (x,y)=(g \cdot x , g \cdot y)$. \pi is the corresponding permutation representation. Prove that $\pi_{X \times Y} \simeq \pi_X ...
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2answers
60 views

Abelian groups cannot have characters of degree 2

I was attempting the following exercise: Assume that $G$ is a simple group. Let $\chi$ be an irreducible character of degree $2$, and $g \in G$ be an element of order $2$. Prove that $\chi (G) ...
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0answers
20 views

Perfect subset of the group of characters of an additive dense subgroup of $\mathbb(R)$

Suppose that $\Gamma$ is a dense subgroup of the group of real numbers $\mathbb{R}$ and let $G$ be the group of characters of $\Gamma$. The problem is to show that for any $a\in\Gamma$ the set ...
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1answer
64 views

Thm 5.19 (Brauer) in Isaacs Martin - Character Theory of Finite Groups

I would please like some help to understand the proof of thm 5.19 (Brauer) in Isaacs Martin - Character Theory of Finite Groups. It states: Let $\chi$ be a character of G with $[\chi,1_G]=0$. Let ...
4
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2answers
103 views

Character theory - Exercise 5.14

I am trying to solve the exercise 5.14 from the Isaac Martins Character Theory of Finite Groups. Let $G$ be a nonabelian group and let $ f=min\{\chi(1) | \chi \in Irr(G), \chi(1)>1 \}. $ Show ...
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1answer
31 views

Prove that there exists an $m$th root of unity $λ ∈ C$ such that for all $g ∈ G$, $\chi(zg) =λ \chi(g)$

Suppose $\chi$ is an irreducible character of $G$. Suppose $z ∈ Z(G)$ and that $z$ has order $m$. Prove that there exists an $m$th root of unity $λ ∈ C$ such that for all $g ∈ G$, $\chi(zg) =λ ...
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1answer
49 views

Finding irreducible subrepresenations of modular representation in GAP

Recently, I have been fiddling with modular representations in GAP. First from what I can tell, GAP does not have a good way built in to find things like Brauer characters of a given non-solvable ...
1
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1answer
64 views

Complete the character table of group of order $21$

You are given the incomplete character table of a group $G$ with order $21$ which has $5$ conjugacy classes, $C_1,\dots,C_5$, which have sizes $1,7,7,3,3$. $$ \begin{array}{|c|c|c|c|c|} \hline & ...
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2answers
63 views

Finding the character table of $Q_8$

Assume $G$ is a finite group. I am trying to construct the character table of $Q_8$, which is defined by $$Q_8=\{\pm 1,\pm i, \pm j,\pm k \}, \ i^2=j^2=k^2=-1, \ ij=k,jk=i,ki=j$$ By considering the ...
0
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1answer
51 views

How to find the character of $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$?

Let $\mathfrak g$ be a Kac-Moody algebra. Then $$ \mathfrak{n}_{-}=\oplus_{\alpha\in\varPhi_{+}}\mathfrak{g}_{-\alpha} $$ and for $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$ the ...
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1answer
29 views

Subgroups of group of characters of a finite abelian group

Let G be a finite abelian group, H a subgroup of the group of characters of G. Is it true that H is the group of characters of some quotient group of G? Thanks for any help.
2
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1answer
17 views

prove that $N(x^m=a)=N(x^d=a)$.

Let $p>2$ be a prime, $m\in \mathbb N$, $d=$ GCD$(m,p-1)$. Let $N(x^n=a)$ denote the number of solutions to the equation $x^n=a$ in $\mathbb F_p$. prove that $N(x^m=a)=N(x^d=a)$. I am familiar with ...
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0answers
24 views

Non-trivial characters of $SU(2)$

Are there non-trivial characters (or quasi-characters) of the special unitary group $SU(2)$? I couldn't find a straightforward answer by googling.
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0answers
105 views

Degrees of irreducible complex characters of alternating groups

What is sum of degrees of the irreducible complex characters of the alternating groups? The background of this question is to calculate the diminsion of a maximal torus of the associated Lie algebra ...
2
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0answers
71 views

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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1answer
29 views

Additive character : For any field or for a finite field?

