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85 views

On a sum taken over all irreducible characters: is $\sum\limits_{i=1}^t\chi_{V_i}(g)^2$ nonzero for all $g\in G$?

Suppose $G$ is a finite group and $\mathbb{k}$ an algebraically closed field of characteristic zero. Denote the irreducible representations of $G$ over $\mathbb{k}$ by $V_1,\ldots, V_t$. Is it ...
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1answer
318 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 ...
3
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3answers
118 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 ...
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1answer
347 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. ...
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2answers
304 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 ...
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1answer
78 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 ...
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0answers
401 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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1answer
71 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 ...
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1answer
108 views

Weyl character formula and finding the trace.

Let $v$ be a positive integer. I have a representation $\rho_v$ of $USp(4) = \{g\in M_2(\mathbb{H})\,|\,g^{T}\bar{g}\}$, where $\mathbb{H}$ is Hamiltons quaternions. The representation $\rho_v$ has ...
2
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2answers
85 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 ...
4
votes
1answer
390 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 ...
7
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2answers
207 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 ...
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2answers
310 views

Convolution of irreducible characters of a finite group

If $\chi^{\lambda}$ and $\chi^{\mu}$ are the characters of two irreducible representations $V^{\lambda}$ and $V^{\mu}$ of a finite group $G$, is there a simple way of proving that : $$ \chi^{\lambda} ...
2
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1answer
90 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
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1answer
83 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 ...
3
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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) - ...
3
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1answer
353 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 ...
4
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2answers
258 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?
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2answers
208 views

Few questions on Character of representation .

a) What does it mean to say that the Character of a representation is irreducible on its own? b) If Char($K$) is $0$ then kernel of character is a normal subgroup of G , why ?? c) Over a field of ...
3
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1answer
224 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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3answers
169 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 ...
0
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1answer
330 views

How to use a character table to get the centre

I have been given a character table and I need to find from the table the centre of each character. I dont know how to do this. if someone could please explain how i can find the centre by looking at ...
0
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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. ...
0
votes
1answer
261 views

Primitive Dirichlet Character

Let $\chi$ be the trivial Dirichlet character mod $N$. What is the primitive Dirichlet character associated to $\chi$? Is it just the character on $\mathbb{Z}$ that sends all integers to 1?
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0answers
316 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 ...
3
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2answers
695 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 ...
11
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1answer
204 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 $\mathbb{Q}$) ...
17
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2answers
891 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 ...
2
votes
1answer
339 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 ...
6
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1answer
280 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) ...
1
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1answer
75 views

Characters on $C\left( \mathbb{R}^n\right)$

A character on $C\left( \mathbb{R}^n\right)$ (the set of all complex-valued continuous functions on $\mathbb{R}^n$) is a continuous $^*$-algebra homomorphism into $\mathbb{C}$. For any fixed $x_0\in ...
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3answers
558 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 ...
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2answers
249 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 ...
4
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2answers
274 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 ...
2
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1answer
255 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 ...
1
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1answer
293 views

Irreducible representation decomposition

Any faithful finite group representation can be written as a sum of irreducible representations $\rho = \oplus_{i} a_i \rho_i$ such that $Ker(\rho)=0=\bigcap_i Ker(\rho_i)$ - is this sufficient to ...
2
votes
1answer
98 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 ...
4
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1answer
135 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 ...
8
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1answer
220 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 ...