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

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3
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1answer
49 views

A class function $f$ is a character if and only if $(f,\chi_{q_i})_G $ is a non-negative integer, for all irreducible characters $\chi_{q_i}$

I'm currently revising representation theory and I'm a bit stuck trying to prove the converse of the above statement. $(\Rightarrow)$ is straight forward because if $f$ is the character of a ...
1
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1answer
23 views

Showing that compact metrizable abelian group has enough characters

I wanted to know if anybody has some clue as why a compact metrizable abelian group has enough characters to sepatare points. (a character on a topological group is a continuous homomorphism from the ...
1
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1answer
56 views

Frobenius Reciprocity and a character theory problem

How Frobenius Reciprocity can help us to solve these two problems: Let $ H $ be a subgroup with index $ m $ in the finite group $ G $. Let $ F $ be an algebraic closed field of characteristic $ 0 $. ...
1
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1answer
32 views

Gauss sum of character $\psi \neq 1$

I am trying to solve Let $1 \neq \psi$ be a charachter of $\mathbb{F}_p$ and define $$G(\psi) = \sum_{x\in \mathbb{F}_p} \psi(x^2) $$ Proof that $|G(\psi)|^2 = p$. What I tried so far: ...
1
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1answer
72 views

Sum of an irreducible character over all $G$

Let $\phi: G\to GL_n(\mathbb C)$ be an irreducible representation of a finite group $G$. Let $\chi: G\to \mathbb C$ be the character of $G$. Prove that: $$\displaystyle \sum_{g\in G}{\chi(g)}=0 $$ I ...
18
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0answers
485 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 ...
11
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427 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 ...
4
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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
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29 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
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0answers
101 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
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0answers
91 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
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0answers
436 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 ...
4
votes
0answers
127 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
votes
0answers
47 views

Linear independence of primitive Dirichlet characters and convolution

This is not an exercise but merely a question I have. Fix $N \in \mathbb{N}$ and suppose there exist some values $a_k \in \mathbb{C}$, for $k \in \mathbb{Z}_N$, such that $$ \sum_{k \in \mathbb{Z}_N} ...
3
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0answers
59 views

Relation between the characters of subgroups of a finite group

Let $ H $ and $ K $ be subgroups of a finite group $ G $. Let $ \chi_1(H) $ and $ \chi_1(K) $ denote the trivial characters of $ H $ and $ K $ over an algebraically closed field of characteristic $ 0 ...
3
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0answers
63 views

How much we know about the Group from its Complex character table?

Suppose $G$ is a finite group and suppose that complex character table of $G$ is given.It is well known that from character table we cannot determine the Group uniquely (For example $Q_8$ and $D_8$ ...
3
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0answers
72 views

Characters of a finite group

Recently, I have been studying about Character Theory of Finite Groups, mostly from "Groups and Representations" by J. Alperin & R. Bell. In the aforementioned textbook, the characters of a finite ...
2
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0answers
19 views

Character sum of a type of “almost linear” surjective mappings over finite fields

Let $F$ be a finite field of characteristic a prime $p$ with $q$ elements and let $E/F$ be a finite extension of degree $n > 1$ over $F$. Let $\chi$ be the additive canonical character on $F$ ...
2
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0answers
36 views

Solving the Character table for $A_4$ and derived algebra characteristics.

Say I am given a group with 4 conjugacy classes $C_1, C_2, C_3, C_4$ with orders 1,3,4,4 respectively. I will call the conjugacy class characters $\chi_1,\chi_2,\chi_3,\chi_4$ respectively. I am ...
2
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0answers
32 views

Find the character of $\mathbb{C}[S_4/D_8]$

Find the character of $\mathbb{C}[S_4/D_8]$. I am assuming with this question that the first step will be to compute the (left) cosets $\{ gD_8: g \in S_4 \}$. Then I'm assuming that then it will ...
2
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0answers
105 views

Degrees of irreducible complex characters of alternating groups

What is sum of degrees of the irreducible complex characters of the alternating groups? The background of this question is to calculate the diminsion of a maximal torus of the associated Lie algebra ...
2
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0answers
71 views

a finite group of order $360$ never has an irreducible character of degree $15$

The following problem is an exercise in character theory: "Prove that a finite group of order $360$ never has an irreducible character of degree $15$." Could you help me for it?
2
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0answers
72 views

A group of order $2^67$

Let $G$ be a finite group of order $2^6\times 7$. Also $G$ has at least $5$ irreducible characters of degree $7$. I want to prove that with this condition there exists no character of even degree in ...
2
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0answers
21 views

What is the basic concept behind extending a character?

I'm trying to do my homework (the overall problem is about $A$ a finite abelian group and showing $A \cong A^\vee \cong A^{\vee\vee}$, but you can ignore that), and I think I have a fundamental lack ...
2
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0answers
66 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 $g \in H$?
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62 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 ...
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0answers
41 views

What values can a character $\psi$ take on an element of order $2$?

