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

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2
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0answers
11 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 ...
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1answer
23 views

Invertibility of character table

Corollary. The character table of a group is an invertible square matrix. The theorem that is a corollary to states that the character table is a square matrix and the explanation for invertibility ...
2
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1answer
40 views

Characters of transitive finite permutation group

I know that Frobenius reciprocity helps us to solve this problem, but I don't know why: Let $ G $ be a transitive finite permutation group with permutation character $ \pi $. If $\chi $ is an ...
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1answer
51 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 $. ...
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1answer
39 views

Facts on $ \mathbb{C} $-characters

My assumption: $ G $ is a finite group & $ \chi $ is a faithful $ \mathbb{C} $-character of $ G $ with degree $ n $ and $ r $ is the number of distinct values assumed by $ \chi $. Now is it true ...
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1answer
30 views

Characters of a compact group with uniform positivity over $G$

Let $G$ be a compact group and let $\widehat{G}$ denote the set of all equivalence classes of irreducible representations of $G$. For each $\pi \in \widehat{G}$, we use $\chi_\pi$ to denote the ...
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2answers
50 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 ...
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2answers
52 views

irreducible characters of a group

I am currently attempting a past exam paper and am stuck on the following question for part a) $\mu$ is an irreducible character iff it is equal to the character of an irreducible representation, ...
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1answer
72 views

character tables and solubility

I am currently going through a past exam paper for a group theory module and am unable to answer the following section of a question. The copy of my lecture notes doesn't seem to have a section on ...
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1answer
42 views

Consequences of $G$ having exactly $3$ irreducible characters

Suppose a finite group $G$ has exactly irreducible characters $\chi_1 = \mathbb{I}, \chi_2,\chi_3$. i) Show that G is soluble and deduce it has a non-trivial one-dimensional character $\chi_2$. ii) ...
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34 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 ...
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1answer
39 views

Product of a character with an irreducible character a non-negative integer

Why is the inner product of a character with an irreducible character a non-negative integer? I can see that by properties of the inner product it will be non-negative but I cannot see why it would ...
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1answer
27 views

Induced character and center of groups

If $Z$ is a subgroup of the center of G and $|G:Z|=m$, then $\chi^*(g)=m\chi(g)$ if $g\in Z$ where $\chi$ is a character of $Z$ and $\chi^*$ is the induced character of $G$. Let $\phi$ be the ...
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0answers
32 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 $\varphi(x)$ ...
2
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1answer
34 views

Product of characters in representation theory

Is it true that if $\phi_1$ and $\phi_2$ are characters then $\phi_1\phi_2$ is a character of a representation? I think this is true, say for instance if $R_1$ is a matrix representation with ...
0
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1answer
41 views

Permutation representation of $S_4$ acting on $\{1,2,3,4\}$

I am considering the permutation representation of $S_4$ acting on $\{1,2,3,4\}$ with character $\chi_{\pi}$. The answer is $\chi_{\pi}=(4,2,1,0,0)$ Now I know that for the permutation representation ...
3
votes
1answer
50 views

Any 1-dimensional character $\otimes$ irreducible character is irreducible

I would like to prove that any 1-dimensional character $\otimes$ irreducible character is again irreducible. (I think this is character but there is chance I have misinterpreted my notes and it may be ...
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0answers
49 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 ...
0
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1answer
48 views

Considering transitive $G$-set

Question. Suppose that $X$ is a transitive $G$-set of size greater than $1$ and let $\pi$ be the associated permutation representation with the character $\chi$. Show that some element $g \in G$ ...
3
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2answers
58 views

Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers.

Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers. I get that if $G$ is non abelian it must have an irreducible ...
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2answers
46 views

Show that $Z(θ^G)≤H$

Suppose $H ≤ G$ and $θ \in Char(H)$. Show that $Z(θ^G)≤H$. ($Z$ is the centre and $θ^G$ is the character induced by $G$)
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1answer
43 views

Compute the character $Ind_{H}^G \mathbb{I}$ by computing the character of $\mathbb{C}[G/H]$

Let $G=S_3$ and $H=\langle (12) \rangle < G$. Compute the character $Ind_{H}^G \mathbb{I}$ by computing the character of $\mathbb{C}[G/H]$. I have computed the character using the induction ...
3
votes
1answer
80 views

Proof of the Frobenius Schur indicator

I am trying to prove the Frobenius-Schur indicator for $\chi$ irreducible character. \begin{equation} i_{\chi} = \begin{cases} 0, & \text{if $\chi$ is not real valued} \\ \pm1, & ...
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1answer
30 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: ...
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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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2answers
54 views

2nd half of proof of $\dim V^G=\frac{1}{|G|}\sum_{g \in G}\chi(g) $

Lemma. Let $\rho: G \rightarrow GL(V)$ be a representation, character $\chi$. Then $$\dim V^G=\frac{1}{|G|}\sum_{g \in G}\chi(g) $$ Proof. RHS: $$\frac{1}{|G|}\sum_{g \in G}tr ...
2
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1answer
50 views

Can two different characters of $S_n$ have the same _multiset_ of values?

As I was going through various representation-theory posts in the site, I stumbled upon this one: Characters of the symmetric group corresponding to partitions into two parts. Now, that question ...
1
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1answer
50 views

How to prove that a certain set is a coset of a subgroup of a group of characters mod $m$?

