0
votes
1answer
23 views

Summing the traces of matrix powers

Let $G=\langle h\rangle_n\subset{\rm GL}(m,\mathbb{C})$ be a cyclic group of order $n$. I wonder if there is a good formula for calculating the sum $\sum_{g\in G}{\rm Tr}(g)$ via ${\rm Tr}(h)$, for ...
1
vote
1answer
32 views

Number of degree-$d$ representations of a perfect group?

It seems to be a standard result that the number of degree-1 representations of a group $G$ is equal to $[G : G']$ where $G'$ is the commutator subgroup (e.g. Lemma 6.2.7 in the 2012 textbook ...
2
votes
0answers
27 views

Rational representation of a 136 order group

Let G be the group of order 136 = 8 * 17 with presentation $$<x,y : y^{8}=x^{17}=1 \quad yxy^{-1}=x^{4} >$$ Find the simple summands of the group ring Q[G].
7
votes
6answers
195 views

The use of conjugacy class and centralizer?

This is more or less for a conceptual and better-understanding question in group theory and in representation theory: (1) Why are conjugacy class and centralizer important concepts in the group / ...
0
votes
0answers
36 views

Matrix coefficients of representations of finite groups

In finite-dimensional complex representations of finite groups, I would like to understand what I can learn by looking at a single matrix coefficient. In particular, I would like to look at "diagonal" ...
0
votes
1answer
25 views

When would a finite group be cosidered as Fp G -Module?

What conditions are necessary to think of a finite group as Fp G -Module ?
1
vote
1answer
87 views

Irreducible representations (over $\mathbb{C}$) of dihedral groups

Find number of complex irreps of the group $D_n$. Find dimension of the irreps. I know that The number of complex irreps of a finite group is equal to the number of conjugacy classes of the ...
1
vote
2answers
67 views

On the graph of induction-restriction for group-subgroup representations

Let $G$ be a finite group, and $H$ a subgroup. Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ and $H$ (up to isom.). Consider the graph ...
0
votes
1answer
46 views

character of a representation of the group $S_n$

Let $\Phi$ be a representation of the group $S_n$ in the space with basis $(e_1, \ldots, e_n)$ such that $\Phi(\sigma)e_i=e_{\sigma(i)}.$ Find character of $\Phi$ This is looks like a regular ...
1
vote
2answers
83 views

Let G be a finite group and V be the regular CG-Module. Find a submodule of V which is isomorphic to the trivial CG-Module

As in the question. I have read through different books and articles and they seem to set W=<$\sum_{g \in G}$ g> as a submodule. I can understand that this is unique, but I fail to see how this is ...
1
vote
1answer
39 views

Maximal subgroup and representations (principal part)

Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Question: Is $dim(V^H) \le ...
1
vote
1answer
26 views

Maximal subgroup and representations (dual part)

Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Let $g \in G$, $K = H \cap ...
2
votes
1answer
29 views

Existence of intermediate subgroups and representations theory.

Let $G$ be a finite group, $V$ an irreducible representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Suppose that $dim(V^H)>1$. Then ...
1
vote
1answer
35 views

A question about Character Degrees of G/N

Let $G$ be a finite group such that $p_1p_2\mid |G|$, where $p_1$ and $p_2$ are two primes. We know that there exists an irreducible character $\chi\in Irr(G)$ such that $p_1p_2\mid \chi(1)$. We know ...
1
vote
0answers
29 views

Permutation representations

I need a source for Permutation representations of general linear groups over finite fields. Can anyone introduce some sources?
4
votes
2answers
55 views

Representation of $S_4$

Is there a general method to work out all irreducible complex representation of a group? Describe all the the irreducible complex representation of the group $S_4$. $S_4$ is the symmetric group ...
0
votes
0answers
25 views

How to show that $Ind_K^G \mathbb{C}_\chi$ is natrally isomorphic to $\mathbb{C}[G]e_{\chi}$?

Let $K \subset G$ be finite groups and $\chi: K \to \mathbb{C}^*$ be a homomorphism. Let $\mathbb{C}_{\chi}$ be the corresponding 1-dimensional representation of $K$. Let $$ e_{\chi} = \frac{1}{|K|} ...
2
votes
2answers
82 views

About the converse of Maschke's theorem

The Maschke's theorem say that\ Let $G$ be a finite group and $F$ a field whose characteristic does not divide $\mid G \mid$. Then every $FG$-module is completely reducible (I'm using the notation of ...
0
votes
0answers
20 views

subgroup of character group

Let $B$ be a subgroup of a finite Abelian group $A$ and $$B^0=\{\alpha \in \hat{A}\mid \alpha(b)=1 \text{ as } b \in B\}$$ Prove a. $B^0$ is subgroup of $\hat{A}.$ And for any subgroup ...
1
vote
1answer
70 views

show that the following three statements are equivalent

Let $\Phi$ be an irreducible complex representation of the group $S_n$ and $\Phi'(\sigma)=\Phi(\sigma) \operatorname{sgn}(\sigma).$ $(\sigma \in S_n).$ Prove that 1) $\Phi'$ is ...
3
votes
1answer
74 views

