Tagged Questions

4
votes
1answer
45 views

Group homomorphisms into a field

Let $G$ be a finite group, and let $k$ be a field, which should be algebraically closed, I think. How to describe all homomorphisms $G\rightarrow k^*$ (i.e. one-dimensional representations: ...
1
vote
1answer
26 views

Symmetry of Plancherel measure (for $S_n$)

For each $n \geq 1$ consider the reverse lexicographical order on the set $P(n)$ of partitions of $n$. Example for $n=7$: $$ \begin{pmatrix} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 ...
2
votes
2answers
99 views

Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$.

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 ...
3
votes
1answer
27 views

Different induced representations - same simples?

is the following case possible: $\pi_1, \pi_2$ two simple representations of the same subgroup over an arbitrary field. $\operatorname{Ind}(\pi_1)$ and $\operatorname{Ind}(\pi_2)$ have equal ...
11
votes
1answer
185 views

What does the group ring $\mathbb{Z}[G]$ of a finite group know about $G$?

The group algebra $k[G]$ of a finite group $G$ over a field $k$ knows little about $G$ most of the time; if $k$ has characteristic prime to $|G|$ and contains every $|G|^{th}$ root of unity, then ...
7
votes
0answers
64 views

Clifford theory and induction

in the answer to this post there was the statement that a representation $\vartheta$ of a subgroup $\langle z\rangle$ can extend to a representation of the whole group $D_{2n}$. If I start the other ...
2
votes
2answers
72 views

Isomorphism of faithful representations

Let $G$ be a group and $f,g: G \rightarrow GL(V)$ be two faithful representations over some field $K$ with $f:x\mapsto f(x)$ and $g:x \mapsto f(x^{-1})$. I would like to find out if $f$ and $g$ are ...
13
votes
1answer
196 views

Writing a group element as $ghg^{-1} h^{-1}$ and as $g^2 h^2$

I recently read the elegant paper Generalized Frobenius Schur Numbers, by Bump and Ginzburg, which I learned about here. The results in this paper imply the following: Let $G$ be a finite group ...
3
votes
0answers
60 views

Mackey criterion for normal subgroups

I am wondering how Mackey's criterion works for arbitrary fields. If there is a representation $\vartheta$ of a subgroup of $G$, then the induced representation $\operatorname{Ind}(\vartheta)$ is ...
3
votes
0answers
55 views

Induction from trivial representation and number of irreducibles

Let $G$ be a finite group, $S$ a subgroup and $K$ a field, whose characteristic does not divide the group order. Let $\pi$ be the trivial representation of $S$. Are there criteria in how many ...
4
votes
0answers
42 views

Induction from normal subgroup, problem with degrees

Suppose $K$ is an arbitrary field, $G$ a finite group and $N$ a normal subgroup. If one know all the irreducible representations of $N$ and then form the induced representations, one can use Mackey ...
1
vote
2answers
68 views

irreducible representation of non-abelian p-group

Can someone help with the following problem? Let $G$ be a non-abelian group of prime-power order $p^n$ and $E$ be an irreducible $G$-space over $\mathbb{C}$ giving a faithful representation of $G$. ...
4
votes
1answer
81 views

Intuition behind Maschke's theorem

I'm an undergraduate learning about group representations and Young tableaux, and have came across Maschke's theorem stating; If $G$ is a finite group and $F$ is a field who's characteristic does ...
1
vote
1answer
27 views

Definition of induced representation by tensor product

Suppose there is a finite group $G$ with a subgroup $H$ an some field $K$. If one has a representation of the subgroup, one can construct the induced representation $\rho:=Ind_H^G$ according to ...
3
votes
1answer
62 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
65 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
votes
1answer
80 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 ...
6
votes
2answers
103 views

Do group rings appear outside of representation theory?