We can define an additive character for any field, can't we? The reason why i'm asking this question is that when i google "additive character", all definitions i have seen are for a finite field. If ...
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0answers
84 views

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 $H$. ...
0
votes
1answer
33 views

Characters and a bound

Assume $\chi\neq\chi_{0}^q$ and $\chi$ is a character modulo $q$. At the lecture the following result was introduced: $\vert \sum_{n\leq x} \chi(n) \vert \leq \varphi(q) -1$ I'm not very happy about ...
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0answers
47 views

on the characters of the normal subgroup and its quotient

I read character theory recently and thought about the following proposition, but I do not know is this true or false: Let $G $ be a finite group such that two distinct primes $ p $ and $ q $ ...
2
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0answers
72 views

A group of order $2^67$

Let $G$ be a finite group of order $2^6\times 7$. Also $G$ has at least $5$ irreducible characters of degree $7$. I want to prove that with this condition there exists no character of even degree in ...
0
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1answer
131 views

Character Table Dihedral group of $D_6$

I'm having real troubles with finding the character table of the dihedral group $D_6$ of order 12: $D_6 = \langle a,b |a^6 = 1 , b^2 = 1, aba = b \rangle$. I've already found the conjugacy classes: ...
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1answer
71 views

Character theory question. [closed]

Let $\chi$ be a nontrivial irreducible character of finite group $G$, and $G$ has odd order. Then, $\chi$ isn't equal to $\bar \chi$.
2
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1answer
53 views

Probability of a sequence of characters within a random sequence of characters

I've been trying to work out a formula for the probability of a specified sequence of upper-case English characters of length $n$ appearing in a random sequence of upper-case English characters of ...
2
votes
1answer
42 views

How to find all Dirichlet characters

I want to know all the Dirichlet characters modulo $m$. I know that the number of such characters are $\phi(m)$. But how do find each and every character. for small moduli I could do it using some ...
3
votes
1answer
62 views

Bounds for general character sums over finite fields

Let $\mathbb{F}_q$ be a finite field with $q$ elements, let $\chi$ be the canonical additive character of $\mathbb{F}_q$, let $\psi$ be a non-trivial multiplicative character of $\mathbb{F}_q$, and ...
1
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1answer
30 views

Product involving Dirichlet characters $\prod_{i=1}^{\phi(m)}(1 - \frac{\chi_i(p)}{p^s}) =(1 - \frac{1}{p^{fs}})^{\frac{\phi(m)}{f}}$

While working with divisors in cyclotomic extensions of $\mathbb{Q},$ I came across this identity: Given a prime $p$ and an integer $m$ such that $(p,m) =1$, let $f$ be the smallest such the $p^f ...
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1answer
49 views

character of irreducible representations of odd-ordered groups

I want to prove that if $G$ is a group and the order of $G$ is odd, and $\chi$ is a real-valued irreducible character of $G$, then $\chi$ must be the trivial representation, $\chi = \epsilon$. So ...
3
votes
2answers
127 views

exponential function, lie group homomorphism

Let $f: \mathbb{R} \to \mathbb{C}^*$ be a continuous map satisfying for all $x, y \in \mathbb{R}$: $f(x + y) = f(x)f(y)$. $f(x) = 1$ for all $t = 2\pi n, n \in \mathbb{Z}$. Show that there exists ...
3
votes
1answer
89 views

An irreducible character of degree prime affords a faithful representation.

This is a long question, but hopefully someone can give me a suggestion, as I've been hitting my head against the wall... Take a non-abelian group $P$, of order $p^3$, $p$ prime. We've already ...
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1answer
35 views

Character of the algebra $\mathbb{C}[G]$ as $G \times G $-module

Let $G$ be a finite group. We can define an action of $G\times G$ on the group algebra $\mathbb{C}[G]$ in the following way: If $x \in \mathbb{C}[G]$ then $(g,h)\cdot x=gxh^{-1}$. Now, what about ...
4
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1answer
67 views

Decompose the following representation of $A_5$ in irreducible representations

Denote the space of functions on the set of faces of the icosahedron by $V_f$, this is a 20-dimensional representation of $A_5$ which acts here on as the group of rotational symmetries. Here follows ...
1
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1answer
40 views

Dimension of subspace stabilized by group and principle character

Let $\phi: G \rightarrow GL(V)$ be a representation with character $\chi$. Let $W$ be the subspace $\{v \in V : \phi(g).v = v$ for all $g \in G \}$ of $V$. Prove that $\dim W = \langle \chi, ...
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1answer
48 views

Characters of Finite Abelian Groups

I am studying this proof in my algebra notes, and I would like some help regarding the requirements of the proof. The statement of the proof is: For each finite abelian group G and each h in G with ...
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0answers
44 views

Character of a Representation

Suppose that we have a representation $V$ of a group $G= SU(2)$ . Is it true that if $ \chi_V \cdot c \neq 0$ for some non-zero constant c, then the trivial representation must be one of the ...
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1answer
68 views

Why is the character group defined as $\mathsf{Hom}(G,\mathbb T)$, i.e why is the codomain specifically $\mathbb T$?