Decomposing the tensor product representation of $S_3$ in terms of irreducibles

I have a theorem which says that: If $\rho_1,...\rho_n$ are a complete set of irreducible $K$-representations of $S_n$ then we have that: $V^{\otimes n}=\bigoplus_1^k(V^{\otimes n}_{\rho_i})$ as ...
1
vote
2answers
55 views

Prove that $\forall g \in G$ $ \exists$ an irreducible non-trivial character $\chi$ of the group $G$ such that $\chi(g)\neq 0$

Let $G$ be a non-trivial finite group. Prove that $\forall g \in G$ $ \exists$ an irreducible non-trivial character $\chi$ of the group $G$ such that $\chi(g)\neq 0$ This is my attempt so ...
1
vote
2answers
81 views

The average value of irreducible character of a non-trivial finite group

Let $G$ be a non-trivial finite group. Let $\chi$ be an irreducible character of the group $G$. Find $$\frac{1}{|G|}\sum_{g \in G} {\chi( g)}$$ I try. But I think that I am wrong. ...
3
votes
1answer
56 views

Irreducibility of complex 2-dimensional character of the group $ S_3 $

Let $\chi$ be 2-dimensional complex character of the group $ S_3 $. Prove that $\chi$ is irreducible character iff $\chi((123))=-1$. There is hint in my book: " Use the Maschke's theorem and ...
2
votes
2answers
63 views

Prove that a subset $\{a \in A \mid \chi (a) = 1\}$ is an $(n - 1)$-dimensional subspace of $V$.

Let $A$ be the additive group of n-dimensional vector space $V$ over the field $\mathbb{F}_p$. Let $\chi$ - a nontrivial irreducible complex character of A. Prove that a subset $\{a \in A \mid ...
4
votes
1answer
56 views

Reading a Character Table in Magma

These are the outputs of two character tables from Magma. The first is for $A_5$. The second is for $GL_3(2)$. What is the significance of the '$+$' and '$0$' symbols? I can produce more tables if ...
2
votes
1answer
68 views

representations of the dihedral group

Let $\rho_\epsilon(a)=\begin{bmatrix}\epsilon & 0\\0 & \epsilon^{-1}\end{bmatrix}$ and $\rho_\epsilon(b)=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$ I can prove that $\rho_\epsilon$ is ...
0
votes
1answer
22 views

irreps of $p^3$-group is faithful representation

Let $A$ be an irreps of $p^3$-group. Prove that $A$ is faithful representation. I know that $p^2$-group and $p$-group are abelian. I have to show, that $Ker A=e$ I have no idea how to start it
0
votes
1answer
38 views

decompose a real representation of a group $C_2\times C_2

Let $\Phi$ be a real representation of the group $C_2\times C_2=\{e,a\} \times \{e,b\}$ such as $ \Phi(a)=\begin{bmatrix}5 & -4 & 0\\6 & -5 & 0 \\0 & 0 & 1\end{bmatrix}$ and $ ...
3
votes
3answers
50 views

irrep of a non unit element in the finite group

Let $G$ be a finite group. Prove following statement. $\forall g \in G$ such that $g \neq e$ $ \exists$ a complex irrep $\rho$ such that $\rho(g) \neq E$ I have no idea how to start it. I can prove ...
0
votes
1answer
32 views

number of complex irreps

Can irreducible complex representation of a finite group of be exhausted by a) two 1-dimensional and two fifth-dimensional representations? b) five one-dimensional and 1 five-dimensional? My ...
2
votes
1answer
36 views

The dimension of space of morhisms as the number of orbits

All groups are finite, all representations are over $\mathbb{C}$ (just in this context, of course), $G$ is a group, $K,H\subset G$ - its subgroups. By $\mathbb{C}$ we denote the trivial representation ...
0
votes
0answers
53 views

Irreducible representations of groups of order $pq$: induction from normal subgroups

Consider a group $G$ of order $pq$ ($p$ and $q$ are distinct primes and also $p<q$). It is easy to show that the dimension of each irreducible representation of $G$ is $1$ or $p$. Also, it can be ...
2
votes
1answer
86 views

Irreducible representations of group of order $pq$

There is the problem to describe dimensions of irreducible representations of a group of order $pq$, where $p$ and $q$ a distinct primes. I am doing it as follows: Suppose $p>q$. Then by the Sylow ...
1
vote
0answers
65 views