I have the following question: Let $m>1$ be a positive integer and $G:=G(m):={(\mathbb{Z}/m\mathbb{Z})}^{*}$. Let $p\in\mathbb{P}$ with the property $p\nmid m$. Let $\widehat{G}:=\text{Char}(G)$ ...
1
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1answer
90 views

Question concerning the Dirichlet density of a subset of the set of primes

I have the following question: I am reading Serre's book "A Course in Arithmetic" (see http://www.math.purdue.edu/~lipman/MA598/Serre-Course%20in%20Arithmetic.pdf). On page 75, it is stated that the ...
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1answer
32 views

Show that $ (φ^G )_K = (φ_{H∩K})^K $ with Mackey's theorem

Suppose H,K ≤ G e θ $ ϵ $ Char(H). Show that Z(θ)≤H. Suppose H,K ≤ G and HK = G. Se $ φ $ ϵ Char(H) show that $ (φ^G )_K = (φ_{H∩K})^K $. For the proof I have to use the Mackey's theorem. How do I ...
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0answers
37 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 ...
2
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1answer
50 views

Product of chracter

From Isaac's character theory book; $3.12$ Let $x\in Irr(G)$ and $g,h\in G$. Show that $$\chi(g)\chi(h)=\dfrac{\chi(1)}{|G|}\sum_{z\in G}\chi(gh^z)$$ I had thought that it was related to $3.9)$; ...
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1answer
39 views

Sum of characters modulo k

I want to find $\sum_{n \leq x} \chi(n) $, where $\chi$ is a non-principal character modulo $k$. I am trying to find $\sum_{n \leq x} \chi (n) n$ using Abel's summation formula, where the series $a_n ...
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1answer
37 views

Isomorphism between an group and its double dual

I wanted to prove that for an abelian group $G$ , $\phi : G \rightarrow \hat{\hat{G}}$ is an isomorphism where $\hat{G}$ is a set of all irreducible characters of $G$ for $x \in G$, ...
3
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0answers
57 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 $D_8$ ...
0
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1answer
31 views

Groups and Characters exercise hint

I am trying to make my way through Grove's Groups and Characters but am having some trouble with the following seemingly benign exercise: If a group $G$ has a normal $p$-complement $K$ show that ...
0
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1answer
90 views

Proving that $\pi_{X \times Y} \simeq \pi_X \otimes \pi_Y$

If $X$ and $Y$ are $G$-sets and $X \times Y$ is a G-set by $g \cdot (x,y)=(g \cdot x , g \cdot y)$. \pi is the corresponding permutation representation. Prove that $\pi_{X \times Y} \simeq \pi_X ...
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2answers
55 views

Abelian groups cannot have characters of degree 2

I was attempting the following exercise: Assume that $G$ is a simple group. Let $\chi$ be an irreducible character of degree $2$, and $g \in G$ be an element of order $2$. Prove that $\chi (G) ...
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0answers
17 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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1answer
62 views

Thm 5.19 (Brauer) in Isaacs Martin - Character Theory of Finite Groups

I would please like some help to understand the proof of thm 5.19 (Brauer) in Isaacs Martin - Character Theory of Finite Groups. It states: Let $\chi$ be a character of G with $[\chi,1_G]=0$. Let ...
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1answer
31 views

Prove that there exists an $m$th root of unity $λ ∈ C$ such that for all $g ∈ G$, $\chi(zg) =λ \chi(g)$

Suppose $\chi$ is an irreducible character of $G$. Suppose $z ∈ Z(G)$ and that $z$ has order $m$. Prove that there exists an $m$th root of unity $λ ∈ C$ such that for all $g ∈ G$, $\chi(zg) =λ ...
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1answer
34 views

Finding irreducible subrepresenations of modular representation in GAP

Recently, I have been fiddling with modular representations in GAP. First from what I can tell, GAP does not have a good way built in to find things like Brauer characters of a given non-solvable ...
1
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1answer
55 views

Complete the character table of group of order $21$

You are given the incomplete character table of a group $G$ with order $21$ which has $5$ conjugacy classes, $C_1,\dots,C_5$, which have sizes $1,7,7,3,3$. $$ \begin{array}{|c|c|c|c|c|} \hline & ...
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2answers
56 views

Finding the character table of $Q_8$

Assume $G$ is a finite group. I am trying to construct the character table of $Q_8$, which is defined by $$Q_8=\{\pm 1,\pm i, \pm j,\pm k \}, \ i^2=j^2=k^2=-1, \ ij=k,jk=i,ki=j$$ By considering the ...
0
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1answer
51 views

How to find the character of $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$?

Let $\mathfrak g$ be a Kac-Moody algebra. Then $$ \mathfrak{n}_{-}=\oplus_{\alpha\in\varPhi_{+}}\mathfrak{g}_{-\alpha} $$ and for $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$ the ...
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1answer
27 views

Subgroups of group of characters of a finite abelian group

Let G be a finite abelian group, H a subgroup of the group of characters of G. Is it true that H is the group of characters of some quotient group of G? Thanks for any help.
2
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1answer
16 views

prove that $N(x^m=a)=N(x^d=a)$.

Let $p>2$ be a prime, $m\in \mathbb N$, $d=$ GCD$(m,p-1)$. Let $N(x^n=a)$ denote the number of solutions to the equation $x^n=a$ in $\mathbb F_p$. prove that $N(x^m=a)=N(x^d=a)$. I am familiar with ...
0
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0answers
21 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.
2
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0answers
88 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
68 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?