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

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1answer
21 views

For a semisimple algebra and two $M$- and $W$-homogeneous parts for $M \ncong W$, why we have $M(A)W(A) = 0$.

Let $F$ be a field and $A$ be an $F$-vector space which is also a ring with $1$. Suppose for all $c \in F$ and $x,y \in A$ we have $$ (cx)y = c(xy) = x(cy) $$ Then $A$ is called an $F$-algebra. If ...
0
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0answers
56 views

Product of standard and sign representation of $S_5$

I am able to work out the sign representation of $S_5$ and standard representation of $S_5$. How do I compute the product of standard and sign representation of $S_5$? What kind of product do I need ...
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2answers
59 views

Standard representation of $S_5$

I am trying to determine the standard representation of $S_5$. I understand that it will be a map from group elements to $\mathbb{C}^4$. The character table is as follows. I understand that the ...
2
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1answer
60 views

Character table for $G:= \langle x,y \mid x^5=y^4=1\text{ and }yx=x^2y\rangle$

Let $G$ be the group of order $20$ defined in terms of generators and relations: $$G:= \langle x,y \mid x^5=y^4=1\text{ and }yx=x^2y\rangle.$$ Can anyone help me to derive the character table? ...
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0answers
57 views

Extending a homomorphism from a subgroup to whole group where the target is not a divisible group

I was reading this post of stack exchange. So in the question if the circle group is replaced by $\mu_{p-1}$ which is the group of $(p-1)^{th}$ root of unity and if the group $G/H$ is assumed to a ...
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0answers
46 views

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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2answers
46 views

Exactly two irreducible characters of dimension 1

I've been working through Artin's Algebra on my own time, and I'm stuck on one of the questions, namely 10.5.3: Suppose that a group G has exactly two irreducible characters of dimension 1, and ...
2
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1answer
107 views

why the column sums of character table are integers?

There is a well-known result of Solomon which states that sum of entries of any row in $\mathbb{C}$-character table of a group $G$ is an integer number. It is mentioned in Martin Isaacs Character ...
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0answers
50 views

Average of additive character sum over all polynomials of degree at most $d$ over $GF(q)$

Let $F$ be a finite field with $q$ elements, let $E$ be its degree $n \geq 2$ extension and let $\psi$ be the canonical additive character of $E$. For $d \leq n $ let $$ \mathcal{P}_{\leq d} = \{f(x) ...
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0answers
38 views

Faithful irreducible character of a group with exactly two minimal normal subgroups

Prove that any finite group with exactly two minimal normal subgroups has a faithful irreducible $\mathbb{C}$-character. What I have tried: Let $N_1$ and $N_2$ be two minimal normal subgroups of $G$....
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0answers
18 views

Is there any therem or application in character theory for Frattini subgroup?

By using Character Theory , there are many theorems and application for $G'$ and $Z(G)$. Is there any theorems or applications in character theory for $\Phi(G)$ which is the intersection of all ...
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0answers
21 views

Additive character sum over additive subgroups of finite fields, with special monomial arguments

Let $F$ be a finite field with $q = p^n$ elements, let $\psi$ be a non-trivial additive character of $F$, let $m$ be an integer coprime to $q-1$, and let $K$ be a large subspace of $F$, say $K$ is the ...
3
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0answers
45 views

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 ...
0
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1answer
21 views

A definition in Character theory?

I would like to know the meaning of the term Character Field used by B. Huppert in his book Endliche Gruppen 1. For example they have used the notation $K(\chi)$. I dont know what it stands for?
3
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1answer
48 views

A more general Kloosterman-type sum

Let $\mathbb{F}_q$ be a finite field and let $a,b \in \mathbb{F}_q$ not both zero. Let $\psi$ be the canonical additive character on $\mathbb{F}_q$. The classical Kloosterman sum is given by $$ K(a,b) ...
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0answers
49 views

Relation of induced character with inflation of characters

I'm trying to prove the following exercise: If $G$ is a group and $N$ is a normal subgroup of $G$, then $$1_N^G = \sum_{\chi \in \text{Irr}(G/N)} \chi(1) \text{ Inf}^G_{G/N}\ \chi,$$ where $1_N^G$ is ...
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0answers
51 views

Counter-examples in representations of associative algebra

Here are something well-known. $V \cong W \Rightarrow \chi_V=\chi_W$ holds for finite representations of arbitrary associative algebra. But $ \chi_V=\chi_W \Rightarrow V \cong W $ is true only for ...
2
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1answer
40 views

Short exact sequence of linear representations of a group

Let $G$ be a group with $V',V,V''$ being representations of $G$. Let $$0 {\longrightarrow}V' {\longrightarrow}V\overset{v}{\longrightarrow}V''\longrightarrow 0\tag{1} $$ be a short exact sequence of $...
1
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1answer
51 views

