# Tagged Questions

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

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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### How to bound the order of a finite group under the following hypotheses?

In the book Character Theory Of Finite Groups by I.Martin Issacs as exercise 2.14 Let $G$ be a finite group with commutator subgroup $G'$. Let $H \subset G' \cap Z(G)$ be cyclic of order n and ...
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### Intuition on characters of topological groups

I am coming to the end of a series of lecture notes on representations of $S_n$ and $GL(V)$. Near the end, it attempts to introduce the notion of the "character of a topological group", but doesn't ...
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### 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) - ...
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### Notation: Character of a Finite Field

This is my first post on StackExchange. I had a quick question about notation (appearing in research literature) that I was unable to find by repeated searches, and I was hoping that someone would be ...
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### Two non-isomorphic groups with the same complex character table

Could you give me an example of two non-isomorphic groups with the same complex character table?
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### Few questions on Character of representation .

a) What does it mean to say that the Character of a representation is irreducible on its own? b) If Char($K$) is $0$ then kernel of character is a normal subgroup of G , why ?? c) Over a field of ...
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### Why unitary characters for the dual group in Pontryagin duality if $G$ is not compact?

In harmonic analysis, for any locally compact abelian group, one constructs the dual group as the group of homomorphisms into the unit circle with the compact open topology. In other words, unitary ...
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### Orthogonality relations of Characters

Could somebody please help me understand the jump from Proposition 10 to Proposition 11 in the following http://www.ms.uky.edu/~pkoester/research/charactersums.pdf Note: The orthogonality relations ...
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### How to use a character table to get the centre

I have been given a character table and I need to find from the table the centre of each character. I dont know how to do this. if someone could please explain how i can find the centre by looking at ...
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### Primitive Dirichlet Character

Let $\chi$ be the trivial Dirichlet character mod $N$. What is the primitive Dirichlet character associated to $\chi$? Is it just the character on $\mathbb{Z}$ that sends all integers to 1?
Sorry for any mistakes I make here, this is my first post here. I have a group $G$ which has an abelian subgroup $A<G$. I also know there is a irreducible character $\chi$ with the degree of ...