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

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23
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0answers
622 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 some ...
11
votes
0answers
501 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 $F_2/F=\text{Fit}...
9
votes
0answers
81 views

Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
6
votes
0answers
173 views

$G$ be a finite group and $G'$ be its commutator subgroup and $\widehat G$ be the character group of $G$ ; then $G/G' \cong \widehat G$?

Let $G$ be a finite group and $G'$ be its commutator subgroup and $\widehat G$ be the character group of $G$ ; then is it true that $G/G' \cong \widehat G$ ? I am completely stuck , please help . ...
5
votes
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135 views

On Applications of the Murnaghan-Nakayama rule

The question is located below. In short, I am looking for an accessible explanation of the Murnaghan-Nakayama rule in relation to the following problem. Pardon the long setup. Let $Y$ be a standard ...
4
votes
0answers
59 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} ...
4
votes
0answers
92 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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39 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
100 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
443 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 Hamilton'...
4
votes
0answers
160 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
25 views

Twisted Kloosterman Sums

A twisted Kloosterman sum is a character sum of the form $$S(\chi, \psi, \eta)=\sum_{t\in (\mathbb{F}_q)^{\times}} \chi(t) \psi(t) \eta(t^{-1}).$$ Here, $\chi$ is a multiplitive character of $(\mathbb{...
3
votes
0answers
37 views

Show that over finite fields of characteristic two the finite group $C_3$ has no nontrivial representation

Let $G = \mathbb Z / 3\mathbb Z$ be the cyclic group of order three. I conjecture that every irreducible representation over a finite field of characteristic $2$ must be trivial. But how to prove this?...
3
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0answers
43 views

Additive character sum over intersection of additive and multiplicative subgroups of finite fields

Let $H$ be a multiplicative subgroup of the finite field $\mathbb{F}_q$ with $q$ elements, say $H$ is the subgroup of $d$-th powers, $d \mid q-1$. Let $L$ be a subspace of $\mathbb{F}_q$ over some ...
3
votes
0answers
63 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
votes
0answers
87 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
votes
0answers
76 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?
3
votes
0answers
106 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
votes
0answers
35 views

Orbits of the permutation action of a subgroup on its cosets

Consider a finite group $G$ and a subgroup $H \subseteq G$. There is a transitive group action of $G$ on the set of left cosets $gH$ by left multiplication, and the stabilizer of $gH$ is $gHg^{-1}$. ...
2
votes
0answers
32 views

Properties of non-abelian characters

I'm looking for some (short of) non-abelian generalization of the following result: Let $G$ be a finite abelian group and let $f$ be a function on $G$ with values in some field of characteristic ...
2
votes
0answers
38 views

If $A$ is a semisimple algebra, and $M_1 \ncong M_2$ as irreducible $A$-modules, why we have that every ideal of $M_1(A)$ is an ideal of $A$

Let $F$ be a field and $A$ be an $F$-vector space which is also a ring with $1$. Suppose for all $c \in F$ and $x,y \in A$ we have $$ (cx)y = c(xy) = x(cy) $$ Then $A$ is called an $F$-algebra. If ...
2
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0answers
22 views

Characters appearing naturally in arithmetic functions

Let $r_2(n)$ denote the number of representations of $n$ as a sum of 2 squares. It is well-known that $$r_2(n) = 4\sum_{d \mid n} \chi(d),$$ where $\chi$ is the non-principal character modulo $4$. ...
2
votes
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
votes
0answers
137 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
votes
0answers
75 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
votes
0answers
25 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
votes
0answers
76 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 $g \in H$?
2
votes
0answers
77 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 ...
1
vote
0answers
20 views

Character of an Induced Representation

In Fulton's book, page 34, it is stated that, To compute the character of $V = Ind\space W$, note that $g \in G$ maps $\sigma W $ to $g \sigma W$, so the trace is calculated from those cosets $\...
1
vote
0answers
37 views

Is character of a group representation the same as trace?

If so, why cannot the Klein group's character be zero? The group element of Klein group matrices can be traceless, right?
1
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0answers
21 views

Kloosterman Sums and K3 Surfaces

In R. Livne's Motivic Orthogonal Two-Dimensional Representations of $\mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, he provides a technique for computing the $\zeta$-function of the $K3$ surface $$X=\{...
1
vote
0answers
30 views

Is there a non-trivial character on any locally compact Abelian group?

