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5
votes
2answers
178 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 ...
4
votes
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 ...
0
votes
1answer
35 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? ...
0
votes
0answers
38 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 ...
1
vote
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 ...
0
votes
2answers
38 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 ...
2
votes
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 $g \in H$?
0
votes
0answers
25 views

Characters and ''twisted dimensions''

Can someone shed some light at this part of wiki article about character theory: http://en.wikipedia.org/wiki/Character_theory#.22Twisted.22_dimension ? It kind of just stands there without any ...
5
votes
1answer
238 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 ...
0
votes
0answers
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 ? ...
0
votes
1answer
48 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 ...
7
votes
1answer
139 views

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 ...
2
votes
1answer
83 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 ...
2
votes
1answer
44 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 ...
2
votes
2answers
82 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 $H$ ...
2
votes
0answers
47 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 ...
1
vote
2answers
70 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 ...
4
votes
2answers
112 views

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: ...
1
vote
1answer
113 views

percentage increase in performance

What is : by how much to how much efficiency of algorithm is increased , if Initially it was executing in 20 seconds, after improvement (Final Time) it is executing only in 5 seconds. My Try: ...
1
vote
1answer
32 views

Number fields containing the values of a character

This is a basic question- Let $K$ be an algebraic number field, $\Gamma$ a finite group, and $R(\Gamma)$ the ring of virtual characters of $\Gamma$ with values in the algebraic closure $\mathbb{Q}^c$ ...
2
votes
3answers
172 views

Constructing the character table of a group

I am aware that, given a group, there is no simple general procedure to construct the character table of the group (over complex numbers). However, for specific groups, we could use helpful additional ...
7
votes
1answer
240 views

Computing Brauer characters of a finite group

I am studying character theory from the book "Character Theory of Finite Groups" by Martin Isaac. (I am not too familiar with valuations and algebraic number theory.) In the last chapter on modular ...
1
vote
0answers
28 views

Values of virtual characters

Let $G$ be a finite group. Let $K$ be a number field and $K^c\subset\mathbb{C}$ its algebraic (separable) closure. Denote by $R_G$ the additive group of functions generated by the characters of ...
1
vote
2answers
154 views

Show that ${\theta}^G \in Irr(G)$ iff $I_G(\theta) = N$, where $N \unlhd G$ and $ \theta \in Irr(N)$.

Let $N \unlhd G$ and $ \theta \in Irr(N)$. Show that ${\theta}^G \in Irr(G)$ iff $I_G(\theta) = N$. Where $I_G(\theta)$ is the stabilizer of $\theta$ in the action of $G$ on $Irr(N)$ defined by ...
1
vote
1answer
77 views

$\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$ for faithful characters $\phi \in Irr(H)$ and $\theta \in Irr(K)$ .

Let $G = H \times K$. Let $\phi \in \operatorname{Irr}(H)$ and $\theta \in \operatorname{Irr}(K)$ be faithful. Show that $\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$. Problem 4.3 of ...
4
votes
3answers
95 views

Fourier analysis on finite abelian groups

Can someone help me show if $f$ is a character of a finite abelian group then for all $a\in G$, $$\sum_{[f]}f(a)\stackrel{}{=} \begin{cases} |G| & \text{if $ a$ is the identity} \\ 0 & ...
3
votes
0answers
85 views

On Applications of the Murnagham Nakayama rule

The question is located below. In short, I am looking for an accessible explanation of the Murnagham Nakayama rule in relation to the following problem. Pardon the long setup. Let $Y$ be a standard ...
4
votes
0answers
75 views

Bounds for multi-dimensional Kloosterman Sums

I'm looking for a general bound (in terms of $p$) for the Kloosterman sum, working in $\mathbb{F}_{p}$, $$\sum\limits_{x_{1} \dots \ x_{n} = a} \psi(x_{1} + \dots + x_{n})$$ for $\psi$ a nontrivial ...
2
votes
2answers
86 views

If $\chi\in\operatorname{Irr}(G)$, $N\unlhd G$, and $\langle\chi_{N},1_{N}\rangle\ne 0$, then $N\subset \operatorname{Ker}(\chi)$.

Let $N \unlhd G$ and $\chi \in \operatorname{Irr}(G)$. Suppose that $\langle\chi_{N},1_{N}\rangle\ne 0$. Show that $N\subset \operatorname{Ker}(\chi)$. Hint: Use that, for any character ...
1
vote
1answer
121 views

Irreducible characters form orthonormal basis of set of class functions

I am reading Serre's book (Linear Representations of Finite Groups). Theorem 6 in chapter 2 says that the irreducible characters $\chi_1,\dotsc,\chi_h$ of a finite group $G$ form an orthonormal basis ...
3
votes
2answers
34 views

Representations of $SO_3(\mathbb{R})$ from $SU_2(\mathbb{C})$

Define $V_n$ as the linear space of all homogeneous polynomials of degree $n$ in two variables $x$ and $y$. Define also the representation $\rho_n$ of $SL_2(\Bbb{C})$ on $V_n$ by: ...
1
vote
1answer
29 views

