2
votes
1answer
28 views

left regular representation of a group thru group action

Let G be a group and let $g\cdot:G\to G$(i.e.$g\cdot g'=gg'$) . This induces a permutation representation of the group. I was trying to walk thru the problems in dummite and foote. One of the problem ...
3
votes
0answers
43 views

The Heisenberg group over $\mathbb{Z}/2\mathbb{Z}$

This is inspired by a problem from from Dummit and Foote. It asked me to calculate the order of every element in the Heisenberg group over $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$, which is defined as ...
1
vote
0answers
29 views

Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and let $H$ be ...
2
votes
0answers
25 views

Sum of degrees of irreducible complex characters for certain groups

The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant do determine the dimension of a maximal torus in the group algebra. I have ...
1
vote
1answer
86 views

Verify that two linear representations are equivalent

I've a problem in verifying that two linear representations are equivalent. First of all, I have two permutation representations of the group $G=\langle\alpha ,\beta ,\gamma\rangle$ on the set ...
2
votes
0answers
31 views

Matrix representation associated to a permutation representation

I have just begun studying group representation theory. I don't understand how I can find the matrix representation associated to a permutation representation. Should I identify each permutation ...
1
vote
0answers
31 views

Representation of a group, and finite index subspaces

Been working on this for a while and haven't gotten anywhere. I would really appreciate some hints. Let $G$ be a group, not necessarily finite. $V=\mathbb C [G] $ a vector space with basis $(e_g, ...
4
votes
1answer
122 views

Reduced Group algebras

Take a finite group and a field of characteristic zero. The group algebra is due to Maschke's theorem semisimple so that its a finite direct sum of matrix algebras over division algebras. I like to ...
2
votes
1answer
21 views

Explicit formula for invariant inner product of the standard representation of $S_3$

Let $V$ be a representation of a group $G$ over $\mathbb{C}$. Given the standard Hermmitian inner product $\langle\cdot,\cdot\rangle$ on $V$ we can always define a $G$-invariant inner product by ...
2
votes
0answers
23 views

Is $B(w_1 w_2)B = (Bw_1B)(Bw_2B)$?

Let $B$ be a Borel subgroup of $GL_n$ and $W$ the Weyl group of $GL_n$. Let $w_1, w_2 \in W$. Is $B(w_1 w_2)B = (Bw_1B)(Bw_2B)$? If this is true. How to prove it? Thank you very much. Edit: If $l(w_1 ...
2
votes
0answers
37 views

What are the main differences among representations of $GL(n, \mathbb{R})$, $GL(n, \mathbb{C})$, and $GL(n,k)$?

What are the main differences among representations of $GL(n, \mathbb{R})$, $GL(n, \mathbb{C})$, and $GL(n,k)$? Here $k$ is a non-archimedean local field. Thank you very much.
1
vote
0answers
47 views

Positive definite functions generated by irreducible representations — what do people call them?

Let $G$ be a group and $\pi:G\to B(H)$ be its irreducible unitary representation (one can endow $G$ with topology and claim that $\pi$ is continuous in some sense, this doesn't matter). For a given ...
0
votes
0answers
21 views

Irreducible representations of group

I'm basically interested in $C^*$-algebras $A$, where the following conditions for a $^*$-representation $\pi$ on Hilbert space $H$ are all equivalent: 1. $\pi$ is irreducible i.e. there are no ...
0
votes
1answer
46 views

Group Theory- S3 table

$\begin{matrix} & e& a& b& c& d&f \\ e& e& a & b& c& d&f \\ a& a& b& e& d& f&c \\ b& b& e& ...
1
vote
0answers
44 views

Family of equivalent unitary representations is not a set.

I have recently come across a statement in the book: Kazhdan's property (T) by B. Bekka, P. de la Harpe, A. Valette at the beginning Appendix F.2. Fell topology on sets of unitary representations. ...
3
votes
2answers
136 views

The definition of the right regular representation

I'm having difficulties understanding the definition of the right regular representation as it appears in Dummit & Foote's Abstract Algebra text. On page 132 it says Let $\pi:G \to S_G$ be the ...
-1
votes
1answer
86 views

Elements whose orders are multiple of $p$ [closed]

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
102 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 $(nāˆ’1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...
3
votes
0answers
70 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 $p$, $G$ ...
1
vote
1answer
25 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
34 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
31 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
150 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 ...
2
votes
1answer
49 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
20 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
21 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
35 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
33 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
56 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 ...
8
votes
0answers
97 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
113 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
29 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
27 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
48 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
81 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
72 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
91 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
33 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
44 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
86 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
45 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
68 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
23 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
57 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
266 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 ...