I am particularly concerned with finite groups. I have seen group rings used in the fundamentals of representation theory as the dual notion to representations. I haven't ever seen them anywhere ...
3
votes
1answer
93 views

Irreducible subgroups of $\text{GL}(2,p)$

I just wonder any sources that provides the list of all minimal irreducible subgroups of $\text{GL}(2,p)$. Here the term "minimal" means there is no proper subgroup that is irreducible. A list of ...
5
votes
0answers
50 views

Subgroup Structure of $\mathrm{SL}(2, p^2)$, and Its Irreducible Characters

I am taking a course in representation theory of finite groups,and somehow I ended up getting assigned to write a report on subgroup structure and irreducible characters of $\mathrm{SL}(2,7)$ and ...
4
votes
1answer
44 views

Irreducible Representations of Finite Coxeter Groups

The Coxeter group is defined as $$S = \langle s_i : s_i^2 = (s_i s_j)^{m_{ij}} = 1 \rangle $$ Does it have an irreducible representation of dimension >2 for $S$ finite? Is there a reference on ...
3
votes
3answers
41 views

$D_6$ as permutation group

I am trying to solve some exercises for a course in representation theory. We are studying finite groups and I have an exercise about the dihedral groups $D_n = ...
1
vote
1answer
60 views

Relation between a representative of a conjugacy class and corresponding irreducible character value

Is there a relation between the representative order of a conjugacy class and the corresponding irreducible character value? Thanks in advance.
1
vote
1answer
90 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 ...
3
votes
0answers
52 views

Induced representation and degrees

Let $G$ be a finite group, $S$ a subgroup, $K$ an arbitrary (not necessarily algebraically closed) field which characteristic does not divide the group order. 1) Let $Ind_S^G(f)$ have the degree $n$ ...
1
vote
0answers
41 views

elementwise conjugate but not conjugate homomorphisms

Does there exist a finite group $G$ and two group homomorphisms $\rho_1,\rho_2:G\to PGL(2,\mathbb{C})$ such that (i) For all $g\in G$ there exists $M=M(g)\in PGL(2,\mathbb{C})$ such that ...
1
vote
1answer
58 views

If $f$ is an irreducible representation, what can we say about $g:x\mapsto f(x^{-1})$?

Let $G$ be a finite group, $K$ a field which characteristic does not divide the group order and $V$ a $K$ vector space. Suppose there is an irreducible representation $f: G \rightarrow GL(V)$, $x ...
3
votes
4answers
97 views

What is a minimal polynomial of a group element, and why would we care if it was quadratic?

EDIT: the $p$-stable definition I give below is incorrect. I have included the correct definition as an answer to this question. I am trying to understand the definition of a p-stable group. The ...
4
votes
0answers
49 views

What is the centralizer of the Young symmetrizer?

I have read a lot about idempotents, several important facts were about central idempotents. Now, the Young symmetrizer is a constant away from an idempotent, but I don't think it's central. ...
0
votes
1answer
33 views

Prime divisors of irreducible degrees

I would like to ask that : assume that $N$ is a normal subgroup of a finite group $G$, if $p$ is a prime divisor of $\chi(1)$ for some $\chi\in Irr(N)$, does it imply that $p$ divides $\varphi(1)$ for ...
6
votes
0answers
75 views

Induced representation, Ind(Res(U))

I am reading a book of Fulton and Harris "Representation theory, a first course". Now it's all about representation theory of finite groups, and there is one exercise, which I can't solve: If $U$ is ...
5
votes
1answer
100 views

Existence of a finite group having a certain kind of 2-dimensional representation.

Is there a finite group $G$, an element $c$ of order 2 in $G$, and an irreducible 2-dimensional complex representation $\rho$ of $G$ such that all the following are true: 1) $\rho(c)$ has trace zero ...
0
votes
1answer
48 views

Simple faithful $KA$-modules over finite fields

Let $V$ be a vector space of dimension $n$ over finite field $K=\mathbb{F}_q$ and let $A$ be an abelian group such that $V$ is simple, faithful $KA$-module. Then $A$ is cyclic. Moreover, for every ...
1
vote
1answer
70 views

Isomorphism types of simple $KG$-modules with $G=S_3$

I am trying an exercises on determining all isomorphism types of simple $KG$-modules with $G=S_3$, the symmetric group of degree $3$. If $K$ is algebraically closed then we can use the following ...
4
votes
1answer
78 views

Connection between $GL(\mathbb{Z}_p)$ and $GL(\mathbb{F}_p)$

Suppose there is a finite group $G$. Is there a connection between indecomposable representations over $\mathbb{Z}_p$ and $\mathbb{Z}/p \mathbb{Z}$. I know what to do if $G$ is cyclic. But if not? If ...
6
votes
1answer
77 views

Do field automorphisms of a character imply outer automorphisms of the group?