$SU_2(\mathbb{C})$ and the characters

i can prove that the irreducible characters $\chi_n$ of $SU_2(\mathbb{C})$ are equal to: $$\chi_n(e^{i\phi})=\frac{\sin((n+1)\phi)}{\sin(\phi)}$$ If i want to give the dimension of the representation ...
0
votes
1answer
81 views

Standard representation of $O_h$ in $\mathbb{R}^3$

I want to give the standard representation of the complete octaedergroup $O_h$ in $\mathbb{R}^3$. To which group is $O_h$ isomorphic, and how to obtain a standard representation of the group? What ...
0
votes
0answers
52 views

Formula for checking the probability of a character appearing multiple times consecutively in an encrypted string

I am a young CS student with a specific interest in Cryptography, but I am relatively new to the field. Yesterday I came across a question I could not answer by myself, so I thought I'd ask some more ...
3
votes
1answer
123 views

Is the quadratic character, unique multiplicative character over $\mathbb Z_{p^n}$, for odd $p$?

Let $p$ be odd and $\mathbb Z_{p^n}$ denote the ring of integers modulo $p^n$. Let the quadratic character, $\eta$, be the function defined on $\mathbb Z_{p^n}^*$ (multiplicative group of $\mathbb ...
0
votes
1answer
104 views

Martin Isaacs's exercise 3.7 (character theory of finite groups)

I would need some help with this exercise: Let $\chi\in{Irr(G)}$ be faithful, and suppose $\chi(1)=p^a$ for some prime p. Let $P\in{Syl_{p}(G)}$, and suppose that $C_{G}(P)\nsubseteq{P}$. Show that ...
1
vote
1answer
166 views

Martin Isaacs's exercise 3.6 (character theory of finite groups)

I'm trying to solve this exercise, can anyone help me? Let $G$ be a p-group, and suppose $\chi\in{Irr(G)}$. Show that $\chi(1)^2$ divides $|G:Z(\chi)|$ Thanks a lot.
1
vote
1answer
129 views

Martin Isaacs's exercise 3.5 (character theory of finite groups)

I need some help with this exercise: Suppose $A\subseteq{G}$ is abelian, and $|G:A|$ is a prime power. Show that $G'\lt{G}$ Thank you very much in advance.
11
votes
5answers
297 views

Applications of Character Theory

Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, , Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character ...
4
votes
3answers
332 views

Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$, where $G$ is abelian and $\mathbb{T}$ is the circle group.

Suppose $G$ is an abelian group and $a\in G$ and $$f:\left<a \right>\to\Bbb T$$ is a homomorphism. Can $f$ be extended to a homomorphism on $G$: $$g:G\to \Bbb T$$ ? $\Bbb T$ is the circle ...
5
votes
1answer
81 views

characters of a $C^*$-algebra

I have read that a state $\rho$ on a unital $C^*$-algebra $A$ is a character (i.e. multiplicative) if and only if, for all unitary $u\in A$, $|\rho(u)| = 1$. Is there an easy proof, or can someone ...
3
votes
1answer
110 views

Some questions about representation theory in the modular case

I'm working on a paper which uses representation theory in order to compute some characters and deduce arithmetical statements about certain field extensions. Let $\Delta$ be a group of order prime ...
1
vote
1answer
55 views

Iwaniec Kowalski Notation

On page 532 of the book analytic number theory by Iwaniec and Kowalski, the following notation is used: $C^{~\infty}$ and $\tau(n,\chi)$. Could anyone tell me what these represent? (the former is ...
3
votes
1answer
179 views

Exercise 2.15 M.Isaacs' Character theory of finite groups

I'm beggining to study character theory, and i'm doing some problems from Isaacs' Character theory book. I would need some help with this one: (2.15): Let $\chi\in \operatorname{Irr}(G)$ be ...
1
vote
1answer
206 views

Exercise 2.8 M.Isaacs' Character theory of finite groups

I'm a starter at character theory. I'm trying to do this exercise: (2.8) Let $\chi$ be a faithful character of a group $G$. Show that $H\subseteq G $ is abelian if and only if every irreducible ...
5
votes
1answer
199 views

Character theory exercises [closed]

I'm doing the exercises from chapter 2 of M.Isaacs' Character theory of finite groups, and I'm having problems with some of them. In particular, I would need help with these ones. Thank you very much ...
6
votes
0answers
84 views

Characters of the symmetric group corresponding to partitions into two parts

Let $n\in\mathbb N$ be a natural number and $\lambda=(a,b)\vdash n$ a partition of $n$ into two parts, i.e. $a\ge b$ and $a+b=n$. In this special case, is there a simple description of the character ...
3
votes
1answer
303 views

Character of a permutation representation

I am self-studying representation theory, and I would like to make sure my proofs are complete. Following Serre's notation, let $X$ be a finite set, and let $G$ be a group that acts on $X$. Let ...
2
votes
1answer
81 views

Vantage point of character theory

I am not sure whether I can frame my question properly, or whether at this point my understandings permit me to comprehend the perspectives of the answers to come, but somehow I find it pretty amazing ...
15
votes
2answers
480 views

Formula for number of solutions to $x^4+y^4=1$, from Ireland and Rosen #8.18.

There is a sequence of three exercise in Ireland and Rosen's Introduction to Modern Number Theory, Chapter 8, page 106. I can do the first two, but can't finish the third. I can include the proofs to ...