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

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8
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$n-1$ 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
1answer
76 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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vote
0answers
51 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
31 views

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

In proposition 4.17, why is $P$ an central element?
2
votes
2answers
48 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 ...
2
votes
1answer
31 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
40 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
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0answers
62 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
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0answers
54 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
53 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
61 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
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0answers
43 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 ...
2
votes
1answer
59 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
84 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 $H$. ...
1
vote
1answer
165 views

How getting the unitarized irreducible representations with GAP?

The function IrreducibleRepresentations on GAP gives non-necessarily unitary representations, for example: ...
2
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1answer
44 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
228 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
55 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
44 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
36 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
13 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
77 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
165 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 ...
3
votes
2answers
553 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
29 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
30 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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vote
0answers
43 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
131 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
8 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
44 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. ...
1
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1answer
84 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
vote
1answer
31 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
106 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 ...
3
votes
2answers
67 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
32 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
117 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
131 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
137 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 ...
1
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0answers
112 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
40 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)\}, ...
2
votes
1answer
36 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 $L^2(G)$ ...
5
votes
1answer
135 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
61 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 ...
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votes
0answers
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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 ...
2
votes
0answers
59 views

Over which fields is a $G$-module reducible?

Let $K$ be a field of characteristic zero, or if this is too general, an algebraic number field. Let $G$ be a finite group and $V$ an irreducible and finite-dimensional $KG$-module. Let $\chi$ be the ...
10
votes
1answer
211 views

Group theoretic solution to an IMO problem

Is there a (strictly) group theoretic interpretation (and possibly a solution) to this problem (taken from the 27th IMO)? "To each vertex of a regular pentagon an integer is assigned in such a way ...
4
votes
0answers
50 views

Intuition behind the construction of Young Symmetrizer

I've been studying representation theory of group on Tung's "Group Theory in Physics". I understood Young Symmetrizers of different Young diagrams are essentially primitive idempotents in group ...
6
votes
0answers
38 views

Relationship between exterior power of representation and variance?

I was reading the question: Symmetric and exterior power of representation regarding how to determine the character of an exterior power of a representation from the original representation. One of ...
15
votes
3answers
199 views

any $2$-dimensional rep of a finite, non-abelian simple group is trivial

Let $G$ be a finite, non-abelian simple group. How would I go about proving that any $2$-dimensional representation of $G$ is trivial? If it helps, I know how to do it when we're considering ...