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

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What are the Pontryagin duals of additive and multiplicative group of complex number?

What are the Pontryagin duals of additive and multiplicative group of complex number? So basically what are all characters of $(\mathbb{C},+$) and $(\mathbb{C^*},.)$?
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Understanding the irreducible representations of $D_3$

By the dimensionality theorem, $$\sum_i d_i^2 = |G|,$$ where $d_i$ is the dimension of the $i$th irreducible representation, we can infer that the dihedral group $D_3$ has two one dimensional irreps ...
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22 views

Example of projective rep being used in Clifford theory

I'm trying to understand the use of projective representations in Clifford theory, and I'd like a small example where projective representations really help, and the ingredients are actually ...
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85 views

Nilpotent groups are monomial

I'm trying to show that a nilpotent group $G$ is monomial; i.e., that every irreducible representation $\rho$ of $G$ satisfies $\rho = \text{Ind}_H^G(\tau)$ for some $H \leq G$, $\tau$ a one ...
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31 views

A class function $f$ is a character if and only if $(f,\chi_{q_i})_G $ is a non-negative integer, for all irreducible characters $\chi_{q_i}$

I'm currently revising representation theory and I'm a bit stuck trying to prove the converse of the above statement. $(\Rightarrow)$ is straight forward because if $f$ is the character of a ...
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1answer
79 views

Sum of irreducible character values in a row of the character table

If $\chi$ is a nontrivial irreducible character of $G$ (a finite group), define $S_{\chi}:= \sum_{x \in G} \chi(x)$. In terms of conjugacy classes $\mathcal{C}$, this is $\sum_{\mathcal{C}} ...
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63 views

Isaacs exercise 10.1 (Character Theory of Finite groups)

I need help on this problem. (10.1) Let $H \le G$, $\theta \in \operatorname{Irr}(H)$ and $\chi \in \operatorname{Irr} (G)$. Suppose $F \subseteq \mathbb{C}$. (a) If $\chi_H = \theta$, show that ...
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208 views

Why does the tensor product of an irreducible representation with the sign representation yield another irreducible representation?

I was writing this question, and I came up with an answer, so I thought I would answer it myself: In considering representations of $S_n$, among others, we have the "sign representation", that is the ...
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1answer
41 views

Proof of convergence of $L'\left(1,\chi\right)$

can someone give me a good reference for a clear proof of the convergence of $L'\left(1,\chi\right)$, $\chi$ real-valued, non-principal Dirichlet character? Thanks in advance.
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60 views

Explanation for a theorem pertaining on Dirichlet character sums

A very well known theorem pertaining on Dirichlet characters sums states that if $\chi$ is a Dirichlet character modulo $k$, defining $$ A\left(n\right)=\sum_{d\mid n}\chi\left(d\right) $$ Then ...
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62 views

Character Theory of group of order 125.

I've managed to do the first 4 problems but are really stumped by the final once. Tried to read all weekend to no avail. Am going to hand in on thursday and could really need some hints. Anything is ...
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26 views

Looking explanation for exercise on Dirichlet Characters

I have some problems with the Dirichlet characters and the Apostol is not really helpful as a book, unfortunately. I am looking at the example at page 139, for those who have the book - for those who ...
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34 views

Properties of characters that remain true for infinite compact groups

Which properties of irreducible characters for finite groups still hold for infinite (compact) groups? In particular, is it still true that the irreducible characters form a basis for the space of ...
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101 views

Characters of a faithful irreducible Module for an element in the centre

Basically here is my questions. We have a character $\chi$ which is faithful and irreducible of a group $G$. we have an element $g$ which i needs to show belongs to the centre $Z(G)$, i.e. $gh=hg$ for ...
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285 views

How to show a representation is irreducible?

