Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.

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3
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Frobenius reciprocity, proof if $V$ irrep of $G$ then multiplicity of $V$ in regular rep of $G$ is $\dim(V)$

I know the standard proof of the fact that if $V$ is an irreducible representation of $G$ then the multiplicity of $V$ is the regular representation of $G$ is $\dim(V)$. Does there exist a proof using ...
2
votes
1answer
33 views

exists homomorphism of $G$-representations $\pi: V \to V$ with image $X$

Let $G$ be a group (not necessarily finite) and $F$ a field. Let $V$ be a $G$-representation. Suppose $V$ is isomorphic to a direct sum of irreducible representations. Let $X \subset V$ be any ...
2
votes
1answer
52 views

An example of $_AM$ is simple but $M^\star_A$ is not?

Let $A$ be a finite dimensional $k$ algebra, $k$ is a field. It is evident that the duality functor ($(-)^\star=hom_k(-,k)$) preserves the simplicity in the case of finitely generated $A$- modules ...
1
vote
0answers
14 views

When are all Gorenstein projective also pure-injective?

For an artin algebra of finite global dimension, each Gorenstein projective module is projective then is pure-injective. Are there any other examples having this property? That is, all Gorenstein ...
3
votes
1answer
41 views

Fixed point subspaces $V^B$ and $V^G$ for a Borel subgroup $B\subset G$ coincide

Assume that $G$ is a linear algebraic group, and let $B \subset G$ a Borel subgroup of it. Let $(V,\rho)$ a rational $G$-module. Define $$V^G := \{ v \in V \mid g \cdot v = v \quad \forall g \in G ...
5
votes
2answers
51 views

$n-1$ dimensional permutation module for $S_n$

Say $n \ge 5$. Let $P$ be the $(n-1)$ dimensional permutation module for $S_n$, i.e. the permutation representation on $\{(x_1, \dots, x_n) \in {\bf C}^n: \sum x_i = 0\}$. Prove that: $\wedge^2P$ ...
2
votes
0answers
61 views

How unitarize an irreducible representation of a finite group?

Let $G$ be a finite group acting irreducibly on the space $V$. $$\psi : G \to Aut(V)$$ Question: How unitarize the representation $V$? I'm looking for a computable process.
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0answers
46 views

Simple question on the unitary representation.

Let $\pi_1,\pi_2,\pi_3$ are all irreducible, unitary representation of some algebraic group G. Then is it ture that $$Hom_{G}(\pi_1,\pi_2 \otimes \pi_3) \simeq Hom_{G}(\pi_1 \otimes ...
2
votes
1answer
30 views

Why it is central in $\mathbb {Z}[G]$?

In proposition 4.17, why is $P$ an central element?
2
votes
2answers
42 views

extracting the middle term of $ (z \cos \theta + w\sin \theta )^m(- z\sin \theta + w\cos \theta )^m $

Is there a systematic way to extract the middle term of the following expression? $$ (z \cos \theta + w\sin \theta )^m(- z\sin \theta + w\cos \theta )^m $$ This is homogeneous polynomial of degree ...
1
vote
1answer
26 views

adjoint representations

I am trying to work out the adjoint representations of $$H=\left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right), X = \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} ...
2
votes
0answers
32 views

representations of Lie algebras

I am studying irreducible representations of Lie algebras when our filed is of positive characteristic, I need an explicit explanation with example (or an article) which describes the differences what ...
2
votes
0answers
28 views

An example contradicting the validity of Maschke's theorem for infinite groups.

I am learning representation theory of finite groups and I am in doubt about a homework problem: Let $G = \mathbb{Z}$ and $V = \{(a_1 , a_2 , . . . )|a_i ∈ R\}$ be a vector space of infinite ...
0
votes
0answers
26 views

On the contragredient representation

Let $\pi$ be a representation of group $G$.Then its contragredient representation $\pi^{\vee}$ is defined by $\pi^{\vee}(g)=^{t}\pi(g^{-1})$. (here $^t$ means the transpose) But I heard that it is ...
2
votes
1answer
37 views

$E_6$ lie algebra and its representation

I've just started learning about Lie theory (only just finished up to basic classification of semisimple lie algebras) and I've got the following questions: How do I show that the complex lie algebra ...
0
votes
1answer
55 views

Representations of symmetric groups of order $2n$ and $n$

Background: Denote by $S_n$ the symmetric group of order $n$. There are many ways to embed $S_n$ as a subgroup into $S_{2n}$. Given a symmetric group, we can use Young diagrams to classify all ...
0
votes
0answers
33 views

problem in representation of a finite abelian group

There is problem, asking, find all in-equivalent representations of an abelian group $G$. My attempt: Let $f:G \to GL(V)$ a representation, by maschacke theorem $f_g$ is equivalent to direct sum of ...
1
vote
1answer
43 views

2-transitively, formula [closed]

Let $G$ be a finite group and let $X$ with $|X| \ge 2$ be a set on which $G$ acts. Then $G$ acts on $X \times X$ via $g \cdot (x, y) = (g \cdot x, g \cdot y)$. The action of $G$ on $X$ is called ...
4
votes
0answers
75 views

Every irreducible character of $G$ is an irreducible character of $H$?

