1
vote
1answer
15 views

Conjugacy classes and centralizers of a SmallGroup

What is the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)? Would someone be willing to provide the complete lists of conjugacy classes and centralizers of ...
2
votes
1answer
46 views

Which non-Abelian finite groups contain the two specific centralizers? - part II

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other ...
2
votes
0answers
44 views

Which finite groups contain the two specific centralizers?

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers both of these two groups: i. the elementary group $Z_2^4$, ...
1
vote
0answers
7 views

About representations and transformations under an $SU(n)$ Lie Group

I think my problem is that I misunderstand what "transforms under" really means. Let's take $SU(3)$, for the $\mathbf{3}$ with Dynkin indices $(1,0)$, a state transforms like : $ψ→gψ$. For the ...
3
votes
2answers
76 views

What are the finite groups with 8 or 16 conjugacy classes?

What are the list of finite groups with 8 or 16 conjugacy classes? I learned that dihedral groups $D_{10}$ and $D_{13}$ have 8 conjugacy classes. (Here the order of these groups are $|D_{10}|=20$, ...
0
votes
0answers
21 views

Conjugacy classes of right cosets

Is it true that all elements of a right coset $Hx$, for a subgroup $H$ of $G$, contained in a unique $G$-conjugacy class? I mean if $Hx=\lbrace{x_1,...,x_s}\rbrace$, then is it true that ...
1
vote
1answer
41 views

what is the conjugate of irreducible character of $G\wr S_n $?

Assume $G$ is any finite group and field as a complex field. The index set of irreducible representations of $G\wr S_n$ is set of all $k$-tuble of partitions ...
1
vote
0answers
25 views

Schur-Weyl duality from Double Commutant Theory

Let $V$ be a finite dim complex vector space. Then $V^{\otimes n}$ carries an action by $S_n$ by permuting factors $\sigma(\pi)(v_1\otimes...\otimes v_n)=v_{\pi^{-1}(1)}\otimes...\otimes ...
3
votes
0answers
29 views

~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
4
votes
1answer
96 views

How to prove that $\zeta*\zeta=\zeta$?

Let $F$ be a non-archimedean local field and $\mathcal{O}_F$ the ring of integers in $F$. Let $G_F=GL_2(F)$. Let $\pi_i$, $i=1,\ldots,n$,be non-equivalent finite dimensional irreducible ...
0
votes
0answers
33 views

Isaacs exercise 10.1 (Character Theory of Finite groups)

I need help on this problem. (10.1) Let $H \le G$, $\theta \in \operatorname{Irr}(H)$ and $\chi \in \operatorname{Irr} (G)$. Suppose $F \subseteq \mathbb{C}$. (a) If $\chi_H = \theta$, show that ...
5
votes
2answers
82 views

Characters and conjugacy classes [duplicate]

This comes up in reading David Speyer's answer to this question. Given a finite group $G$ and two non-conjugate elements $x, y,$ how does one construct a unitary representation $\rho$ of $G$ such that ...
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 ...
5
votes
1answer
78 views

Sums of products of average character values on cosets

Consider a finite group $G$, a subgroup $H\leq G$, and a transversal $G/H = \{t_1H, t_2H,\ldots,t_rH\}$. Given three characters $\chi_1,\chi_2$ and $\chi_3$ of $G$, I'd like to compute: $$ \sum_{i}^r ...
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 ...
7
votes
6answers
202 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 / ...
3
votes
1answer
54 views

Building invariants of non-fundamental $SU(2)$

Suppose you have two objects, $ \phi _i $ and $ \psi _j $ that form representations of $ SU(2) $. With both fields in the fundamental representation, I believe there is one invariant, \begin{equation} ...
0
votes
0answers
11 views

How to transform the following direct product of the group representations?

Let's have 4-vector $A_{\mu}$ which transforms as $\left(\frac{1}{2}, \frac{1}{2}\right)$ representation of the Lorentz group. So the product $A_{\mu}B_{\nu}$ refers to the direct product $$ \tag 1 ...
2
votes
2answers
63 views

Characters of a faithful irreducible Module for an element in the centre

Basically here is my questions. We have a character $\chi$ which is faithful and irreducible of a group $G$. we have an element $g$ which i needs to show belongs to the centre $Z(G)$, i.e. $gh=hg$ for ...
5
votes
1answer
125 views

How to show a representation is irreducible?