If $\psi$ is the character of a degree $2$ complex representation $\varphi\colon G\to GL_2(\mathbb{C})$, and $x\in G$ has order $2$, then $\psi(x)=0,\pm 2$. I noticed this by seeing $\varphi(x)$ ...
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0answers
56 views

Character group of $\mathbb{Z}$

I am trying to compute the character group of $\mathbb{Z}$ which contains homomorphisms that map into $\mathbb{C}^\times$. I have determined that each homomorphism $\phi \in \hat{\mathbb{Z}}$ may be ...
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0answers
41 views

Understanding restriction in $S_3$

Let $G=S_3$. \begin{array}{|c|c|c|} \hline & e & (123) & (12) \\ \hline \chi_0 & 1 &1 & 1\\ \hline \chi_1 & 1 & 1 & -1 \\ \hline \chi_2 & 2 & -1 & 0 \\ ...
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0answers
44 views

Character of a Representation

Suppose that we have a representation $V$ of a group $G= SU(2)$ . Is it true that if $ \chi_V \cdot c \neq 0$ for some non-zero constant c, then the trivial representation must be one of the ...
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0answers
33 views

Orthogonality Relations for Character Groups

I'm trying to understand a part of a proof of orthogonality relations for character groups and finite abelian groups, and I don't quite get this part from the below link: ...
1
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0answers
53 views

Character as sum with regular representation

Suppose $G$ is a group and $\chi$ is a character of $G$ with $\chi(g_1)=\chi(g_2)$ for all non-identity $g_1,g_2 \in G$, and let $\chi_{reg}$ denote the regular representation character. I read that ...
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0answers
47 views

Summation of Legendre symbol

Let $\chi_{2,q}$ be the real Dirichlet character modulo a prime $q>2$, which is not the principal one (the so-called Legendre symbol). Is it true that $$ \sum_{n=0}^{+\infty} ...
1
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0answers
79 views

Bounds for the norm of certain additive character sums

Let $q$ be a power of a prime, let $\alpha$ be a primitive element in the finite field with $q$ elements and let $\chi$ be the canonical additive character of the field. Recently I became interested ...
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0answers
80 views

Understanding the irreducible representations of $D_3$

By the dimensionality theorem, $$\sum_i d_i^2 = |G|,$$ where $d_i$ is the dimension of the $i$th irreducible representation, we can infer that the dihedral group $D_3$ has two one dimensional irreps ...
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0answers
79 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). ...
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0answers
137 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 ...
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39 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 ...
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38 views

Can each of the following be a character of a finite group $G$

Can each of the following be a character of a finite group $G$? $(i) \ \ (2,0,\frac{1}{2},\frac{1}{2},0)$ $(ii) \ \ (3,-1,0,4,0)$ $(iii) \ \ (2,2,2,2)$ $(iv) \ \ (1,0,-1,0)$ I think that the ...
0
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0answers
39 views

Characters of Linear Algebraic Groups

Reading about the semi-invariants of quivers, I see a fact which is frequently referred to in the literature, and is assumed to be trivial. However, I don't see that very easily. So, I was wondering ...
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0answers
20 views

Perfect subset of the group of characters of an additive dense subgroup of $\mathbb(R)$

Suppose that $\Gamma$ is a dense subgroup of the group of real numbers $\mathbb{R}$ and let $G$ be the group of characters of $\Gamma$. The problem is to show that for any $a\in\Gamma$ the set ...
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0answers
26 views

Non-trivial characters of $SU(2)$

Are there non-trivial characters (or quasi-characters) of the special unitary group $SU(2)$? I couldn't find a straightforward answer by googling.
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47 views

on the characters of the normal subgroup and its quotient

I read character theory recently and thought about the following proposition, but I do not know is this true or false: Let $G $ be a finite group such that two distinct primes $ p $ and $ q $ ...
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0answers
48 views

Dirichlet L series estimation

let $\chi$ be a non-principal character modulo $q$, $M\geq 1$. I have to prove that, if $\vert \sigma - 1 \vert \leq \frac{1}{\log M}$, then $\vert \sum_{n=1}^M \chi (n)n^{-s}\vert\leq 1+e\log M$ and ...
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0answers
15 views

Can this matrix be said to consist of Dirichlet characters?

When: $j=1$ the following formula: $$T(n,k)=\prod\limits_{m=1}_{m \mid n}^{\text{n}} \left(\exp ^{-\mu \left(\frac{n}{m}\right)}\left(\Lambda \left(\frac{n}{m}\right)\right) \chi _{\exp ^{-\mu ...
0
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0answers
36 views

Conductor of Dirichlet character divides every quasiperiod

Let $\chi$ be a Dirichlet character which has quasiperiods $d_1, d_2$. I.e., if $(n(n + kd_i), q) = 1$ then $\chi(n + d_i) = \chi(n)$ for any $k \in \mathbb{Z}$. Supposedly we can then show that ...
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0answers
38 views

Is the Transpose of a Representation an Equivalent Representation?

Suppose we are working over $\mathbf{Z}[G]$ where $G$ is finite. Suppose further we have two representations $\rho$ and $\rho^\prime$ such that $\rho^\prime=(\rho)^T$. Can we say that these two ...
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0answers
49 views

Show that 2 representations are not equivalent and find all the irreducible representations of G.

Show that 2 representations are not equivalent and find all the irreducible representations of $G$. The group $G=T_{16}$ has order 16 and presentation given by $G=\langle a,b : a^8=b^2=1, ...
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74 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 ...
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0answers
41 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 ? ...