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

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18
votes
0answers
491 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 ...
17
votes
2answers
1k views

exercise in Isaacs's book on Character Theory

I'm stuck on an exercise in Isaacs's book "Character Theory of Finite Groups" - it relates to something I'm looking at as part of ongoing research, but I guess it belongs here rather than on ...
15
votes
2answers
610 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 ...
12
votes
1answer
250 views

Do all algebraic integers in some $\mathbb{Z}[\zeta_n]$ occur among the character tables of finite groups?

The values of irreducible characters of a finite groups are always sums of roots of unity; do all sums of roots of unity (i.e. algebraic integers in the maximal abelian extension of $\mathbb{Q}$) ...
11
votes
5answers
356 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 ...
11
votes
0answers
432 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 ...
10
votes
2answers
338 views

Functional equation of irreducible characters

I am preparing to an exam in representations of finite groups. I am trying to tackle a problem regarding a characterization of irreducible characters: Let $f$ be a complex-valued function on a finite ...
9
votes
1answer
170 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 ...
9
votes
1answer
306 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 ...
8
votes
2answers
380 views

What is an irrreducible character of a finite group?

Let $S_n$ be the group of permutations of $\{1, 2, \ldots, n\}$. A “character” for $S_n$ is a function $\chi\colon S_n \to \mathbb{C} \setminus \{0\}$ with $\chi(ab) = \chi(a)\chi(b)$ for all $a, b ...
8
votes
3answers
787 views

Character Table From Presentation

I've recently learned about character tables, and some of the tricks for computing them for finite groups (quals...) but I've been having problems actually doing it. Thus, my question is (A) how to ...
8
votes
1answer
252 views

An exercise involving characters

Suppose $p$ is a prime, $\chi$ and $\lambda$ are characters on $\mathbb{F}_p$. How can I show that $\sum_{t\in\mathbb{F}_p}\chi(1-t^m)=\sum_{\lambda}J(\chi,\lambda)$ where $\lambda$ varies over all ...
8
votes
1answer
119 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 ...
7
votes
2answers
294 views

Estimates on conjugacy classes of a finite group.

In Character Theory Of Finite Groups by I Martin Issacs as exercise 2.18, on page 32. Theorem: Let $A$ be a normal subgroup of $G$ such that $A$ is the centralizer of every non-trivial element ...
7
votes
1answer
421 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 ...
7
votes
1answer
155 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 ...
7
votes
1answer
270 views

How does Pontryagin duality fit into the general cohomology theory framework?

Pontryagin duality implies the isomorphic relation of the function space $C(G)$ on a locally compact group $G$ to the function space on it's dual group $\hat G \overset{\sim}{=}\text{Hom}(G,T)$, ...
6
votes
2answers
400 views

Two non-isomorphic groups with the same complex character table

Could you give me an example of two non-isomorphic groups with the same complex character table?
6
votes
2answers
315 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 ...
6
votes
1answer
487 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 ...
6
votes
1answer
429 views

Question about Weyl character formula

In the book of Humphreys, page 139, Weyl character formula is $$\left(\sum_{w\in W} \operatorname{sn}(w)\epsilon_{w\delta}\right) * \operatorname{ch}_{\lambda} = \sum_{w\in W} \operatorname{sn}(w) ...
6
votes
1answer
231 views

Some irreducible characters of the Symmetric group $S_n$

I want to have characters of some irreducible $S_n$-modules corresponding to certain partitions $\lambda$ of $n$, the computations using Frobenius formula get complicated and I am unable to find in ...
6
votes
1answer
677 views

What is the relationship between Mackey's theorem in character theory and Mackey's theorem in transfer theory?

Here are the statements of the two theorems. The first statement I took from a paper I have been reading, but I believe can also be found in Isaacs' Character Theory of Finite Groups as an exercise. ...
6
votes
1answer
78 views

Does anyone know the Burnside Matrices?

For $G$ a fine group with conjugacy classes $C_1,\dots C_k$ we introduced the Burnside Matrices $A_r$ where $1<r<k$ with entries: $$A_r := \Big(\sqrt{\frac{|C_t|}{|C_s|}}a_{rst}\Big)_{1\leq ...
5
votes
2answers
140 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 ...
5
votes
1answer
116 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 ...
5
votes
1answer
249 views

Generalizing Artin's theorem on independence of characters

Artin's theorem says that for any field $K$ and any (semi) group $G$, the set of homomorphisms from $G$ into the multiplicative group $K^*$ is linearly independent over $K$. Can this theorem be ...
5
votes
1answer
175 views

Formula for evaluation of character on a transposition

Let $\lambda\vdash n$ be a partition of $n\in\mathbb N$ and $\chi=\chi_\lambda$ the corresponding irreducible character of the symmetric group $S_n$. Denote by $\lambda^t$ be the transpose of ...
5
votes
1answer
577 views

Sum of squares of dimensions of irreducible characters.

