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

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55 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
18 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 ...
4
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0answers
74 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 $H$. ...
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votes
1answer
31 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
37 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 $ ...
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0answers
67 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
77 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
59 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
33 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
20 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
42 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 ...
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1answer
26 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
37 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 ...
0
votes
2answers
91 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
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1answer
62 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 ...
1
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1answer
27 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
votes
1answer
45 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
vote
1answer
28 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, ...
0
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1answer
37 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
41 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
58 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
35 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
15 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
43 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 ...
1
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1answer
26 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
41 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
29 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 ...
0
votes
0answers
11 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
49 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
25 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 ...
0
votes
0answers
40 views

The irreducible representation of $S_n$ of degree $n-1$ [duplicate]

So I understand it's not hard to show that the standard $(n-1)$-dimensional irreducible representation of $S_n$ is the only irreducible representation of $S_n$ of degree $n-1$ using characters/Young ...
2
votes
1answer
68 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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vote
2answers
23 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
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0answers
17 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
62 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 ...
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0answers
34 views

Summation of Legendre symbol

Let $\chi_{2,q}$ be the real Dirichlet character modulo a prime $q>2$, which is not the principal one (the so-called Legendre symbol). Is it true that $$ \sum_{n=0}^{+\infty} ...
2
votes
2answers
33 views

Summation of non-principal real Dirichlet character

Let $q > 3$ be a prime and $$ S_q := \sum_{k=1}^{q-1} \chi_{2,q} (k) \, k, $$ where $\chi_{2,q}$ is the real Dirichlet character modulo $q$ which is not the principal one. I have to prove that ...
0
votes
1answer
52 views

Expressing a sum involving a nontrivial character as a Jacobi sum

Let $\chi$ be a multiplicative non trivial character of $F_p$ and $\rho$ be a character of order 2. Show that $\sum\chi(1-t^2)=J(\chi,\rho).$ [Hint: Evaluate $J(\chi,\rho)$ using the relation ...
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1answer
54 views

Sum of an irreducible character over all $G$

Let $\phi: G\to GL_n(\mathbb C)$ be an irreducible representation of a finite group $G$. Let $\chi: G\to \mathbb C$ be the character of $G$. Prove that: $$\displaystyle \sum_{g\in G}{\chi(g)}=0 $$ I ...
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votes
0answers
26 views

Is the Transpose of a Representation an Equivalent Representation?

Suppose we are working over $\mathbf{Z}[G]$ where $G$ is finite. Suppose further we have two representations $\rho$ and $\rho^\prime$ such that $\rho^\prime=(\rho)^T$. Can we say that these two ...
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votes
2answers
73 views

Consider $\mathbf{Z}G$, $G$ finite. If the characters of two $\mathbf{Z}G$-modules are equal, does it follow that the modules are isomorphic?

So I have recently started to delve into integral representation theory and I was wondering if a particularly useful theorem survives the transition to integral rep theory. Basically, suppose we have ...
5
votes
1answer
43 views

Characters of a Group: two definitions

If $G$ is an abelian group, the characters associated to the rapresentations of $G$ over $\textrm{GL}_1(\mathbb C)=\mathbb C^\ast$ are simply the group homomorphisms: $$\chi:G\longrightarrow\mathbb ...
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0answers
57 views

Bounds for the norm of certain additive character sums

Let $q$ be a power of a prime, let $\alpha$ be a primitive element in the finite field with $q$ elements and let $\chi$ be the canonical additive character of the field. Recently I became interested ...
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1answer
55 views

Character Theory Exercise

I am having trouble with the following exercise in character theory: If $\chi, \psi, \zeta$ are irreducible characters of a finite group then $\langle \chi\psi, \zeta \rangle \leq \zeta(1)$. I can ...
2
votes
1answer
71 views

Decomposing direct product of irreps

I know characters of two 2-dimentional irreps (U and V) of a group with 6 conjugate classes. The characters are: $\begin{pmatrix} 2&-1&-1&2&0&0\end{pmatrix}$ and $\begin{pmatrix} ...
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votes
1answer
25 views

Characters of a unipotent group.

Let $\def\C{\mathbb{C}}T = (\C^*)^n$. A character of $T$ is defined to be a homomorphism from $T$ to $\C^*$. The characters of $T$ is of the form $f(t_1,\ldots,t_n)=t_1^{a_1}\cdots t_n^{a_n}$ for some ...
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1answer
24 views

Discontinuous characters on locally compact abelian groups

I'm doing some reading out of Rudin's Fourier Analysis on Groups and Folland's Abstract Harmonic Analysis for research. Particularly I am on characters and dual groups and I've noticed that neither ...
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2answers
73 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 ...
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1answer
65 views

How to tell whether a representation of a group is faithful or unfaithful?

From just the character table and the basis functions of the irreducible representations, how do I know whether a representation is faithful or unfaithful? For the 1-D representation it is trivial to ...
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0answers
34 views

Show that 2 representations are not equivalent and find all the irreducible representations of G.

Show that 2 representations are not equivalent and find all the irreducible representations of $G$. The group $G=T_{16}$ has order 16 and presentation given by $G=\langle a,b : a^8=b^2=1, ...