I have a professor who says that I should be able to show a representation is irreducible simply by looking at its trace (with other possible conditions), but after researching this for a while, I ...
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20 views

Fourier transform of a quasi character on a local field

Why is the twice iterated fourier transform of a quasi-character $c$ on local field $k$, $c$ itself? That is, why is $\hat{\hat c}(\alpha) = c(\alpha)$? In general, when we apply the fourier transform ...
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95 views

Reference Request: Characters of Finite General Linear Groups

I've been looking at J.A. Green's article The Characters of Finite General Linear Groups and it seems that Green in this article comes up with a way of calculating all irreducible characters of a ...
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$\chi(g)$ group character $\Rightarrow$ $\chi(g^m)$ group character

Let $G$ be a group of order $n$ and and $\gcd(m,n)=1$. Let $\chi:G\rightarrow\mathbb{C}$ be a class function and define $\chi^m\!: g\mapsto\chi(g^m)$. How can one show that $\chi^m$ is a character iff ...
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47 views

A question about Character Degrees of G/N

Let $G$ be a finite group such that $p_1p_2\mid |G|$, where $p_1$ and $p_2$ are two primes. We know that there exists an irreducible character $\chi\in Irr(G)$ such that $p_1p_2\mid \chi(1)$. We know ...
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154 views

Character Table of the Monster

Does anyone have a link to a website having the character table of the Griess Monster in characteristic $0$?
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91 views

$\Psi_g(A)=\Phi(g)A^t\Phi(g)$; express $\chi_\Psi$ through $\chi_\Phi$

Let $\Phi$ be a matrix n-dimensional representation of the group G. We construct a representation $\Psi$ of $G$ on the space of square matrices of order n, such that ...
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25 views

subgroup of character group

Let $B$ be a subgroup of a finite Abelian group $A$ and $$B^0=\{\alpha \in \hat{A}\mid \alpha(b)=1 \text{ as } b \in B\}$$ Prove a. $B^0$ is subgroup of $\hat{A}.$ And for any subgroup ...
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69 views

Suzuki exceptional characters with $\epsilon = -1$

I have a question about Suzuki's theory of exceptional characters of finite groups. If you are familiar with this theory, then I'm just asking: can we always choose $\epsilon=1$? If not, here is a ...
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1answer
103 views

show that the following three statements are equivalent

Let $\Phi$ be an irreducible complex representation of the group $S_n$ and $\Phi'(\sigma)=\Phi(\sigma) \operatorname{sgn}(\sigma).$ $(\sigma \in S_n).$ Prove that 1) $\Phi'$ is ...
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78 views

Prove that $\forall g \in G$ $ \exists$ an irreducible non-trivial character $\chi$ of the group $G$ such that $\chi(g)\neq 0$

Let $G$ be a non-trivial finite group. Prove that $\forall g \in G$ $ \exists$ an irreducible non-trivial character $\chi$ of the group $G$ such that $\chi(g)\neq 0$ This is my attempt so ...
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118 views

The average value of irreducible character of a non-trivial finite group

Let $G$ be a non-trivial finite group. Let $\chi$ be an irreducible character of the group $G$. Find $$\frac{1}{|G|}\sum_{g \in G} {\chi( g)}$$ I try. But I think that I am wrong. ...
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74 views

Irreducibility of complex 2-dimensional character of the group $ S_3 $

Let $\chi$ be 2-dimensional complex character of the group $ S_3 $. Prove that $\chi$ is irreducible character iff $\chi((123))=-1$. There is hint in my book: " Use the Maschke's theorem and ...
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52 views

if $g^k=e$ then $\chi(g)=\sum_j^n \zeta_k$

Let $G$ be a group. Let $g \in G$ and $g^k=e.$ Let $\chi$ be an $n$-dimensional character of the group $G.$ Let $\zeta_k$ be $k$-th root of unity. Prove that $\chi(g)$ is equal to sum of a ...
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96 views

Reading a Character Table in Magma

These are the outputs of two character tables from Magma. The first is for $A_5$. The second is for $GL_3(2)$. What is the significance of the '$+$' and '$0$' symbols? I can produce more tables if ...
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69 views

If $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$.

Let $G$ be finite abelian group and $\hat G$ be its character group. I need hint proving that if $a\in G$ and $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$ (the identity element). I can prove it ...
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65 views

Is there any Relationship between Eigenvectors and Characters?

