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If $\chi\in\operatorname{Irr}(G)$, $N\unlhd G$, and $\langle\chi_{N},1_{N}\rangle\ne 0$, then $N\subset \operatorname{Ker}(\chi)$.
Let $N \unlhd G$ and $\chi \in \operatorname{Irr}(G)$. Suppose that $\langle\chi_{N},1_{N}\rangle\ne 0$. Show that $N\subset \operatorname{Ker}(\chi)$.
Hint: Use that, for any character ...
1
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1answer
15 views
Irreducible characters form orthonormal basis of set of class functions
I am reading Serre's book (Linear Representations of Finite Groups). Theorem 6 in chapter 2 says that the irreducible characters $\chi_1,\dotsc,\chi_h$ of a finite group $G$ form an orthonormal basis ...
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2answers
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Representations of $SO_3(\mathbb{R})$ from $SU_2(\mathbb{C})$
Define $V_n$ as the linear space of all homogeneous polynomials of degree $n$ in two variables $x$ and $y$. Define also the representation $\rho_n$ of $SL_2(\Bbb{C})$ on $V_n$ by:
...
1
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1answer
24 views
$SU_2(\mathbb{C})$ and the characters
i can prove that the irreducible characters $\chi_n$ of $SU_2(\mathbb{C})$ are equal to:
$$\chi_n(e^{i\phi})=\frac{\sin((n+1)\phi)}{\sin(\phi)}$$
If i want to give the dimension of the representation ...
0
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1answer
26 views
Standard representation of $O_h$ in $\mathbb{R}^3$
I want to give the standard representation of the complete octaedergroup $O_h$ in $\mathbb{R}^3$.
To which group is $O_h$ isomorphic, and how to obtain a standard representation of the group?
What ...
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0answers
20 views
Formula for checking the probability of a character appearing multiple times consecutively in an encrypted string
I am a young CS student with a specific interest in Cryptography, but I am relatively new to the field. Yesterday I came across a question I could not answer by myself, so I thought I'd ask some more ...
3
votes
1answer
30 views
Is the quadratic character, unique multiplicative character over $\mathbb Z_{p^n}$, for odd $p$?
Let $p$ be odd and $\mathbb Z_{p^n}$ denote the ring of integers modulo $p^n$. Let the quadratic character, $\eta$, be the function defined on $\mathbb Z_{p^n}^*$ (multiplicative group of $\mathbb ...
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1answer
52 views
Martin Isaacs's exercise 3.7 (character theory of finite groups)
I would need some help with this exercise:
Let $\chi\in{Irr(G)}$ be faithful, and suppose $\chi(1)=p^a$ for some prime p.
Let $P\in{Syl_{p}(G)}$, and suppose that $C_{G}(P)\nsubseteq{P}$. Show that ...
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0answers
54 views
Martin Isaacs's exercise 3.4 (character theory of finite groups)
I need some help with this:
Let $G$ be a simple group and suppose $\chi\in{Irr(G)}$ with $\chi(1)=p$, a prime.
Show that a Sylow $p-$subgroup of G has order p.
Thanks a lot in advance.
1
vote
1answer
47 views
Martin Isaacs's exercise 3.6 (character theory of finite groups)
I'm trying to solve this exercise, can anyone help me?
Let $G$ be a p-group, and suppose $\chi\in{Irr(G)}$. Show that $\chi(1)^2$ divides $|G:Z(\chi)|$
Thanks a lot.
1
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1answer
61 views
Martin Isaacs's exercise 3.5 (character theory of finite groups)
I need some help with this exercise:
Suppose $A\subseteq{G}$ is abelian, and $|G:A|$ is a prime power. Show that $G'\lt{G}$
Thank you very much in advance.
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5answers
206 views
Applications of Character Theory
Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, , Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character ...
2
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2answers
113 views
Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$, where $G$ is abelian and $\mathbb{T}$ is the circle group.
Suppose $G$ is an abelian group and $a\in G$ and
$$f:\left<a \right>\to\Bbb T$$
is a homomorphism. Can $f$ be extended to a homomorphism on $G$:
$$g:G\to \Bbb T$$
?
