The characters tag has no wiki summary.
17
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2answers
769 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 ...
13
votes
2answers
368 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 ...
10
votes
5answers
185 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 ...
10
votes
1answer
178 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 $\mathbb{Q}$) ...
10
votes
0answers
224 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 ...
8
votes
1answer
198 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
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0answers
198 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 ...
7
votes
3answers
348 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 ...
7
votes
2answers
181 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 ...
7
votes
0answers
101 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
1answer
165 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
74 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 ...
6
votes
1answer
92 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
129 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 $\hat G \overset{\sim}{=}\text{Hom}(G,T)$, ...
5
votes
2answers
211 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 ...
5
votes
2answers
116 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 ...
5
votes
1answer
51 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 ...
4
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2answers
225 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
254 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
153 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?
4
votes
1answer
184 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) ...
4
votes
1answer
80 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
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0answers
43 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
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0answers
55 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 ...
3
votes
3answers
139 views
Orthogonality relations of Characters
Could somebody please help me understand the jump from Proposition 10 to Proposition 11 in the following
http://www.ms.uky.edu/~pkoester/research/charactersums.pdf
Note: The orthogonality relations ...
3
votes
2answers
322 views
Condition for abelian subgroup to be normal
Sorry for any mistakes I make here, this is my first post here. I have a group $G$ which has an abelian subgroup $A<G$. I also know there is a irreducible character $\chi$ with the degree of ...
3
votes
1answer
43 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 ...
3
votes
3answers
90 views
Character of $S_3$
I am trying to learn about the characters of a group but I think I am missing something.
Consider $S_3$. This has three elements which fix one thing, two elements which fix nothing and one element ...
3
votes
1answer
62 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 ...
3
votes
1answer
51 views
Subspace spanned by powers of a faithful character
The following well known theorem can be found in many books on character theory:
Let $\chi$ be a faithful character of a finite group $G$ and suppose that $\chi(g)$ takes on exactly $m$ different ...
3
votes
1answer
185 views
Notation: Character of a Finite Field
This is my first post on StackExchange. I had a quick question about notation (appearing in research literature) that I was unable to find by repeated searches, and I was hoping that someone would be ...
3
votes
1answer
43 views
Characters of subrepresentation
Given an algebra $A$ with finite dimensional representation $V$ with action $\rho$, I want to prove the following statement:
If $W\subset V$ are finite dimensional representations of A, then ...
3
votes
1answer
80 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 ...
3
votes
1answer
75 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 ...
3
votes
1answer
144 views
Why unitary characters for the dual group in Pontryagin duality if $G$ is not compact?
In harmonic analysis, for any locally compact abelian group, one constructs the dual group as the group of homomorphisms into the unit circle with the compact open topology. In other words, unitary ...
3
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1answer
128 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 ...
3
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0answers
288 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
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0answers
74 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
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1answer
49 views
Characters of affine algebraic groups and the determinant
Let $G$ be an affine algebraic group (i.e. a $k$-variety which is also a group and the group multiplication and inversion are morphisms of varieties). A character of $G$ is a morphism of algebraic ...
2
votes
1answer
206 views
Setting up Brauer character theory
My question relates to p. 147 of Serre's Linear Representations of Finite Groups, where he is setting up the definitions relevant to Brauer character theory.
Having fixed an algebraically closed ...
2
votes
2answers
91 views
Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$.
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 ...
2
votes
2answers
70 views
Equality Involving Character Sums
I'm reading through a paper involving character sums, and I have run into an equality that I am unsure how to justify. Here is the set-up:
Suppose $\chi$ is a multiplicative character of ...
2
votes
1answer
60 views
How to bound the order of a finite group under the following hypotheses?
In the book Character Theory Of Finite Groups by I.Martin Issacs as exercise 2.14
Let $G$ be a finite group with commutator subgroup $G'$. Let $H \subset G' \cap Z(G)$ be cyclic of order n and ...
2
votes
1answer
61 views
Intuition on characters of topological groups
I am coming to the end of a series of lecture notes on representations of $S_n$ and $GL(V)$. Near the end, it attempts to introduce the notion of the "character of a topological group", but doesn't ...
2
votes
1answer
221 views
character tables for groups of order $pq^2$
What is the character table for groups or order $pq^2$? The classification of order $pq^2$ groups has already been discussed in relation to Sylow theory.
For the Abelian groups, $\mathbb{Z}_p ...
2
votes
1answer
46 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 ...
2
votes
1answer
57 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 ...
2
votes
1answer
67 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 ...
2
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1answer
133 views
What is a rational character?
Let $G$ be the group of $F$-points of a connected, reductive group over a $p$-adic field $F$. The unramified character of $G$ are $\chi\circ\psi$ where $\chi$ is an unramified character of ...
2
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1answer
94 views
Is the estimation of number's name's length and comma-grouping feasible?
I am thinking in a mathematical problem that probably is already formulated and even solved. It is about big integers and someting else.
Let n be an integer positive number.
For n := 1,000 we have ...
