-1
votes
1answer
56 views

Elements whose orders are multiple of $p$ [on hold]

Let $G$ be a non-solvable group, $N$ an abelian minimal normal $p$-subgroup of order $p^r$ with $p\notin \pi(G/N)$, $N=C_G(N)$ and $K=G/N\cong A_5$. By these assumption we can conclude that $G$ has ...
3
votes
1answer
65 views

Computing values of centralizers in a non-solvable group with a given property

A finite group G satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n−1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...
3
votes
0answers
65 views

The order of the representative elements of conjugacy classes

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ ...
2
votes
1answer
24 views

Question in Fulton and Harris regarding induced representation.

I'm confused by the following paragraph: I don't see why $g\cdot W$ depends only on the left coset $gH$. What does he mean precisely by that? Why is it true that $gh\cdot W = g\cdot(h\cdot W) = ...
1
vote
1answer
30 views

Adjoint Lie algebra homomorphism

I have a problem deriving the adjoint action $ad_X(Y)=XY-YX$ from the adjoint transformation of the group on the Lie algebra. Background: The adjoint action of the Lie algebra on itself is given by ...
0
votes
0answers
27 views

Formal proof of Clebsch Gordon sum

physicist here. When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be ...
10
votes
1answer
136 views

Why is the Plancherel measure interesting?

One can average a class function $f:G\to\Bbb C$ for a finite group $G$ by interpreting $f$ as a complex-valued function on the space ${\rm cl}(G)$ of conjugacy classes and computing the expectation ...
0
votes
0answers
34 views

Action of an irreducible representation

Suppose $\rho:G\to GL(V)$ is a finite-dimensional irreducible representation of $G$ a finite group. Are there properties of the group $\rho(G)\leq GL(V)$ or of the induced $G$-action on $V$ that ...
2
votes
1answer
42 views

Decomposing direct product of irreps

I know characters of two 2-dimentional irreps (U and V) of a group with 6 conjugate classes. The characters are: $\begin{pmatrix} 2&-1&-1&2&0&0\end{pmatrix}$ and $\begin{pmatrix} ...
0
votes
0answers
21 views

Real or complex representation

How can one know for a given algebra $\frak{g}$ if a specific representation is real or complex? For example if $\frak{g}=so(10)$ how can one know that the representation $\underline{16}$ is complex? ...
0
votes
0answers
19 views

Ordinary irreducible representations of semi-direct products

What is the best source for learning about constructing irreducible representations of semi-direct product $G=N \rtimes_\phi H$ from irreducible representations of $H$ and $N$ over field of complex ...
0
votes
1answer
19 views

Characters of a unipotent group.

Let $\def\C{\mathbb{C}}T = (\C^*)^n$. A character of $T$ is defined to be a homomorphism from $T$ to $\C^*$. The characters of $T$ is of the form $f(t_1,\ldots,t_n)=t_1^{a_1}\cdots t_n^{a_n}$ for some ...
4
votes
1answer
28 views

Characters on a torus.

Let $T = (\mathbb{C}^*)^n$. It is said that the characters on $T$ must be of the form $f(t_1,\ldots,t_n)=t_1^{a_1}\cdots t_n^{a_n}$ for some $a_1,\ldots, a_n \in \mathbb{Z}$. Is it possible that $a_i ...
0
votes
1answer
22 views

Uniqueness of a map.

Let $f: \mathbb{Z}^n \to \mathbb{C}^*$ be a homomorphism. Where $(\mathbb{Z}^n,+)$ is considered as an additive group and $(\mathbb{C}^*$ is considered as an multiplicative group. Fix $b_1,\ldots, b_n ...
2
votes
1answer
34 views

Conjugacy class name of the product in ATLAS

Well, I'm trying to read "ATLAS of Finite Groups". To be more precise, I'm interested in character tables of some Weyl groups. Is it possible to determine the conjugacy class name of the product of ...
2
votes
1answer
32 views

Understanding the structure of a module over a group algebra

Suppose one has a permutation group $G$ acting on the set $[n] = \{1, 2, \ldots, n\}$, which extends naturally for any field $F$ to a $FG$-module structure on the set $F[n]^k$ of formal $F$-linear ...
3
votes
1answer
54 views

the representation of a free group

A group $G$ is generated by $\begin{pmatrix}1&n\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\n&1\end{pmatrix}$, then we know $G\cong \mathbb{F}_2$ which is a free group generated by two ...
7
votes
0answers
47 views

The division algebras arising in the Wedderburn decomposition of a finite group modulo its radical in characteristic $p$

The following question is probably straightforward for those who know. However, I am used to working either over splitting fields or in characteristic zero. Question. Let $G$ be a finite ...
3
votes
1answer
97 views

Infinite abelian group counterexample [duplicate]

A finite group $G$ is abelian iff all its irreducible representation $\rho$ have dimension 1. I'm looking for a counter-example when $G$ is an infinite group. Are there any? EDIT We're dealing ...
2
votes
1answer
18 views

Show that the map defined by $\sigma(g)$=$p(g^{-1})$ is a representation.

Suppose G is abelian. Show that the map $\sigma : G -> GL(n,F)$ defined by $\sigma(g)=p(g^{-1})$ for all g in G is a representation of G. I think I have done this I would just like to check my ...
0
votes
0answers
28 views

representation of $D4$

Let the representations of $D4$ : $ T: D4 \to GL(2,C)$ , $T(a)=\begin{bmatrix} 0 & -1 \\ 1 & 0 \\ \end{bmatrix}$ , $ T(b)=\begin{bmatrix} 0 & 1 \\ ...
0
votes
0answers
23 views

Show that 2 representations are not equivalent and find all the irreducible representations of G.