I have a professor who says that I should be able to show a representation is irreducible simply by looking at its trace (with other possible conditions), but after researching this for a while, I ...
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
0answers
41 views

Is $\widehat{\mathbb{R}/\mathbb{Z}} = \mathbb{Z}$? [duplicate]

Let $\widehat{\mathbb{R}/\mathbb{Z}}$ be the set of all homomorphisms from $\mathbb{R}/\mathbb{Z}$ to $\mathbb{C}$. Is $\widehat{\mathbb{R}/\mathbb{Z}} = \mathbb{Z}$? I think that ...
4
votes
1answer
103 views

Recognition of positive integral projections in a group algebra

Let $G$ be any finite Group and $e \in \mathbb{C}G$ be a central idempotent element which decomposes $\mathbb{C}G = R \times S$ into a direct product of rings $R = \mathbb{C}Ge$ and $S = ...
1
vote
1answer
89 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 ...
2
votes
1answer
41 views

Sum of representations and characters of the symmetric group

Hi I was wondering if I could have some help to go in the right direction. I want to show that $\displaystyle\sum\limits_{\sigma \in S_n} (sgn(\sigma)*\chi(\sigma)) =0$ where $sgn(\sigma)$ : $S_n ...
6
votes
0answers
91 views

A little bit of Intuition for Corepresentations from Representations

Hi folks I am trying to prove what I think should be a straightforward enough result but I am having to make a somewhat unnatural definition to do it. This unnatural definition is hinted at in a paper ...
1
vote
0answers
28 views

Irreducible representations of $SO(5)$

I am looking for irreducible representations of the group $SO(5)$ that can be described by a tensor of at most rank two. My own considerations have brought me to the conclusion that there is a ...
1
vote
1answer
26 views

Irreducible Representation and the center of a group

Hi I was wondering if someone could help me/hint along the right path. Let $\rho:G \rightarrow GL(V)$ be an irreducible representation. Let $Z(G)$ be the center of $G$. Show that if $a\in Z(G)$, then ...
2
votes
1answer
64 views

Standard represention of $S_3$

I am wondering how to extract the standard representation from the permutation representation? I want to obtain the permutation rep matrices $\Gamma((1,2)), \Gamma((1,3))$ and $\Gamma((1,3,2))$ in 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 ...
1
vote
1answer
57 views

Representation of $SL(3,\mathbb Z)$

I read the paper "Real-analytic actions of lattices", it says that any representation of any finite-index subgroup of $SL(3,\mathbb Z)$ into $GL(2,\mathbb R)$ has finite image, so how to prove it? ...
1
vote
1answer
63 views

Isaacs exercise 5.4 (Character Theory of Finite groups)

Any advice/hints how to prove the following statement? If $G$ is a finite group and $b(G)=\max\{\chi(1); \chi\in \mathrm{Irr}G\}$ is the maximal degree of irreducible characters and $H\leq G$, then ...
0
votes
0answers
18 views

How to decompose a representation of $so(n)$ into representations of a subalgebra

In some cases, it is possible. For instance the representation $16$ of $so(9)$ decomposes as $8_c+8_s$ of $so(8)$. Now I would like to do the same with representations of $so(8)$ into a sum of ...
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 ...
8
votes
1answer
120 views

$\chi(g)$ group character $\Rightarrow$ $\chi(g^m)$ group character

Let $G$ be a group of order $n$ and and $\gcd(m,n)=1$. Let $\chi:G\rightarrow\mathbb{C}$ be a class function and define $\chi^m\!: g\mapsto\chi(g^m)$. How can one show that $\chi^m$ is a character iff ...
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 ...
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
1answer
80 views

representation of abelian which is noncyclic

Let $G$ be a noncyclic abelian group acting by conjugation on an elementary abelian $p$-group $V$, where $p$ is a prime not dividing the order of $G$. (a) Prove that if $W$ is an irreducible ...
1
vote
1answer
32 views

Classification of $G$-modules

Suppose that I work only on vector spaces over $\mathbb C$. If I want to classify all $n$-dimensional modules over a finite group $G$, is it enough to choose a vector space $V$ with dimension $n$ and ...
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 ...
12
votes
3answers
177 views

Proving facts about groups with representation theory.

I was enrolled in a representation theory (of finite groups) course in the fall and throughout the class we focused on properties of representations and paradigms built around them. The whole time, I ...
1
vote
1answer
35 views

Why $\dim V^G = Trace(\varphi)$?

Let $G$ be a finite group. Let $V^G = \{v \in V: \pi(g)v = v, \forall g \in G \}$. Here $(\pi, V)$ is a representation of $G$. Let $\varphi = \frac{1}{|G|} \sum_{g \in G} \pi(g)$. How to show that ...
3
votes
1answer
57 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 ...
1
vote
0answers
33 views

What does “The Hilbert space carries a representation of […] group” means?

Often, in quantum mechanics I found the sentence "The Hilbert space carries a representation of $SU(2)$ group" (in particular when dealing with anglar momenta). Effectively, I know that this means ...
0
votes
1answer
42 views

if $g^k=e$ then $\chi(g)=\sum_j^n \zeta_k$

Let $G$ be a group. Let $g \in G$ and $g^k=e.$ Let $\chi$ be an $n$-dimensional character of the group $G.$ Let $\zeta_k$ be $k$-th root of unity. Prove that $\chi(g)$ is equal to sum of a ...
1
vote
0answers
20 views

Irreducible Decompositions of the Space of Sections of an Equivariant Vector Bundle

Let $G$ be a compact semi-simple Lie group, and $X = G/H$ a homogeneous space of $G$. If $V$ is a vector bundle over $X$, then is it true that $\Gamma^\infty(V)$ always has a decomposition into finite ...
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 ...