Definition of auxiliary function in proof of Maschke's theorem

Let $K$ be a field and $G$ be a finite group, and denote by $K[G]$ its group ring, then Maschke's theorem is: Suppose $\mbox{char}(K)$ does not divide $|G|$. Let $V$ be a $K[G]$-module. If $W \...
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1answer
66 views

For simple $\mathbb C[G]$-modules is the representation unique

Let $R$ be a ring, a $R$-module is called simple if it has no proper, nontrivial submodules. Let $G$ be a finite group, and denote by $\mathbb C[G]$ the free vector space over $G$, with the product ...
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0answers
61 views

Character representation of right regular representation as sum of irreducible characters

Let $G$ be a finite group acting on itself by the action $g \ast x := xg^{-1}$. Then this corresponds to an representation $\rho : G \to GL(L^2(G))$, where $L^2(G)$ denotes the space of function on $G$...
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1answer
80 views

Proof that the induced class function $\theta^G$ is a character if $\theta$ is a representation on subgroup

In these lecture notes by Daniel Bump on Induced Characters I have a question on the proof of Theorem 2.5.1. If $H$ is a subgroup of the finite group $G$ and $(\pi, V)$ a representation of $H$, i.e. a ...
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1answer
78 views

In what sense are the linear characters among the irreducible characters

Let $G$ be a finite group and denote by $\mathbb C^{\times}$ the multiplicative group of the complex numbers. A linear character is a homomorphism $\chi : G \to \mathbb C^{\times}$. A presentation of $...
1
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1answer
51 views

Proving the Fourier inversion formula in finite abelian groups

The following exercise is exercise 2.2.3 from these lecture notes by Daniel Bump. If $G$ is a finite abelian group, then denote by $G^{\ast}$ the set of its characters, further if $x \in G$ let $\...
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1answer
33 views

Fourier inversion formula on finite abelian groups

The following exercise is exercise 2.2.3 from these lecture notes by Daniel Bump. Let $\mathcal F : L^2(G) \to L^2(G^{\ast})$ be the Fourier transform, defined by $\mathcal{F}f = \hat f$, where $\hat{...
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0answers
46 views

Relation between finite abelian group and its set of linear characters

Let $G$ be a finite abelian group, and denote by $G^{\times}$ its set of linear characters, i.e. homomorphisms $\phi : G \to \mathbb C^{\times}$, where $\mathbb C^{\times}$ denotes the multplicative ...
0
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1answer
24 views

Norm of the projection onto a maximal ideal

Let $A$ be a complex Banach algebra and $\chi \ne 0$ be a complex character. Consider the quotient space $\hat A = \dfrac A {\ker \chi} \simeq \Bbb C$. If $\hat x \in \hat A$, how can one quickly ...
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1answer
65 views

One dimensional representations of the plane orthogonal group $O(2)$.

I recently thought about representations of the orthogonal group $O(2)$ and found the one dimensional representations a bit confusing. We have that $SO(2)$ is a normal subgroup of $O(2)$. In fact, we ...
3
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1answer
50 views

An Equality Involving Character Sums in Characteristic 2

For context: Let $E = \mathbb{F}_{2^n}$, and let $\alpha \in E$. Define $\chi(a) := (-1)^{Tr(a)}$, where $Tr$ is the absolute trace from $E$ to $\mathbb{F}_2$. For the purposes of this question, we ...
0
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1answer
53 views

Character Table for $D_4$ and what I can get from linear characters

I know there is a lot that a character table can tell me but while I am figuring out a lot there are a few points that I just can not seem to figure out and any help and/or general reasoning is ...
2
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2answers
109 views

Irreducibility of a character if and only if inner product equals 1

I am working through the following representation theory notes: Notes I am having some trouble understanding why the Corollary to Theorem 15 is true. Theorem 15 Suppose $G$ is a finite group with ...
3
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1answer
116 views

Characteristics of a Character table and what it tells me.

I am trying to solve the character table and some related questions. The questions are below, and what I have done is below that. Any help on any pieces I am sure will enlightening. For parts c and ...
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0answers
35 views

Kronecker symbol vs. Koblitz symbol

In Koblitz, Introduction to Elliptic Curves and Modular Forms on page 188 it is defined $$\left( \frac{-1}{j}\right)=0$$ in case $j$ is even. Apart from that definition $\left( \frac{c}{d}\right)$ is ...
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0answers
173 views

$G$ be a finite group and $G'$ be its commutator subgroup and $\widehat G$ be the character group of $G$ ; then $G/G' \cong \widehat G$?