For anyone familiar with Artin's Algebra book, I just worked through the proof of the following theorem, which can be seen here: (5.9) Theorem Let $G$ be a group of order $N$, let ...
5
votes
2answers
88 views

Character of $\text{Hom}_{\mathbb{C}}(\mathbb{C}G,\mathbb{C})$

I've been given the following two questions, and for both I'm really unsure what to do (I'm a beginner with the theory of characters). Let $G$ be a finite group, $\mathbb{C}G$ its group algebra over ...
5
votes
4answers
114 views

Representation of an abelian group

Without using the structure theorem, how do I prove b? I struggle with the proof of injectivity. Any tips? Problem: Let $G$ be a finite Abelian group. (a) Prove that the group homomorphisms $\chi ...
5
votes
1answer
44 views

Characters of a Group: two definitions

If $G$ is an abelian group, the characters associated to the rapresentations of $G$ over $\textrm{GL}_1(\mathbb C)=\mathbb C^\ast$ are simply the group homomorphisms: $$\chi:G\longrightarrow\mathbb ...
5
votes
1answer
286 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 ...
5
votes
1answer
273 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 ...
4
votes
2answers
349 views

How to generalise $(\wedge^2 \chi)(g) = \frac{1}{2}(\chi(g)^2-\chi(g^2))$?

One can decompose $\bigotimes^2 V = \bigvee^2 V \oplus \bigwedge^2 V$, getting a corresponding decomposition for representations, say when $V$ is a module for some finite group $G$. One then has the ...
4
votes
1answer
613 views

Are two groups isomorphic if they have the same character table and each $|\chi| \leq 1$?

Suppose two groups have the same character table of complex representations. Also, all the entries in this character table have absolute value at most $1$. Does this imply that the two groups are ...
4
votes
2answers
56 views

Finding the character table of this group

if $ G = <a,b| a^9 = b^3 = 1, bab^{-1} = a^4> $ of order 27 then know the following, that any element can be written as $b^ka^n$ with n $\in [0,8], k\in[0,2]$ and that the 11 conjugacy classes ...
4
votes
1answer
109 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 ...
4
votes
3answers
466 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 ...
4
votes
1answer
128 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 ...
4
votes
3answers
103 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 & ...
4
votes
1answer
163 views

Is there some relation between characters in representation theory and multiplicative characters?

A character of a group representation is obtained by taking trace of each matrix in this representation. The word character is often used in the sense that it is a homomorphism from a group to ...
4
votes
2answers
161 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: ...
4
votes
2answers
105 views

Character theory - Exercise 5.14

I am trying to solve the exercise 5.14 from the Isaac Martins Character Theory of Finite Groups. Let $G$ be a nonabelian group and let $ f=min\{\chi(1) | \chi \in Irr(G), \chi(1)>1 \}. $ Show ...
4
votes
1answer
71 views

Decompose the following representation of $A_5$ in irreducible representations

Denote the space of functions on the set of faces of the icosahedron by $V_f$, this is a 20-dimensional representation of $A_5$ which acts here on as the group of rotational symmetries. Here follows ...
4
votes
1answer
142 views

The natural inclusion of an infinite abelian group $G$ into $\widehat{\widehat{G}}$

I was recently trying to think of a simple example that demonstrates that the natural inclusion of an abelian group $G$ into ...
4
votes
0answers
84 views

Every irreducible character of $G$ is an irreducible character of $H$?

Let $H$ be a proper subgroup of $G$ such that for all $\chi\in Irr(G)$, $\chi_H\in Irr(H)$. That is, the restriction of every irreducible character of $G$ to $H$ is an irreducible character of $H$. ...
4
votes
0answers
30 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 ...
4
votes
0answers
111 views

On Applications of the Murnaghan-Nakayama rule

The question is located below. In short, I am looking for an accessible explanation of the Murnaghan-Nakayama rule in relation to the following problem. Pardon the long setup. Let $Y$ be a standard ...
4
votes
0answers
93 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 ...