Questions tagged [characters]

Question 1

I've been reading Serre's Linear Representations of Finite Groups (GTM 42), but I’m a bit confused about the motivation and applications of two classical theorems: Artin’s Theorem: Each character of $...

Question 2

Elements of finite commutative group $G$ are colored in three colours. Number of elements of a fixed color in not greater than $|G|/2$. Let $A$ be the set of ordered same colored four elements $(x,y,...

Question 3

so the problem I'm interested in is to show that all sufficiently large finite fields contain an arithmetic progression of 9 distinct perfect squares. A professor in my department had some previous ...

Question 4

Let $G$ be a finite group and $S\subseteq G$. Let $\rho$ be an irreducible representation of $G$ and $\chi$ be its (irreducible) character. Define $$\rho(S):=\sum_{s\in S}\rho(s),$$ $$\chi(S):=\sum_{s\...

Question 5

I'm trying to determine all Dirichlet Characters modulo 15. I know how to determine Dirichlet Characters mod 3 and mod 5, and CRT tells us that $\mathbb Z_{15}^* \cong \mathbb Z_{3}^* \times \mathbb ...

Question 6

Let $G$ be a finite group and $H\le G$. Let $\lbrace{\rho_1,\rho_2,\ldots,\rho_t}\rbrace$ be the set of all inequivalent, irreducible representations of $G$. Consider the sum $$T:=\sum_{g\in G\...

Question 7

I've partially read Linear Representations of Finite Groups and Modular Representation Theory of Finite Groups, by Jean-Pierre Serre and Peter Schneider, respectively. However, both textbooks seem to ...

Question 8

Let $G$ be a finite group and let $\mathbb{Q}$ be the field of rational numbers and $\chi$ be an irreducible character of $G$. Moreover with $m_{\mathbb{Q}}(\chi)$ we denote the Schur index of $\chi$ ...

Question 9

Let $G$ be a finite group, $K$ be a field with characteristic zero, and $C$ be an algebraic closure of $K$. The algebras $CG$ and $KG$ are semisimple. Let $\chi$ be a $C$-character of $G$ ...

Question 10

In this book in $\S2.5$ (pg 21) the author states this following theorem without proof: Let $G$ be a locally compact abelian group and $\hat{G}$ its dual. Let $$ U_g, g\in G, $$ be a continuous ...

Question 11

Let $G$ be a finite group. (1) It is well-known that every ordinary representation of $G$ is defined over some cyclotomic field. The values of an orindary character, being a sum of roots of unity, is ...

Question 12

I am having trouble understanding a particular reduction in a proof of Manz & Wolf's book "Representations of Solvable groups". The setup is as follows. Let $N \leq G \unlhd \Gamma$ be ...

Question 13

Let $\chi^{j}(g):=Tr[D^{j}(g)]$ the character of $g\in SU(2)$ in the representation of spin $j$. I need to prove that if $\chi^{j}(g)=2j+1$ then $g=\pm I$ if $j$ is integer, and $g=+I$ if $j$ is half-...

Question 14

Suppose $G$ is a finite group, $F$ is a field of characteristic zero, and $L/F$ is a field extension. Let $\varphi$ be an irreducible representation of $G$ over $F$, with character $\chi$, and in $L$ ...

Question 15

I'm doing an old example sheet for a first course in representation theory. In part (a) of this question the full character table is found from knowing some of the rows, so we have the full character ...

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Questions tagged [characters]

Motivations of Artin's theorem and Brauer's Theorem on Characters

Inequality for a colored finite commutative group

Understanding a character calculation for arithmetic progressions of squares in finite fields

The sum of an irreducible representation over a subset of a finite group

Dirichlet Characters modulo 15

Eigenvalues of a sum of representation matrices.

How do I determine the maximal indecomposable characters based on the ordinary characters?

Schur index and Field of values: does there exists a list of related finite groups?

Character theory of finite groups and Schur group of fields with characteristic zero

What is the character on a LCA group corresponding to a group element?

ordinary vs. modulare representations [closed]

Proof of Lemma 0.17 of Manz & Wolf "Representations of Solvable Groups"

Specific values for SU(2) characters.

Dimension of decomposition of an irreducible representation on a non-algebraically closed field

Deducing generators of $PSL_2(\Bbb{F}_7)$ from its character table

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