In the paper Category Theory Applied to Pontryagin Duality by Roeder, the character group of an lca group is defined as the topological (under the compact-open topology) abelian group of continuous ...
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1answer
44 views

Character Table of $C_6$

$$\begin{array}{rrrrrrrrrrr} & 1 & g & g^2 & g^3 & g^4 & g^5\\ \phi_0 & 1 & 1 & 1 & 1 & 1 & 1 \\ \phi_1 & 1 & \zeta_6^1 & & & ...
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0answers
33 views

Orthogonality Relations for Character Groups

I'm trying to understand a part of a proof of orthogonality relations for character groups and finite abelian groups, and I don't quite get this part from the below link: ...
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0answers
53 views

Character as sum with regular representation

Suppose $G$ is a group and $\chi$ is a character of $G$ with $\chi(g_1)=\chi(g_2)$ for all non-identity $g_1,g_2 \in G$, and let $\chi_{reg}$ denote the regular representation character. I read that ...
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1answer
38 views

Existence of Irreducible Character s.t. $\chi(g) \neq 0, \chi(1) \neq 0 \text{ mod } |C(g)|$ for Elements in Conjugacy Class of Prime Order

Given a finite group $G$, and a non-identity representative $g$ in a conjugacy class of prime order $p$, I'm trying to show that some nontrivial irreducible character of $G$ must have $\chi(g) \neq 0$ ...
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0answers
48 views

Dirichlet L series estimation

let $\chi$ be a non-principal character modulo $q$, $M\geq 1$. I have to prove that, if $\vert \sigma - 1 \vert \leq \frac{1}{\log M}$, then $\vert \sum_{n=1}^M \chi (n)n^{-s}\vert\leq 1+e\log M$ and ...
2
votes
1answer
49 views

Dirichlet character modulo p

How can I prove that if $\chi$ is a non-principal character modulo $p$ prime, then $\chi (-1) = \overline{\chi} (-1)= \pm 1$ and $\sum_{x=1}^p \chi (x) e^{2\pi i x}=0$? For the first question, I just ...
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0answers
15 views

Can this matrix be said to consist of Dirichlet characters?

When: $j=1$ the following formula: $$T(n,k)=\prod\limits_{m=1}_{m \mid n}^{\text{n}} \left(\exp ^{-\mu \left(\frac{n}{m}\right)}\left(\Lambda \left(\frac{n}{m}\right)\right) \chi _{\exp ^{-\mu ...
3
votes
1answer
55 views

Representation is reducible

Suppose $V$ is a representation of a finite group $G$ over a field $k$ of characteristic $0$, and suppose dim$V=3$ and $\wedge ^2V$ is reducible. Then $V$ is reducible. I was trying to do it by ...
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votes
0answers
35 views

Conductor of Dirichlet character divides every quasiperiod

Let $\chi$ be a Dirichlet character which has quasiperiods $d_1, d_2$. I.e., if $(n(n + kd_i), q) = 1$ then $\chi(n + d_i) = \chi(n)$ for any $k \in \mathbb{Z}$. Supposedly we can then show that ...
2
votes
1answer
72 views

Largest ideal of a local field on which a character is trivial

Let $K$ be a nondiscrete locally compact field. Then fixing a character $\chi$ on $K$, any character on $K$ can be written as $t \mapsto \chi(xt)$ for some $x \in K$. For $E \leq K$ a closed ...
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2answers
33 views

What is the name, meaning and function of the circle(s) in this composition of functions?

This is the S-DES encryption algorithm. I don't recognize this character. Sidebar: How can I write this in LaTeX/MathJax?
2
votes
0answers
21 views

What is the basic concept behind extending a character?

I'm trying to do my homework (the overall problem is about $A$ a finite abelian group and showing $A \cong A^\vee \cong A^{\vee\vee}$, but you can ignore that), and I think I have a fundamental lack ...
3
votes
0answers
72 views

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 ...