Let $H$ be a proper subgroup of $G$ such that for all $\chi\in Irr(G)$, $\chi_H\in Irr(H)$. That is, the restriction of every irreducible character of $G$ to $H$ is an irreducible character of $H$. ...
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0answers
95 views

How getting the unitarized irreducible representations with GAP?

The function IrreducibleRepresentations on GAP gives non-necessarily unitary representations, for example: ...
2
votes
1answer
35 views

Fundamental weights of $A_n$

I have the following problem: Let $\mathfrak{g}$ be the Lie algebra of type $A_n$. We choose $e_i^*-e_{i+1}^*$ as simple roots. Is there a closed formula for the fundamental weights? Thank you!
0
votes
1answer
38 views

Character Table S4

I am trying to understand how to build a character table of S4. I've already read many articles about it but I am stuck at one point. S4 has 5 conjugacy classes and therefore 5 irreducible ...
0
votes
2answers
32 views

Automorphism group of vector space

I was trying to understand definition of representation and trivial representation thus came across the case where $ V= K $ here $V$ is a vector space over a field $K$ and thus $Aut_K (V) \cong ...
3
votes
1answer
24 views

$G$-invariant subspaces in $K[G]$

Let $K$ be an algebraically closed field and $G$ an linear algebraic group (i.e. a group object of the category of affine varieties over $K$). Denote by $A$ the coordinate ring of $G$. Then the right ...
3
votes
0answers
23 views

Tensor product via the diagonal action of a Hopf algebra

Let $H$ be a Hopf algebra and $V$ and $W$ two left $H$-modules, then $V\otimes W$ is also a left $H$-module via the comultiplication of $H$. I now want to consider the functor $-\otimes_H (V\otimes ...
2
votes
0answers
12 views

$[L_+^m, L_y^n]$ in the $SO(3)$ Lie Algebra

Let $SO(3)$ be generated by infinitesimal rotations $L_x, L_y, L_z$ such the typical relations $ [L_x, L_y] = L_z $ and similar. Let $L_\pm = L_x \pm i L_y$ be the raising and lowering operators. Is ...
2
votes
1answer
65 views

Nontrivial example of an artin algebra R such that R is pure-injective as an R-module

Give a nontrivial example of an artin algebra $R$ such that $R$ is pure-injective as an $R$-module. Clearly $0$-Gorenstein (self-injective) artin algebra has this property. Can anyone give me ...
6
votes
3answers
98 views

Textbooks on permutation groups?

I need good texts on group theory that cover the theory of permutation groups. I think there is a book called Wielandt. Is it good? are there newer alternatives? Can I find books that are not ...
0
votes
1answer
54 views

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions?

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions? Needed for the irrep decompositon of $3\otimes 3\otimes 3$ in here. No idea where to start to ...
1
vote
1answer
19 views

Natural Lie algebra representation on function space

There is natural Lie group representation of $GL(n)$ on $C^\infty(\mathbb{R}^n)$ given by \begin{align} \rho: GL(n) & \rightarrow \text{End}(C^\infty(\mathbb{R}^n)) \\ A & \rightarrow ...
2
votes
1answer
28 views

Notation in Kac Problem 3.2

I'm working through Kac's book, "Infinite Dimensional Lie Algebras", and have come across some notation I find confusing. Here, $e$, $f$, and $h$ are the Cartan generators of ...
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0answers
31 views

Irreducible Representations of Nilpotent Lie algebras

By Lie's theorem all irreducible representations of a solvable Lie algebra over $\mathbb C$ are one dimensional. What are all irreducible representations of a nilpotent Lie algebra ?
2
votes
0answers
103 views

Harmonic Analysis on the Affine Group

In my previous question, I asked about harmonic analysis on the group $\operatorname{SL}(3, \mathbb{R})$. The representation theory of this group appears to be quite complicated, so I am now looking ...
0
votes
0answers
7 views

How to write a polyhedra formula explicitly?