$\Bbb T$ is the circle ...
5
votes
1answer
55 views
characters of a $C^*$-algebra
I have read that a state $\rho$ on a unital $C^*$-algebra $A$ is a character (i.e. multiplicative) if and only if, for all unitary $u\in A$, $|\rho(u)| = 1$. Is there an easy proof, or can someone ...
3
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1answer
53 views
Some questions about representation theory in the modular case
I'm working on a paper which uses representation theory in order to compute some characters and deduce arithmetical statements about certain field extensions.
Let $\Delta$ be a group of order prime ...
1
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1answer
42 views
Iwaniec Kowalski Notation
On page 532 of the book analytic number theory by Iwaniec and Kowalski, the following notation is used:
$C^{~\infty}$ and $\tau(n,\chi)$.
Could anyone tell me what these represent? (the former is ...
3
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1answer
74 views
Exercise 2.15 M.Isaacs' Character theory of finite groups
I'm beggining to study character theory, and i'm doing some problems from Isaacs' Character theory book.
I would need some help with this one:
(2.15): Let $\chi\in \operatorname{Irr}(G)$ be ...
1
vote
1answer
78 views
Exercise 2.8 M.Isaacs' Character theory of finite groups
I'm a starter at character theory. I'm trying to do this exercise:
(2.8) Let $\chi$ be a faithful character of a group $G$. Show that $H\subseteq G $ is abelian if and only if every irreducible ...
4
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1answer
82 views
Character theory exercises [closed]
I'm doing the exercises from chapter 2 of M.Isaacs' Character theory of finite groups, and I'm having problems with some of them.
In particular, I would need help with these ones. Thank you very much ...
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0answers
47 views
Characters of the symmetric group corresponding to partitions into two parts
Let $n\in\mathbb N$ be a natural number and $\lambda=(a,b)\vdash n$ a partition of $n$ into two parts, i.e. $a\ge b$ and $a+b=n$. In this special case, is there a simple description of the character ...
3
votes
1answer
54 views
Character of a permutation representation
I am self-studying representation theory, and I would like to make sure my proofs are complete.
Following Serre's notation, let $X$ be a finite set, and let $G$ be a group that acts on $X$. Let ...
2
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1answer
58 views
Vantage point of character theory
I am not sure whether I can frame my question properly, or whether at this point my understandings permit me to comprehend the perspectives of the answers to come, but somehow I find it pretty amazing ...
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2answers
372 views
Formula for number of solutions to $x^4+y^4=1$, from Ireland and Rosen #8.18.
There is a sequence of three exercise in Ireland and Rosen's Introduction to Modern Number Theory, Chapter 8, page 106. I can do the first two, but can't finish the third. I can include the proofs to ...
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vote
1answer
95 views
Irreducible Representations and Maschke's Theorem
$\mathscr L(V,W)^G = \mathscr L(V_1,W)^G \oplus...\oplus\mathscr
L(V_k,W)^G$ and for each irreducible represtation of G on a space W,
the number of $j\in (1,...,k)$ for which $V_j \cong W$ is ...
1
vote
1answer
79 views
Proving $g(\chi\rho)^6=(-1)^{(p-1)/2}p(\overline{\chi(2)J(\chi,\rho)})^4$, from Ireland and Rosen.
Suppose $p\equiv 1\pmod{3}$, $\chi$ is a cubic character, and $\rho$ is the quadratic character on $F_p$.
If $\chi\rho$ is a character of order $6$, why does the Guass sum ...
4
votes
1answer
74 views
Formula for evaluation of character on a transposition
Let $\lambda\vdash n$ be a partition of $n\in\mathbb N$ and $\chi=\chi_\lambda$ the corresponding irreducible character of the symmetric group $S_n$. Denote by $\lambda^t$ be the transpose of ...
2
votes
2answers
63 views
Are “FG-module characters” sometimes used, too?
