Questions tagged [characters]

For questions about characters (traces of representations of a group on a vector space).

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35 views

Existence of subgroups of index $2$

I would like to seek some clarification to the solution to the following problem from Representations and Characters of Groups (Gordon, Liebeck): Let $\rho$ be a representation of the finite group $G$...
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1answer
44 views

Basis change of $Z(\mathbb C [G])$

Let $G$ be a finite group with . Consider the two bases of $Z(\mathbb C [G])$: One is the elements of the form $e_{g} =\Sigma_{x\in c_g} x$, where $c_g$ is a conjugacy class, and another is elements ...
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5 views

Algebraic characters versus analytic characters for algebraic Tori over non-archimedean field

Given an split torus $T$ over a field $K$ complete with respect to a non-archimedean absolute value, the (algebraic) character group can be defined as $$ X(T):=\text{Hom}_{K-gr}(T,\mathbb{G}_m)$$ ...
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Confirmation of Relation between Degree of Induced Character and Degree of Initial Character

I have been reading about induced characters, and I haven't found explicitly written anywhere, the relation between degree of the induced character of a character and the dimension of the character ...
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1answer
25 views

Absolute value of sum of additive characters of $\mathbb{F}_p$

Consider the absolute value of the following exponential sum: $\left|\sum_{x \in \mathbb{F}_p} \sum_{y \in \mathbb{F}_p}e^{\frac{2\pi i}{p}(ux+vy-wxy)}\right|$ for given $u,v,w\in\mathbb{F}_p$ with ...
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1answer
63 views

A special type of Gauss sum

In the work of my thesis I came up with a problem that is elementary, but I can't figure out its proof. Let $p$ be an odd prime, let $(\mathbb{Z}/p^n\mathbb{Z})^\times$ denote the multiplicative ...
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22 views

How can I easily find character table of Sergeev group (finite)?

I am looking for the character table of the Sergeev group S_d for small d (say, 'd' up to 10 or up to whatever is possible). The Sergeev group $S_d$ is defined as follows: Let $\mathfrak{S}_d$ be ...
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1answer
32 views

Irreducible character of degree $n$ is $|\chi(g)|=n$ for every element in the center

G is a finite group. Let $\chi$ be an irreducible character of degree $n$ (that is $\chi(1)=n$). Show that $|\chi(g)|=n$ for every element in $Z(G)$. I am basically asking to prove that $Z(G)\...
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1answer
67 views

When an irreducible representation is an induced representation

So I've been trying to answer this exercise to much of my frustration: Let $G$ be a finite group and $S$ a normal subgroup. Let $\rho$ be an irreducible representation of $G$ over $\mathbb{C}$. ...
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36 views

A function of the characters of a representation that determine how many orbits there are

Let $G$ be a finite (even cyclic group generated by $g$) and $X$ be a set on which it acts and considering the permutation representation induced by this, let $\chi_i$ be the set of characters. Then, ...
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1answer
33 views

Find an element of a Finite Group based on a table of characters

Given $G$ is a group of order 24 with the following table of characters Find the value for element A. These are all the instructions that are given. I guess I am not really getting this question. ...
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1answer
17 views

Irreducible characters of k[G] when k is not algebraically closed and char k divides order of G.

Let k[G] be the group algebra where char(k) divides |G| with G being a finite group. Assume k is not algebraically closed. How can one show that the characters associated with the irreducible ...
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Find a irrep from Character Table

I have been studying Group Theory from P. Ramond's book and I already googled a lot but I could not find the answer or a procedural to find what procedure Ramond uses in the image below. So, my ...
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1answer
40 views

Determining the conductor of an induced Dirichlet character

Say I have a Dirichlet character $\chi$ mod $N$ and I know that $\chi$ is induced by a Dirichlet character $\chi'$ mod $M$ with $M|N$. I want to then say that the conductor of $\chi$ divides $M$, ...
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1answer
31 views

Why do characters of groups have modulus $1$?

This is a fairly simple question, and it is supposed to be a definition-level question. But it really bothers me as I couldn't see the reason. Let $f$ be a character of an (arbitrary) group $G$, then $...
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1answer
36 views

Dirichlet $L$-function of primitive character in function field setting

Let $q=p^k$ be a prime power, and let $Q \in \mathbb{F}_q[t]$ be a polynomial. A Dirichlet character $\varphi$ of modulus $Q$ is a group homomorphism $$ \varphi \colon (\mathbb{F}_q[t]/Q\mathbb{F}_q[t]...
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Character is algebraic number

I'm reading the book "Representation Theory of finite groups" the chapter about characters. Suppose that $V$ is a KG module where $G$ is a group and $K$ is a field. Let $\chi$ be the group character. ...
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38 views

Show that characters are irreducible.