I noticed a strong similarity in the proofs of the linear independence of characters (homomorphisms from a given group $G$ to units of $\mathbb C$) and of eigenvectors with different eigenvalues). ...
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134 views

Possible small mistakes in Springer's *Linear Algebraic Groups*

In the 2nd edition of Springer's Linear Algebraic Groups, the proof of 16.2.2(i) (p. 271) begins with the assertion that the reductive group G is quasi-split over F iff the restriction map of ...
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48 views

conjugacy class of a dicyclic group

I have given a group and I have to prepare the character table of this given group. I know that firstly I have to find the conjugacy classes of the given group. The group is below: ...
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225 views

finding normal subgroups of a dihedral group whose character table is given

my question is about character theory. for example i have a character table of the dihedral group D12 which has 6 irreducible characters.so the character table is a 6*6 matrix.i also know all ...
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219 views

p-adic numbers and group characters

The wiki article on p-adic numbers has this wonderfully charming and pretty graphic: This is supposed to represent "the 3-adic integers, with selected corresponding characters on their Pontryagin ...
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1answer
100 views

multiplicative character evaluated at -1 (from Ireland and Rosen's number theory book)

I'm self studying from Ireland and Rosen's "A Classical Introduction to Number Theory" Second edition. Near the beginning (page 153 in my book) of Chapter 11 the authors discuss the number of ...
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37 views

What do these characters mean in RDF/OWL Domain and Range logic?

Does someone know what these strange looking characters are? I would like to learn what they mean. Can you send me a reference/hyperlink so I can understand what they mean? ...
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40 views

If $G$ is solvable, is it true that for any $m,n\in\operatorname{cd}(G)$, there exists a prime $p$ such that $p\mid m,n$?

Let $G$ be a finite group and let $\operatorname{cd}(G)$ be the set of degrees of irreducible characters of $G$. It is known that if for any $m,n \in \operatorname{cd}(G) \setminus \{1\}$, there ...
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40 views

Fourier transform of a function over finite group

Let $G$ be finite abelian group and $\hat G$ be its character group. The Fourier transform of a function $f:G \to \mathbb C$, is the function $\hat{f}:\hat{G}\to \mathbb C$ defined by ...
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Characters of subgroups of finite abelian groups

Let $G$ be a finite abelian group. Let $H$ be a subgroup of $G$. Let $\hat{G}$ be the group of characters of $G$. Is there a character $\chi \in \hat{G}$ such that $\chi(g) = 1$ iff $g \in H$?
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252 views

Holonomy and Differential Characters

This question is going to be rather vague, but I'm just trying to see if there are obvious connections between these two concepts. So the holonomy of a vector bundle with Lie group $G$ is ...
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35 views

a exercise in Berkovich‘ book

in Berkovich' book characters of finite groups there is a exercise in page 59. exercise 15, a group G is a Q-group if and only if for any cyclic subgroup Z of G who can tell me how to prove it ? ...
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49 views

Is $\sum_{c \notin \mathbb Z_q}\psi(c)-\sum_{c \notin \mathbb Z_q}\psi(c^n)$ an integer?

Let $n$ be a positive integer. Let $p$ be an odd prime and $q=p^k$. Let $c \in \mathbb Z_q$. Consider the additive character $\psi:\mathbb Z_q \rightarrow \mathbb C^{\times}$ that is defined as ...
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character degree and solvability

There is an unsolved problem in Berkovich's book "Characters of Finite Groups Part 2" I state here: Is $G$ solvable if $\chi(1)^2$ divides $|G|$ for all $\chi \in \operatorname{Irr}{(G)}$? Can ...
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1answer
113 views

prove on induced and restricted representation

can anybody please help me with this question? I have huge trouble even just start it. Let H, K be subgroups of G and HK=G. If $\psi$ is a character of H, show that $(\psi^G)_K = (\psi_{H \cap ...
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1answer
48 views

Proof involving characters

I'm self studying from a Classical Introduction to Modern Number Theory by Ireleand and Rosen. In the exercises for the chapter on Gauss and Jacobi sums I came across this question. Let $\chi$ be a ...
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2answers
94 views

Induction of characters

This is a very basic question on induction of characters. Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{C}_G$ and $\mathcal{C}_H$ denote the spaces of class functions for $G$ and $H$ ...
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52 views

Simple components and the irreducible characters of the group ring $K[G]$

Let $K$ be an algebraically closed field of characteristic $0$ and $G$ a finite group. I know that the group ring $K[G]$ is a semisimple and so decomposes as a direct sum of $m$ simple components ...
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76 views

linear character of a finite group

I am reviewing my notes of algebra. It's not a long proposition so I tried to prove it by myself but failed. We have a finite group $G$ and a linear character $\chi$ of $G$. I need to show ...
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Computing bicharacters of (small) finite groups

I'm trying to find some finite groups with certain properites (hopefully of small order; no more than 100, I suspect), and one of the things I need to look at are all of its bilinear bicharacters: ...