Show that 2 representations are not equivalent and find all the irreducible representations of $G$. The group $G=T_{16}$ has order 16 and presentation given by $G=\langle a,b : a^8=b^2=1, ...
3
votes
0answers
43 views

Continuous subgroup of SO(3)?

I read from a paperarXiv: cond-mat/0602109 by a theoretical physicist, Prof. Frank Bais, close subgroups of $SO(3)$ is given by ${C_n,D_n,T,O,I,SO(2)\rtimes Z_2}$, where $C_n$ is the cyclic group of ...
0
votes
0answers
13 views

Assign a root to a irreducible representation

Given a root, e.g. $(-1 0 1 00)$ of $\text{SO}(10)$, how can I see/find to which representation of the Lie group it belongs?
1
vote
1answer
79 views

What are the Pontryagin duals of additive and multiplicative group of complex number?

What are the Pontryagin duals of additive and multiplicative group of complex number? So basically what are all characters of $(\mathbb{C},+$) and $(\mathbb{C^*},.)$?
1
vote
0answers
71 views

maximal irreducble subgroups of $SL(2,q)$

If $H$ is a maximal solvable irreducible subgroup of $GL(2,q)$ then intersection $H \cap SL(2,q)$ is maximal solvable irreducible subgroup of $SL(2,q)$. Why is it true? Maybe this is not true, but ...
8
votes
2answers
89 views

the morphism from $SL(2,\mathbb{Z})$ to $SL(2,\mathbb{R})$

For every morphism $\rho: SL(2,\mathbb{Z}) \to GL(2,\mathbb{R})$, then $Im(\rho)\subset SL(2,\mathbb{R})$? Thanks.
1
vote
0answers
29 views

Adjoint Representation of Lorentz Group

I'm thinking about the image under the adjoint representation $\mathrm{Ad}$ of the proper (identity connected component) Lorentz group $SO^+(1,3)$. Since this group has a trivial centre (it contains, ...
1
vote
0answers
41 views

Understanding the irreducible representations of $D_3$

By the dimensionality theorem, $$\sum_i d_i^2 = |G|,$$ where $d_i$ is the dimension of the $i$th irreducible representation, we can infer that the dihedral group $D_3$ has two one dimensional irreps ...
0
votes
1answer
77 views

How can I use Clebsch-Gordan coefficients to decompose this group representation?

Let $G$ be a compact group, $\alpha$ be a unitary irrep of $G$ with carrier space $\mathcal A$, and $\beta$ be a unitary irrep of $G$ with carrier space $\mathcal B$. Then, the action of $G$ on ...
2
votes
1answer
43 views

The only irrep of a group of order p over a field of characteristic p is trivial

I found an answer to my bigger question here, but I'm curious about my attempted proof in the case where $|G|=p$. I'm nearly certain this does not work, but I can still learn something from it. Do I ...
2
votes
2answers
64 views

Nilpotent groups are monomial

I'm trying to show that a nilpotent group $G$ is monomial; i.e., that every irreducible representation $\rho$ of $G$ satisfies $\rho = \text{Ind}_H^G(\tau)$ for some $H \leq G$, $\tau$ a one ...
3
votes
1answer
22 views

A class function $f$ is a character if and only if $(f,\chi_{q_i})_G $ is a non-negative integer, for all irreducible characters $\chi_{q_i}$

I'm currently revising representation theory and I'm a bit stuck trying to prove the converse of the above statement. $(\Rightarrow)$ is straight forward because if $f$ is the character of a ...
1
vote
2answers
55 views

the presentation of $SL(2,\mathbb{Z})$

There is a natural presentation $SL(2,\mathbb{Z})\hookrightarrow GL(2,\mathbb{R})$, are there other presentations in real dimension 2? Or there is a classification of all the presentation of ...
18
votes
5answers
254 views

$\sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 $

I came across this result while doing some representation theory of the permutation group $S_n$ $$ \sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 = 2 n!$$ This can be ...
1
vote
1answer
53 views

Group representations and short exact sequences

Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be a short exact sequence of groups. What can be said about group representations of $B$ if we assume a complete classification of the ...
0
votes
1answer
76 views

The number of a set of irreducible projective characters vs the number of the ordinary characters of a finite group G.

I need valid references to show that the number of a set of irreducible projective characters with non-trivial factor set is always strictly less than the number of the ordinary characters of a ...
1
vote
1answer
41 views

Sum of irreducible character values in a row of the character table

If $\chi$ is a nontrivial irreducible character of $G$ (a finite group), define $S_{\chi}:= \sum_{x \in G} \chi(x)$. In terms of conjugacy classes $\mathcal{C}$, this is $\sum_{\mathcal{C}} ...
1
vote
1answer
61 views

Intertwining map in Schur's Lemma

I am learning Schur's Lemma from page 4 here. It says Schur's Lemma 1. If $(\rho_1, V_1)$ and $(\rho_2, V_2)$ are irreducible representations of a group $G$, then any nonzero homomorphism $\phi : ...
0
votes
0answers
28 views

Quotient braid group as a representation of SU(n)

I am working with the quotient braid group $B_3 (3) = B_3 / \langle\sigma_1 ^3\rangle$, where I construct a vector space $V$ so that every element $a \in B_3 (3)$ has a corresponding basis vector ...
2
votes
1answer
47 views

Representing Groups as matrices

How to represent any group as group of matrices ? Like how to represent dihedral (4) group (order $8$) as group of $2$ by $2$ matrices ? How to represent direct product of $Z_2$ and $Z_2$ as a group ...
2
votes
1answer
57 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
68 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
55 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
14 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
98 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 $|D_{10}|=20$, ...
1
vote
1answer
54 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
27 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
43 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
103 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 ...