I would like to prove the following: We have a group $G$ and: $$\chi:G\rightarrow \mathbb{C}$$$$ \tilde{\chi}: G/N \rightarrow\mathbb{C}$$ characters of groups $G$ and $G/N$ respectively, ...
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1answer
60 views

Primitive roots of unity for primitive characters [closed]

Let $\chi:\mathbb{Z}\to\mathbb{C}^{\times}$ be a primitive Dirichlet character mod $N$. If $\varphi(N)=|(\mathbb{Z}/N\mathbb{Z})^{\times}|$ is it always true that the image of $\chi$ contains a ...
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20 views

Question about the conductor of some character.

I have a character $\chi$, which have conductor $p^n$ where $p$ is an odd prime. The conductor of $\chi^p$ is always $p^{n-1}$?? What about the conductor of $\chi^k$?
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1answer
38 views

For each irreducible representation of $S_4$ determine whether the restriction on $A_4$ is an irreducible representation of $A_4$

Let $H$ be a subgroup of $G$ and $\DeclareMathOperator{\GL}{GL} \rho: G \rightarrow \GL(V)$ a representation of G. The restriction map $\rho |_H$ is a group homomorphism, i.e., a representation of $H$....
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Find all irreducible characters of $C_2\times C_3$.

Find all the irreducible characters of $C_2\times C_3$. Note here that $C_n$ is a cyclic group of order n. I have calculated the character tables for $C_2$ and $C_3$ and this is what I got: $C_2$: $...
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1answer
47 views

Let $χ$ be an irreducible character of finite group $G$. Prove $χ(e)^2\leq \big[G: Z(χ)\big]$ $\iff$ if $χ(g) = 0$ for all $g ∈ G\backslash Z(χ)$.

Let $χ$ be an irreducible character of finite group $G$. Prove that $$χ(e)^2\leq\big[G: Z(χ)\big]\,,$$ and the equality holds iff $χ(g) = 0$ for all $g ∈ G\backslash Z(χ)$. Any help would be ...
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1answer
97 views

Showing $\tilde\chi: G/N \rightarrow \mathbb{C} $ is a character of group $G/N$.

So I need to prove the following: Let $\chi:G\rightarrow \mathbb{C}$ be a character of group $G$, with the property that $N \leq \ker \chi$, show that: $$\tilde\chi: G/N \rightarrow \mathbb{C} \...
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21 views

Dirichlet characters acting on reduced residue classes/residue classes

In proving Dirichlets theorem on arithmetic progressions, we talk about the haracter $\chi$ defined as: 1: $\chi$ is periodic with period $q$. $\chi(n) = \chi(n+q)$ 2: $\chi(a) = 0$ if $(a,q)\...
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1answer
19 views

Character table of $S_4$.

I know there are potentially quite a few posts related to this question, but I guess I had a specific one that wasn't found in the ones that came up. I was reading the notes pertaining to this ...
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32 views

A sum of characters of a cyclic group

For a cyclic group $G$ of order $n$, with $g\in G$ not a generator of $G$, prove that $$\sum_{d\mid n}\frac{\mu(d)}{\phi(d)}\sum_{\chi\in \hat G_d}\chi(g)=0,$$ where $\hat G_d$ is the set of all ...
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18 views

Definition of p-primary component of character and local component of character at p

(Sorry for my poor english..) Let $\chi : (\mathbb{Z}/N\mathbb{Z})^{*}\to \mathbb{C}$ be a Dirichlet character and $p\mid N$ be a prime. I have some questions about the definition of $p$-primary ...
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1answer
52 views

Each irreducible representation of $G$ is of the form $\rho_1\otimes\rho_2$, for some irreducible representations of $G1$ and $G2$

Each irreducible representation of $G_1\times G_2$ is isomorphic to a representation $\rho_1\otimes\rho_2$, where $\rho_i$ is an irreducible representation of $G_i$ for $i=1,2.$ So, this is the ...
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1answer
73 views

Representations of finite group with normal prime index subgroup

The problem is this: Let $G$ be a finite group, $Q$ a normal subgroup of prime index. (i) Let $\psi = \chi_V$ be the character of the $Q$ representation $V$. Show that either $\psi$ can be ...
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24 views

Multiplicative quasi-characters on ℚ

I want references on multiplicative quasi-characters on $\mathbb{Q}$, especially having growth on natural numbers bounded with $|\chi(m)| \le m$. Results in number theory? Use in visualization of data,...
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1answer
68 views

Properties of characters of representations

Suppose $\chi$ is the character of the representation $\sigma:G\rightarrow \mathbb{C}$ (where $G$ could be assumed to be the symmetric group as well), then do we know anything about the following sum $...
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26 views

Prove that: Z(G) = { g∈G ∣ |χ(g)| = χ(1) for all χ ∈ Irr(G)} (Representation Theory)

I'm working on representation theory and I came across a statement that Z(G) = { g ∈ G ∣ |χ(g)| = χ(1) for all χ ∈ Irr(G)}. I'm trying to see why this is the case. (1) For the backwards direction, ...
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17 views

Character of regular representation.

