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2
votes
2answers
108 views

Are “FG-module characters” sometimes used, too?

I am only beginning my study of group representations and characters. So far I have already encountered the regular group algebra $FG$. Although in an FG-module the multiplication is only defined for ...
0
votes
1answer
33 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.
0
votes
1answer
46 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 ...
0
votes
1answer
70 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 ...
14
votes
0answers
313 views

Subgroups as isotropy subgroups and regular orbits on tuples

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In ...
10
votes
0answers
291 views

Character theory of $2$-Frobenius groups.

Edit Summary: I've posted this on MO and received a partial answer there. Can anybody help me expand on this? Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and ...
8
votes
0answers
181 views

Where does this elliptic curve come from?

In Zeta functions of an infinite family of K3 surfaces, Scott Alhgren, Ken Ono and David Penniston compute the zeta functions (given a good reduction restriction mentioned below) of the K3 surfaces ...
6
votes
0answers
118 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 ...
6
votes
0answers
76 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 ...
4
votes
0answers
66 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 ...
4
votes
0answers
397 views

Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.

I posted a question a short while ago on this but got no response. I have worked on this more and so now have a more specific question. To start with we work with the $\mathbb{Q}$ version of ...
3
votes
0answers
80 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 ...
3
votes
0answers
100 views

Character formula for $S_n$ and $GL(V)$

In a set of lecture notes I'm reading, we consider representations of the symmetric group $S_n$ via treatment of Young tableaux, partitions of $n$ etc. (in what I believe is the standard approach) - ...
2
votes
0answers
49 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 $g \in H$?
2
votes
0answers
45 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 ...
2
votes
0answers
67 views

Isomorphism of annihilator of a subgroup in the context of group characters

I am trying to learn about characters of finite abelian groups. A character is a homomorphism from a finite abelian group $G$ into the multiplicative group of complex numbers of absolute value 1. In ...
1
vote
0answers
49 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). ...
1
vote
0answers
125 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 ...
1
vote
0answers
27 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 ...
0
votes
0answers
33 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 ...
0
votes
0answers
10 views

Prove representation irreducible

I have the above 5-dimensional representation of a group of order 125 and need to show that it's irreducible. (epsilon is a primitive 5th root of unity). I figure that i have to calculate the ...
0
votes
0answers
37 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 ...
0
votes
0answers
18 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 ...
0
votes
0answers
11 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 ...
0
votes
0answers
10 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 ...
0
votes
0answers
20 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 ...
0
votes
0answers
30 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: ...
0
votes
0answers
31 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 ...
0
votes
0answers
23 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 ...
0
votes
0answers
22 views

Molien series from character tableof group?

I was reading up on Molien series: http://en.wikipedia.org/wiki/Molien_series and I heard that you can compute this series from the character table of the group (in for instance the Atlas of finite ...
0
votes
0answers
50 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 ...
0
votes
0answers
179 views

Martin Isaacs's exercise 3.4 (character theory of finite groups)

I need some help with this: Let $G$ be a simple group and suppose $\chi\in{Irr(G)}$ with $\chi(1)=p$, a prime. Show that a Sylow $p-$subgroup of G has order p. Thanks a lot in advance.
0
votes
0answers
129 views

How to use a character table to get the kernel

I have been given a character table and I need to find from the table the kernel of each character. I dont know how to do this. I just tried to enter the table here using latex, but its not working. ...