Let $G$ be a locally compact Abelian group. Is there a non-trivial character on $G$? indeed, i want the proof of the existence of a non-trivial character on a locally compact Abelian group.
1
vote
0answers
24 views

Hints in proving the irreducibility of the permutation representation of $S_n$

Can anyone give me any hints of how to proceed: I have scoured this forum and exhausted all online notes. I am trying to prove that there is no room for an irreducible permutation representation, $\...
1
vote
0answers
35 views

A character on a subgroup could be written as a difference with some character on the whole group, help on argumentation

Let $G$ be a finite group, and $U \le G$ of odd order such that $N_G(U) = TU$ with $T = \langle t \rangle$ for some involution $t$ and assume that $U^g \ne U$ implies $U^g \cap U = 1$. Then I can not ...
1
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0answers
23 views

Characters of a Finite Abelian group

Let $C_m$ denote the cylic group of order $m$. Recall $\widehat{C}_m=\{ \psi_j: j\in C_m\},$ where $\psi_j$ is defined by \begin{align} \psi_j(k)=e\Big( \frac{2\pi \sqrt{-1}jk}{m}\Big). \end{align} ...
1
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0answers
35 views

All the characters on $S^1$

Would anyone please give me a hint to find all the characters on $S^1$? By characters I mean complex valued functions on $ S ^ 1 $ that their values have norm $1 $ and they satisfy: $f (x+y)=f (x) ...
1
vote
0answers
17 views

Relation between the sum of the values of a polynomial $f$ over a finite field, and the additive character sum with $f$ as the polynomial argument

Let $F$ be a finite field, let $f(T) \in F[T]$ and let $\psi$ be the canonical additive character of $F$. If $\sum_{x \in F}f(x) = 0$, what can we say of $\sum_{x \in F} \psi(f(x))$?
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0answers
52 views

Extending a homomorphism from a subgroup to whole group where the target is not a divisible group

I was reading this post of stack exchange. So in the question if the circle group is replaced by $\mu_{p-1}$ which is the group of $(p-1)^{th}$ root of unity and if the group $G/H$ is assumed to a ...
1
vote
0answers
44 views

Is there a stable probability distribution on the rational numbers?

Does there exist a (non-trivial) probability distribution on the rational numbers $$\sum_{r\in\mathbb{Q}}p_r=1$$ with $0\leq p_r$, which is stable, meaning that the sum of two i.i.d. random variables ...
1
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0answers
38 views

Faithful irreducible character of a group with exactly two minimal normal subgroups

Prove that any finite group with exactly two minimal normal subgroups has a faithful irreducible $\mathbb{C}$-character. What I have tried: Let $N_1$ and $N_2$ be two minimal normal subgroups of $G$....
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0answers
51 views

Counter-examples in representations of associative algebra

Here are something well-known. $V \cong W \Rightarrow \chi_V=\chi_W$ holds for finite representations of arbitrary associative algebra. But $ \chi_V=\chi_W \Rightarrow V \cong W $ is true only for ...
1
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0answers
61 views

Character representation of right regular representation as sum of irreducible characters

Let $G$ be a finite group acting on itself by the action $g \ast x := xg^{-1}$. Then this corresponds to an representation $\rho : G \to GL(L^2(G))$, where $L^2(G)$ denotes the space of function on $G$...
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0answers
46 views

Relation between finite abelian group and its set of linear characters

Let $G$ be a finite abelian group, and denote by $G^{\times}$ its set of linear characters, i.e. homomorphisms $\phi : G \to \mathbb C^{\times}$, where $\mathbb C^{\times}$ denotes the multplicative ...
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0answers
33 views

Kronecker symbol vs. Koblitz symbol

In Koblitz, Introduction to Elliptic Curves and Modular Forms on page 188 it is defined $$\left( \frac{-1}{j}\right)=0$$ in case $j$ is even. Apart from that definition $\left( \frac{c}{d}\right)$ is ...
1
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0answers
54 views

Gauss sum of a multiplication of two multiplicative characters of a finite field

Let $F$ be a finite field with $q$ elements and characteristic $p$. Let $E$ be a proper extension over $F$ of degree $n$. Let $\psi$ be the canonical additive character of $E$ defined by $\psi(x) = e^{...
1
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0answers
56 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 $\varphi(x)$ ...
1
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0answers
80 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 ...
1
vote
0answers
43 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
58 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 ...
1
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0answers
40 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: http://www.ms.uky.edu/~...