Let $G$ be a finite group and $k$ a finite dimensional field. I am asked to 'find the character' of the regular representation of $G$. And determine what happens in the modular case when ...
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1answer
51 views

Are finite abelian groups isomorphic to k-character groups for arbitrary alg.closed k?

Let $G$ be a finite abelian group, $k$ an algebraically closed field with group of units $k^{\times}$, and let $Ch_{k}(G)$ be the set of $k$-characters of $G$ whose elements are multiplicative ...
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1answer
26 views

Character of representations of SO(3) derived from the irreducible representations of SU(2)

I have been given that there is a 2:1 surjective homomorphism from $SU(2)$ to $SO(3)$ with kernel $\{-I,+I\}$ and that $V_n$ are irreducible representations of $SU(2)$ where $V_n$ is the space of ...
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1answer
55 views

Character induced by the trivial character of a subgroup

We know that if $\{ e\}<G$ is the trivial subgroup and $\chi_0$ is the (necessarily) trivial character of $\{e\}$, then the induced character in $G$ can be written neatly as $$ \textrm{Ind}_{\{ e \...
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22 views

How to determine the irreducible characters for $S_n$

Is there a certain way one can determine the irreducible characters for $S_n$? I am trying to find all the irreducible characters for $S_4$ and $S_5$, but I have no idea how to go about this or where ...
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1answer
55 views

Why is $\chi_\rho - \chi_1$ Always a Character, and Why is it Irreducible?

In my lecture, the professor constructed a character table for $S_3$. There are $3$ conjugacy classes, so there are $3$ irreducible characters. The three characters we used were: $\bullet$ $\chi_1$...
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1answer
80 views

$5$-dimensional irreducible representation of $\mathcal{A}_5$

I know that the character table of $\mathcal{A}_5$ is the following: $$\begin{array}{c|c|c|c|c|c|c|c|c} & 1 & 15 & 20 & 12 & 12\\ \hline \mathcal{A}_5 & id & (12)(34) &...
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1answer
33 views

Prove that the value of $\chi_{(n−1,1)}$ on a permutation $\sigma\in S_n$ is one less than the number of fixed points in $\sigma$.

Fix $n$ such that $n > 2$. Let $\chi_{(n−1,1)}$ denote the character of the Specht module $S_{(n−1,1)}$ How do I prove that the value of $\chi_{(n−1,1)}$ on a permutation $\sigma\in S_n$ is one ...
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1answer
52 views

Character sum involving conjugacy classes

I am tasked with showing $$ \sum_{ x \in C_1, y \in C_2, z \in C_3} \chi(xyz) = \frac{|C_1| |C_2| |C_3| \chi(C_1)\chi(C_2)\chi(C_3)}{\chi(1)^2} $$ where $C_1,C_2,C_3$ are conjugacy classes in a ...
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25 views

Vanishing elements

Definition: We say that $g\in G$ is a vanishing element if there exists $\chi \in \text{Irr(G)}$ such that $\chi(g)=0$. Lemma 1: Let $N \leq M \leq G$ with $N$ and $M$ normal in $G$ and $(|N|, |M/...
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1answer
28 views

Clifford Correspondence

I have encountered a situation in a proof of a result of which I am struggling to figure out. Suppose we have subgroups $H,K$ and $N$ of a group $G$ such that $N\unlhd K$ and $K \leq NH$. If $\psi \...
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51 views

An Exercise from Representation of Finite Groups by Serre (section 10)

I was trying to solve a statement from Serre's Linear Representation of Finite Groups (taken from the hint of Exercise 10.6(b) in the book) : Suppose that $H$ is normal in $G$ and that $G/H$ is ...
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2answers
74 views

Do characters distinguish real representations of a finite group?

Let $G$ be a finite group. Consider two representations $\rho_1, \rho_2: G\to GL_{n}(\mathbb R)$. Suppose that these two representations have the same characters (i.e for any $g\in G$, ${\rm tr}\rho_1(...
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1answer
54 views

Determining conductor of a Dirichlet character

I was reading the book Introduction to Cyclotomic Fields by Lawrence C. Washington and the conductor of a Dirichlet character is defined as: An example is also given to make the definition clear. I ...
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23 views

Characters of sets of representations closed under tensor product

Let $R$ denote the set of all irreducible representations of a group $G$ over a complex vector space. Let $U \subset R$ denote a subset of representations which is closed under tensor product (i.e., ...
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1answer
63 views

How to compute this character?

Let $K$ a field of characteristic $0$ and $K[G]$ a group algebra of a finite group $G$. Let $I$ be a left ideal of $K[G]$ generated by an idempotent $e$ and $\rho: G \longrightarrow GL(I)$ a ...
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1answer
33 views

Is there a way to determine whether a representation of a finite group is faithful? [duplicate]

I want to determine whether a representation of a finite group is faithful from the character table. Is there a such general way or specific way for finite group of small order?

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