Let $m$ be a positive integer and $$ A_m = \{r=(r_1,r_2,r_3,r_4) \in \mathbb{Z}_{+}^4: r_4 \leq r_2, 2r_1+3r_2+3r_3 \leq m \}. $$ Let $$ch_m = \sum_{r \in A} ch((m-r_1-3r_2-3r_3)\omega_1 + (r_2 + r_3 ...
3
votes
1answer
37 views

Are the primitive groups linearly primitive?

A transitive permutation group $G \subset S_n$ is primitive if $G_1 \subset G$ is a maximal subgroup. A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation. ...
0
votes
1answer
19 views

Which inclusions of finite groups are relatively linearly primitive?

This post is a sequel of: Which finite groups have faithful complex irreducible representations? A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation. ...
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1answer
38 views

The finite groups with an irreducible faithful complex representation

All the groups below are supposed finite, and their representations, complex. An abelian group admits an irreducible faithful representation iff it is cyclic. A group has all its non-trivial ...
1
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1answer
30 views

Is comaximal equivalent to simple?

A finite group $G$ is called comaximal if for any non-trival irreducible representations $V$ and $W$ of $G$, it exists $n \in \mathbb{N} \ $ such that $(V^{\otimes n},W)\ge 1$. A finite group $G$ is ...
2
votes
0answers
78 views

Is there a transitive permutation group satisfying these properties?

Let $G \subset S_n$ be a transitive permutation group and let $H=G_1:=\{ g \in G \ \vert \ g(1)=1 \}$. Let $(K_i)_{i \in I}$ be the sequence of minimal overgroups of $H$ in $G$. Note that if $G$ is ...
2
votes
2answers
42 views

Decomposition and harmonic analysis of $L^2(S^n)$

Deitmar and Echterhoff write in their book Principles of Harmonic Analysis that `It follows from the Peter-Weyl theorem that the $SU(2)$ representation on $L^2(S^3)$ is isomorphic to the orthogonal ...
2
votes
1answer
28 views

On the Semisimplicity of a Permutation Module given by a Transitive Group Action

Let $G$ be a finite group acting transitively on a finite set $\Omega$. Let $K$ be field such that its characteristic divides $|\Omega|$. Is it true that $K\Omega$ is not semisimple? I think this ...
2
votes
1answer
42 views

Stone's One Parameter Unitary Group Theorem and the Fourier transform

Stone's theorem on one parameter unitary groups asserts a one-to-one correspondence between strongly continuous one parameter groups of unitary operators $\mathcal{H}\to\mathcal{H}$ on a Hilbert space ...
0
votes
1answer
82 views

Character Table Dihedral group of $D_6$

I'm having real troubles with finding the character table of the dihedral group $D_6$ of order 12: $D_6 = \langle a,b |a^6 = 1 , b^2 = 1, aba = b \rangle$. I've already found the conjugacy classes: ...
3
votes
1answer
82 views

Relation between irreducible and completely reducible representations

While studying representations of finite groups I got confused by the the statement that any irreducible representation is at the same time a completely reducible representation. This doesn't seem to ...
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0answers
43 views

Constructing irreducible representations of quaternion group over $\mathbb{Q}$

I am a beginner in studying the representation theory and I am doing some exercises in this field. So this is not a homework. My question is about constructing all irreducible representations of ...
2
votes
0answers
37 views

Conjugacy class A(4)

I want to find all conjugacy classes of $A(4)$. So basically what I did, I took all elements of $A(4)$ and calculated their conjugates. I had no problems with $$\{e\}, \{(123),(134),(142),(243)\}, ...
1
vote
1answer
24 views

Decomposition of regular representation

Let $G$ be a compact group. Then there is an isomorphism $L^2(G)\simeq \bigoplus_{\tau\in \hat{G}} V_{\tau}\otimes V_{\tau^*}$ which intertwines the conjugation action of $G\times G$ on $L^2(G)$ ...
4
votes
1answer
103 views

Harmonic Analysis on the real special linear group

I would like to understand the representation theory and generalized Fourier transform of $SL(3, \mathbb{R})$ in as concrete a manner as possible. My ultimate goal is to develop an algorithm that can ...
2
votes
1answer
52 views

A question on Schur's lemma and semisimple modules

One variant of Schur's Lemma states that $$ \text{Hom}(S,T) \cong \left\{ \begin{matrix} 0 & \text{if } S \neq T \\ \mathbb{C} & \text{if } S = T \end{matrix} \right. $$ when $S,T$ are ...
0
votes
0answers
25 views

Is the notion of strongly graded algebra a Morita invariant?

Let $G$ be a group and $A$ be a ring. $A$ is a $G$-graded ring if $A=\oplus_{g\in G} A_g$ such that $A_gA_h \subset A_{gh}$ for all $g,h\in A$. Such a ring is said to be strongly graded if ...