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40 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
34 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 ...
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1answer
42 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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1answer
19 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
11 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
56 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
48 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
23 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, ...
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1answer
79 views

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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0answers
41 views

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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0answers
20 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 ...
2
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2answers
64 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 ...
3
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1answer
22 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
41 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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15 views

A question about character of symplectic group

Let $V$ be a vector space with symplectic two form $\Omega$. Then the character $\chi:Sp(V,\Omega)\to U(1)$ must be trivial?
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0answers
41 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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2answers
89 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
36 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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1answer
53 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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0answers
51 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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0answers
22 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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0answers
19 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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2answers
78 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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1answer
157 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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0answers
12 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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1answer
60 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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1answer
131 views

$\chi(g)$ 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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1answer
39 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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1answer
106 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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1answer
73 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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0answers
23 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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1answer
60 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
74 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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2answers
60 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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2answers
92 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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1answer
68 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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1answer
43 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 ...
4
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1answer
68 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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1answer
57 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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0answers
57 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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0answers
130 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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1answer
37 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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1answer
108 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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2answers
159 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
88 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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1answer
32 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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0answers
36 views

Properties and powers of real characters of finite groups

Would anybody please be able to list some main properties of characters of real representations of finite groups? I know some pretty helpful basic ones for characters of complex representations, but I ...
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1answer
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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2answers
37 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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0answers
52 views

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 $g \in H$?