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

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2
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1answer
28 views

Relating a Character sum to a Gauss sum

Let $q$ be a prime power. Consider the mapping $f:(\mathbb{F}_q)^{\times} \to (\mathbb{F}_q)^{\times},$ where $x\mapsto x^2$. I was interested in sums of the form $$\sum_{t\in \mbox{Im}(f)} ...
0
votes
0answers
28 views

Constructing character table of subgroup from character table of whole group

If $\psi : G \to \mathbb C$ is a character and $U \le G$, then $\psi_{|U} : U \to \mathbb C$ is a character, but I guess this might not be irreducible with respect to $U$. But is it possible to ...
0
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0answers
35 views

How to construct a character table? (E.g Klein 4 group)

Could someone explain to me how you make a character table? Say I wanted to give the character table for the Klein $4$ group, $K$. $K$ is isomorphic to the product $\mathbb{Z}/2\mathbb{Z} \times ...
1
vote
1answer
45 views

Conjugacy classes, irreducible representations, character table of $D_{10}$ (order 20)

$D_{10}=\langle r,s \rangle$ is the dihedral group of order 20 . I have been struggling a bit with this question, particularly c, regarding values in the character table (a). Find the conjugacy ...
0
votes
1answer
33 views

Why do we have these values of the generalized character when evaluated with the scalar product?

Let $U \le G$ be a subgroup of odd order of the finite group $G$. Suppose $t \notin U$ is an involution with $u^t \in uU'$ for all $u \in U$, where $U'$ denotes the commutator subgroup of $U$. Set $T ...
0
votes
1answer
35 views

Extending a linear character of $U$ to $TU$, where $T$ is generated by an involution normalising $U$

Let $U \le G$ be a subgroup of the finite group $G$ of odd order. And suppose $t \notin U$ is an involution normalising $U$, i.e. $U^t = U$ and $t^2 = 1$. Assume $t$ centralizes $U / U'$, i.e. $u^tU' ...
2
votes
1answer
36 views

Does same character table imply isomorphic abelianizations?

We know two finite groups with the same character table might not be isomorphic (e.g. $D_4$ and $Q_8$), but the sizes of their abelianizations are equal (in fact equal to the number of linear ...
1
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0answers
22 views

Characters of a Finite Abelian group

Let $C_m$ denote the cylic group of order $m$. Recall $\widehat{C}_m=\{ \psi_j: j\in C_m\},$ where $\psi_j$ is defined by \begin{align} \psi_j(k)=e\Big( \frac{2\pi \sqrt{-1}jk}{m}\Big). \end{align} ...
1
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0answers
34 views

All the characters on $S^1$

Would anyone please give me a hint to find all the characters on $S^1$? By characters I mean complex valued functions on $ S ^ 1 $ that their values have norm $1 $ and they satisfy: $f (x+y)=f (x) ...
0
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0answers
25 views

Inverse of character table

The character table of a group is always invertible, because the rows are orthogonal. Is there a general formula to compute the inverse of the character table?
0
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2answers
36 views

Degree one irreducible representations

In section 2.5 of his Linear representations of finite groups (I have the french copy), Serre gives an example of determination of the character table of a group $G$. The group $G$ is taken to be ...
1
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0answers
17 views

Relation between the sum of the values of a polynomial $f$ over a finite field, and the additive character sum with $f$ as the polynomial argument

Let $F$ be a finite field, let $f(T) \in F[T]$ and let $\psi$ be the canonical additive character of $F$. If $\sum_{x \in F}f(x) = 0$, what can we say of $\sum_{x \in F} \psi(f(x))$?
1
vote
1answer
46 views

Show that character vanishes on specific element $g$ if for $H \le G$ we have $[\chi_H, 1_H] = 0$ and all elements of $Hg$ are conjugate in $G$

Let $G$ be a finite group and $H \le G$ with $g \in G$ such that all elements of the coset $Hg$ are conjugate in $G$. Let $\chi$ be a $\mathbb C$-character of $G$ such that $[\chi_H, 1_H] = 0$. Show ...
2
votes
0answers
38 views

If $A$ is a semisimple algebra, and $M_1 \ncong M_2$ as irreducible $A$-modules, why we have that every ideal of $M_1(A)$ is an ideal of $A$

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
votes
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
50 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 ...
0
votes
2answers
52 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
votes
1answer
57 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? ...
1
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0answers
42 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 ...
1
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0answers
43 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 ...
1
vote
2answers
37 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
votes
1answer
87 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 ...
0
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0answers
46 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) ...
1
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0answers
37 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 ...
0
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0answers
17 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 ...
0
votes
0answers
20 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
votes
0answers
40 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
votes
1answer
20 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?
2
votes
1answer
40 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) ...
0
votes
0answers
37 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 ...
1
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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
votes
1answer
37 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
vote
1answer
45 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 ...
-1
votes
1answer
63 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 ...
1
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0answers
60 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 ...
1
vote
1answer
68 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 ...
1
vote
1answer
57 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
vote
1answer
50 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 ...
1
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1answer
31 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 ...
1
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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
votes
1answer
20 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 ...
0
votes
1answer
54 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
votes
1answer
49 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
votes
1answer
48 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
votes
2answers
90 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
votes
1answer
109 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 ...
1
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0answers
31 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 ...
6
votes
0answers
166 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
votes
1answer
59 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
votes
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 $4$. ...