Let $G$ be a finite group and $G'$ be its commutator subgroup and $\widehat G$ be the character group of $G$ ; then is it true that $G/G' \cong \widehat G$ ? I am completely stuck , please help . ...
2
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1answer
68 views

Why does the Kronecker symbol $\left(\frac{a}{\cdot}\right)$ define a character?

Let $a\not\equiv 3\pmod 4$ and $a\ne 0$. How do you show that $\chi(n):=\left(\frac{a}{n}\right)$ (where $\left(\frac{\cdot}{\cdot}\right)$ denotes the Kronecker symbol) defines a character of modulus ...
2
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0answers
22 views

Characters appearing naturally in arithmetic functions

Let $r_2(n)$ denote the number of representations of $n$ as a sum of 2 squares. It is well-known that $$r_2(n) = 4\sum_{d \mid n} \chi(d),$$ where $\chi$ is the non-principal character modulo $4$. ...
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0answers
57 views

Gauss sum of a multiplication of two multiplicative characters of a finite field

Let $F$ be a finite field with $q$ elements and characteristic $p$. Let $E$ be a proper extension over $F$ of degree $n$. Let $\psi$ be the canonical additive character of $E$ defined by $\psi(x) = e^{...
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2answers
123 views

additive character of a finite field, trace map to middle field

The additive character of a finite field $\mathbb F_q$ is obtained by using the trace function to base field $F_p$. Can we write some of them into a middle field. Using trace function from $\mathbb ...
2
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1answer
63 views

Character sum of a type of “almost linear” surjective mappings over finite fields

Let $F$ be a finite field of characteristic a prime $p$ with $q$ elements and let $E/F$ be a finite extension of degree $n > 1$ over $F$. Let $\chi$ be the additive canonical character on $F$ ...
5
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1answer
104 views

Does anyone know the Burnside Matrices?

For $G$ a fine group with conjugacy classes $C_1,\dots C_k$ we introduced the Burnside Matrices $A_r$ where $1<r<k$ with entries: $$A_r := \Big(\sqrt{\frac{|C_t|}{|C_s|}}a_{rst}\Big)_{1\leq s,t\...
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0answers
60 views

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} ...
5
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4answers
129 views

Representation of an abelian group

Without using the structure theorem, how do I prove b? I struggle with the proof of injectivity. Any tips? Problem: Let $G$ be a finite Abelian group. (a) Prove that the group homomorphisms $\chi :...
2
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1answer
64 views

An identity involving Gauss sums and convolution

For a Dirichlet character $\chi$ modulo $N$, the Gauss sum attached to $\chi$ is given by $$G_\chi(m) = \sum_{k \in \mathbb{Z}_N} \chi(k) e^{2\pi i mk/N}.$$ Suppose one has an $N$-periodic function $...
2
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1answer
80 views

Solving the Character table for $A_4$ and derived algebra characteristics.

Say I am given a group with 4 conjugacy classes $C_1, C_2, C_3, C_4$ with orders 1,3,4,4 respectively. I will call the conjugacy class characters $\chi_1,\chi_2,\chi_3,\chi_4$ respectively. I am ...
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2answers
54 views

Find character table for symmetric group $S_3$

This group contains all permutations of 3 elements, so it has order 3!=6. Its three congruency classes are {1}, {(1,2),(1,3),(2,3)}, {(123),(132)}. As we know that the number of congruency classes ...
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1answer
40 views

Showing that compact metrizable abelian group has enough characters

I wanted to know if anybody has some clue as why a compact metrizable abelian group has enough characters to sepatare points. (a character on a topological group is a continuous homomorphism from the ...
0
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1answer
38 views

Irreducible characters of finite abelian groups

Let $G$ be finite abelian group and $K$ a field such that $char(K)$ does not divide the order $r$ of $G$. For each divisor $d$ of $r$ let $\omega_d$ be a primitive $d$-root of unity and $a_d:=\frac{\...
4
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1answer
262 views

Order of group elements from a character table

Most questions that I can find on here (or anywhere else on the internet) deal with constructing a character table given a description of the group. I'm trying to answer a question which goes the ...
2
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1answer
124 views

Using character table to find normal subgroups

I am working through a question on an old character theory exam. I've answered the first two parts ok, but am now struggling on the third part. Here is the part that I can't do: I've computed ...
3
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1answer
102 views

Inducing representation for groups of order $p^3$

For groups $G$ such that $|G|=p^3$ one can show that $(1)$ $Z=Z(G)\cong C_p$ $(2)$ $G'=Z$ $(3)$ $G/Z \cong C_p \times C_p$ Take any $x \in G/Z$. Then $N=\langle x,Z \rangle$ is an abelian normal ...