Decomposition of representation of symmetric group

Let $V$, $\dim V=n-1$ be the standard representation of the symmetric group $S_n$ and let $V'= \langle x_1,x_2,\ldots,x_n \rangle$ be its natural representation. Then ( see. Fulton, Harris, 4.19) ...
1
vote
0answers
36 views

Each irreducible representation is a subrepresentation of induced one

I'm learning what irreducible representation is and need some examples. One of them is as follows: Let $G$ be any group and $H$ - it abelian subgroup. How to prove that each irreducible representation ...
0
votes
1answer
43 views

how to deduce that finite group $G$ has at most $|G|$ characters

Let $G$ be a finite abelian group. A character of $G$ is a group homomorphism $\chi: G\longrightarrow \mathbb{C}^{\times}$. I have proved by induction that for distinct $\chi_1,\cdots,\chi_r$, they ...
1
vote
0answers
36 views

Irreducible representations and abeliean subgroups

There is a theorem in representation theory which is surprising to me: the dimension of irreducible (complex) representation of finite group is not greater that $(G:H)$ - an index of abelian subgroup ...
0
votes
0answers
53 views

Irreducible and regular representation

I was working on representations of $S_3$ and thought about the following problems: The symmetric group $S_4$ acts on $\mathbb{C}^{4}$ by permuting coordinates. Decompose this representation into ...
0
votes
1answer
65 views

Maschke's theorem and the problem of the irreducible representation

Need to prove the following statement Let $\rho_k:<a>_n\rightarrow GL_2(R)$ is representation. $\rho_k(a)= \left( \begin{array}{cc} \cos {\frac{2 \pi k}{n}} & -\sin{\frac{2 \pi k}{n}} \\ ...
4
votes
0answers
78 views

on the simple group $M_{11}$

As we know, the simple group $M_{11}$ is a important group,it has order $7920$, how can we prove the simple group of order $7920$ is isomorphic to $M_{11}$ ?
0
votes
0answers
67 views

Representation theory

I am trying to study representation theory from the book Algebra by Artin. I came across the following problem which seemed interesting: Prove that the linear operator $T=\sum_{g\in C} \rho_{g}$ is G ...
1
vote
1answer
82 views

Representation theory and characters

I have been studying representation theory for 6 months now. I came across the following question in a graduate course example sheet. Let $\chi$ be the character of a representation $\rho$ of ...
1
vote
1answer
79 views

If every quotient by normal subgroup is abelian, then the irreducible representations are injective

The following is Problem 6 of January 2006 algebra qualifying exam from University of Maryland. See here for the problems. Let $G$ be a finite group. Suppose that for each normal subgroup $K\neq ...
4
votes
1answer
62 views

If an $H\le G$ has an irreducible representation of dimension $d$, then show $G$ has an irreducible representation of at least dimension $d$.

Let $H$ be a subgroup of a group $G$, and let $\rho :H\to GL(V)$ be an irreducible representation of dimension $d$. Prove that there is an irreducible representation of $G$ whose dimension is at least ...
3
votes
2answers
229 views

Existence of a G-invariant matrix

Let $\phi: G \to GL(\mathbb{R}^n)$ be a homomorphism, $G$ finite. Prove that there is a positive-definite matrix $M$ such that $\phi(g)^tM \phi(g) =M$ $\forall g \in G $. This looks really ...
1
vote
0answers
36 views

If an $H\leq G$ has an irreducible representation of dimension $d$, then show $G$ has an irreducible representation of atleast dimension $d$. [duplicate]

Let $H$ be a subgroup of a group $G$, and let $\rho:H\rightarrow GL(V )$ be an irreducible representation of dimension $d$. Prove that there is an irreducible representation of $G$ whose dimension is ...
0
votes
1answer
65 views

The characters of the irreducible representations of a group

Let $G$ be finite group of order $n$ with $s$ conjugacy classes and let $X_1, . . . ,X_s$ be the characters of the irreducible representations of $G$ over $C$. Prove that the sum $ \sum_{g\in G} X_i(g) ...
0
votes
0answers
37 views

how to see that Group Orthogonality Theorem is correct

For the group representation theory, how did people discover the Group Orthogonality Theorem? ...
1
vote
1answer
40 views

If $G$ is solvable, is it true that for any $m,n\in\operatorname{cd}(G)$, there exists a prime $p$ such that $p\mid m,n$?

Let $G$ be a finite group and let $\operatorname{cd}(G)$ be the set of degrees of irreducible characters of $G$. It is known that if for any $m,n \in \operatorname{cd}(G) \setminus \{1\}$, there ...