I am only beginning my study of group representations and characters. So far I have already encountered the regular group algebra $FG$. Although in an FG-module the multiplication is only defined for ...
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1answer
45 views
Characters of subrepresentation
Given an algebra $A$ with finite dimensional representation $V$ with action $\rho$, I want to prove the following statement:
If $W\subset V$ are finite dimensional representations of A, then ...
1
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1answer
83 views
Dirichlet Characters modulo $260$
I want to count the number of Dirichlet characters with given properties:
Number of Dirichlet characters modulo $260$
Number of quadratic Dirichlet characters modulo $260$
Number of primitive ...
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0answers
109 views
Where does this elliptic curve come from?
In Zeta functions of an infinite family of K3 surfaces, Scott Alhgren, Ken Ono and David Penniston compute the zeta functions (given a good reduction restriction mentioned below) of the K3 surfaces ...
3
votes
1answer
81 views
Is there some relation between characters in representation theory and multiplicative characters?
A character of a group representation is obtained by taking trace of each matrix in this representation.
The word character is often used in the sense that it is a homomorphism from a group to ...
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1answer
97 views
Some irreducible characters of the Symmetric group $S_n$
I want to have characters of some irreducible $S_n$-modules corresponding to certain partitions $\lambda$ of $n$, the computations using Frobenius formula get complicated and I am unable to find in ...
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1answer
88 views
Number of permutations given a sequence of 5 letters that are offset from 1-9
If I have a random sequence of letters "AOKNG", and I'd like to find how many permutations of this can be formed given a character offset from 1-9.
So, offset the first character "A" 9 times would ...
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vote
1answer
108 views
Gauss sum for primitive real Dirichlet character
If $\chi$ is a real non-trivial primitive character modulo $q\ge 1$, then how could one show that
$$\sum_{n\in \mathbb Z/q\mathbb Z} \chi(n)e\left(\frac nq\right) = \sum_{n\in \mathbb Z/q\mathbb Z} ...
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0answers
38 views
Isomorphism of annihilator of a subgroup in the context of group characters
I am trying to learn about characters of finite abelian groups. A character is a homomorphism from a finite abelian group $G$ into the multiplicative group of complex numbers of absolute value 1. In ...
2
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1answer
74 views
Sum of squares of dimensions of irreducible characters.
For anyone familiar with Artin's Algebra book, I just worked through the proof of the following theorem, which can be seen here:
(5.9) Theorem Let $G$ be a group of order $N$, let ...
0
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1answer
215 views
Character table of $U_{16}$.
Find the character table of $U_{16}$.
Could you give me a hint or a start?
Thank you.
6
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1answer
135 views
How does Pontryagin duality fit into the general cohomology theory framework?
Pontryagin duality implies the isomorphic relation of the function space $C(G)$ on a locally compact group $G$ to the function space on it's dual group $\hat G \overset{\sim}{=}\text{Hom}(G,T)$, ...
4
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1answer
80 views
Generalizing Artin's theorem on independence of characters
Artin's theorem says that for any field $K$ and any (semi) group $G$, the set of homomorphisms from $G$ into the multiplicative group $K^*$ is linearly independent over $K$.
Can this theorem be ...
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0answers
55 views
Character of $\text{Hom}_{\mathbb{C}}(\mathbb{C}G,\mathbb{C})$
I've been given the following two questions, and for both I'm really unsure what to do (I'm a beginner with the theory of characters).
Let $G$ be a finite group, $\mathbb{C}G$ its group algebra over ...
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209 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 ...
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1answer
61 views
Proving that a Dirichlet character $\chi$ must be the Kronecker symbol $\chi_D = \left ( \frac{D}{\cdot} \right )$
I'm stuck trying to prove that a particular Dirichlet Character $\chi$ must be the Kronecker symbol $\chi_D = \left ( \frac{D}{\cdot}\right )$. The context is the following.
Let $d$ be a square-free ...
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2answers
58 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
142 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
95 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 ...
6
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1answer
175 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
220 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
52 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
302 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
56 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 ...
