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

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53 views

How to prove that a certain set is a coset of a subgroup of a group of characters mod $m$?

I have the following question: Let $m>1$ be a positive integer and $G:=G(m):={(\mathbb{Z}/m\mathbb{Z})}^{*}$. Let $p\in\mathbb{P}$ with the property $p\nmid m$. Let $\widehat{G}:=\text{Char}(G)$ ...
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93 views

Question concerning the Dirichlet density of a subset of the set of primes

I have the following question: I am reading Serre's book "A Course in Arithmetic" (see http://www.math.purdue.edu/~lipman/MA598/Serre-Course%20in%20Arithmetic.pdf). On page 75, it is stated that the ...
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1answer
32 views

Show that $ (φ^G )_K = (φ_{H∩K})^K $ with Mackey's theorem

Suppose H,K ≤ G e θ $ ϵ $ Char(H). Show that Z(θ)≤H. Suppose H,K ≤ G and HK = G. Se $ φ $ ϵ Char(H) show that $ (φ^G )_K = (φ_{H∩K})^K $. For the proof I have to use the Mackey's theorem. How do I ...
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40 views

Characters of Linear Algebraic Groups

Reading about the semi-invariants of quivers, I see a fact which is frequently referred to in the literature, and is assumed to be trivial. However, I don't see that very easily. So, I was wondering ...
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50 views

Product of chracter

From Isaac's character theory book; $3.12$ Let $x\in Irr(G)$ and $g,h\in G$. Show that $$\chi(g)\chi(h)=\dfrac{\chi(1)}{|G|}\sum_{z\in G}\chi(gh^z)$$ I had thought that it was related to $3.9)$; ...
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1answer
50 views

Sum of characters modulo k

I want to find $\sum_{n \leq x} \chi(n) $, where $\chi$ is a non-principal character modulo $k$. I am trying to find $\sum_{n \leq x} \chi (n) n$ using Abel's summation formula, where the series $a_n ...
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1answer
43 views

Isomorphism between an group and its double dual

I wanted to prove that for an abelian group $G$ , $\phi : G \rightarrow \hat{\hat{G}}$ is an isomorphism where $\hat{G}$ is a set of all irreducible characters of $G$ for $x \in G$, ...
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65 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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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 ...
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1answer
105 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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63 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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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
67 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 ...
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2answers
105 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
55 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 ...
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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
66 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 ...
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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
30 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.
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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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29 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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107 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 ...
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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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35 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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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 $H$. ...
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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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48 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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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 ...
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1answer
139 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
72 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 $\bar \chi$.
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1answer
57 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 ...
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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 ...
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1answer
67 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
31 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 ...
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2answers
128 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 ...
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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
36 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 ...
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1answer
70 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 ...
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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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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
70 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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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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54 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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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 ...
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1answer
51 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 ...