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22 views

Molien series from character tableof group?

I was reading up on Molien series: http://en.wikipedia.org/wiki/Molien_series and I heard that you can compute this series from the character table of the group (in for instance the Atlas of finite ...
1
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0answers
27 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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2answers
100 views

Show that ${\theta}^G \in Irr(G)$ iff $I_G(\theta) = N$, where $N \unlhd G$ and $ \theta \in Irr(N)$.

Let $N \unlhd G$ and $ \theta \in Irr(N)$. Show that ${\theta}^G \in Irr(G)$ iff $I_G(\theta) = N$. Where $I_G(\theta)$ is the stabilizer of $\theta$ in the action of $G$ on $Irr(N)$ defined by ...
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1answer
73 views

$\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$ for faithful characters $\phi \in Irr(H)$ and $\theta \in Irr(K)$ .

Let $G = H \times K$. Let $\phi \in \operatorname{Irr}(H)$ and $\theta \in \operatorname{Irr}(K)$ be faithful. Show that $\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$. Problem 4.3 of ...
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3answers
89 views

Fourier analysis on finite abelian groups

Can someone help me show if $f$ is a character of a finite abelian group then for all $a\in G$, $$\sum_{[f]}f(a)\stackrel{}{=} \begin{cases} |G| & \text{if $ a$ is the identity} \\ 0 & ...
3
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0answers
80 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
66 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 ...
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2answers
82 views

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 ...
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1answer
92 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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34 views

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: ...
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1answer
29 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
70 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
50 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
91 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
94 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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179 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.
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1answer
136 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.
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1answer
119 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
269 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 ...
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3answers
307 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 ...
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1answer
73 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 ...
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1answer
102 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 ...
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1answer
51 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
155 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 ...
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1answer
178 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 ...
5
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1answer
169 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
76 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 ...
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1answer
245 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 ...
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1answer
79 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
461 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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1answer
148 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 ...
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1answer
101 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 ...
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1answer
122 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 ...
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2answers
109 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
61 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 ...
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1answer
104 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
181 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 ...
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1answer
127 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 ...
6
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1answer
144 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 ...
1
vote
1answer
150 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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1answer
157 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
67 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 ...
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1answer
225 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 ...
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1answer
256 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
197 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 $\hat G \overset{\sim}{=}\text{Hom}(G,T)$, ...
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1answer
170 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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2answers
79 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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292 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 ...
0
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1answer
100 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
85 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 ...