Apologies for the imprecise wording of the title. In studying the basic representation theory of finite groups, I've been struck by a pair of phenomena present in every example I've worked with but ...
2
votes
2answers
100 views

1 dimensional representations of $S_n$

I want to show that $S_n$ has only two 1 dimensional represnetations. mainly the trivial and sign represnetations. Where I assumed that our Field we're working on is with characteristic $\neq 2$. ...
0
votes
0answers
269 views

How good is “International Journal of Group Theory” [closed]

I have found that there is a new journal in Group Theory, that is International Journal of Group Theory http://www.theoryofgroups.ir/ . However, I dont know how good it is compared to others in ...
4
votes
2answers
136 views

Looking for texts in representation theory

I recently finished a course in representation theory, and while I learned a lot from it, I know that there's a lot more in the subject that I missed. For the course we used Fulton and Harris as a ...
1
vote
0answers
54 views

Exercise 4.2, I. Martin Isaacs' Character Theory

Problem : Let $\mathcal{K}$ be a conjugacy class in a finite group $G$ which is not contained in any proper normal subgroup of $G$. Let $K$ be the corresponding class sum in $\mathbb{C}[G]$ and let ...
2
votes
0answers
35 views

Product of two squares in finite groups

Let $G$ be a finite group and $g$ an element of order $n$ in $G$. Assume that $g$ is a product of two squares. Moreover, assume that $n$ and $k$ are coprime. Prove that $g^k$ is also a product of two ...
2
votes
0answers
98 views

Problem 4.3, I. Martin Isaacs' Character Theory

Let $G=H\times K$ be the direct product of finite groups. Let $\varphi\in Irr(H)$ and $\eta\in Irr(K)$ be faithful. Show that $\varphi\times\eta$ is faithful if and only if $(|Z(H)|,|Z(K)|)=1$. Here, ...
1
vote
1answer
103 views

Multiplicity of a completely reducible representation in another irreducible representation.

I have got the next question that I am pondering the answer to. Let $\tau$ be a completely reducible representation of finite dimension of a group $G$, and let $\pi$ be another irreducible ...
2
votes
2answers
50 views

Checking that $\rho$ is a representation

Let $z$ be a generator of the cyclic group $\mathbb{Z}_3 = \{ 1,z,z^2 \}$. Prove that a representation $\rho$ of $\mathbb{Z}_3$ in the $2$-dimensional complex vector space $\mathbb{C}^2$ can be ...
1
vote
1answer
51 views

Isomorphisms of linear representations of finite groups

Let $G$ be a finite group with representations $\rho_1, \rho_2:G\rightarrow GL(V)$. According to the definition of representation isomorphisms, $\rho_1$ and $\rho_2$ are isomorphic if there exists a ...
2
votes
1answer
69 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
votes
1answer
214 views

Character table of $U_{16}$.

Find the character table of $U_{16}$. Could you give me a hint or a start? Thank you.
4
votes
1answer
97 views

Exaples: Representations over finite rings and Maschke's theorem

Is there a possibility to get the simple $R[G]$ modules, if $R$ is the ring $\mathbb{Z}/n\mathbb{Z}$ and $G$ a finite group and $ord(G)$ and $n$ are relatively prime? For which groups would their ...
3
votes
2answers
115 views

Indecomposable modules

Suppose $q$ is a prime $(\neq 2)$ and $G$ a finite group, for example the cyclic group $C_p$. Is there a way to determine all the $\textbf{indecomposable}$ $\mathbb{F}_{q^n}[G]$ modules for some $n\in ...
2
votes
1answer
112 views

A conjugacy class $C$ is rational iff $c^n\in C$ whenever $c\in C$ and $n$ is coprime to $|c|$.

Let $C$ be a conjugacy class of the finite group $G$. Say that $C$ is rational if for each character $\chi: G \rightarrow \mathbb C$ of $G$, for each $c\in C$, we have $\chi(\sigma) \in \mathbb Q$. I ...

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