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

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Suppose p : G → GL(n, C) is a representation. Suppose that g, h exist in G and that p(g)p(h) = p(h)p(g). Is it then true that gh = hg?

Suppose $p : G → GL(n, C)$ is a representation. Suppose that $g, h$ exist in $G$ and that $p(g)p(h) = p(h)p(g)$. Is it then true that $gh = hg$? I don't know if I am not understanding the question, ...
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102 views

Invariants for the $SU(2)$ representation

The quantities $\delta_{ij}a_ib_j$ and $\epsilon_{ijk}a_ib_jc_k$ are invariant under the transformation of the $j=1$ (fundamental) representation of $SO(3)$. What would be the analogous expressions ...
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What is the Lie algebra of $G=\mathbb{R}$

The question is updated as following. 1. Let $(\Phi,L^2(R))$ be left regular representation of $\mathbb R$ given by $$ \Phi(g)f(x)=f(x-g). $$ It is unitary representation of $\mathbb R$. 2. For ...
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16 views

Combining infinitesimal generators of diferent dimensions

I am reading a paper about ways in which you can get $SU(2)\times{}U(1)\times{}U(1)$ as a subgroup of $SU(3)\times{}SU(2)\times{}U(1)$. At a certain point, it starts considering ways of getting ...
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40 views

Showing that an $\mathfrak{sl}(2,\mathbb{C})$-module is determined by eigenvalues of $h$

This question is essentially exercise 8.4 from the book "Introduction to Lie Algebras" by Erdmann and Wildon: "Suppose that $V$ is a finite-dimensional $\mathfrak{sl}(2,\mathbb{C})$-module. Show that ...
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65 views

Invariants under a transformation

Consider a $j=1,\,SU(2)$ representation (or fundamental $SO(3)$ representation). Suppose that $a_1, b_i, c_i$ with $i=1,2,3$ are vectors transforming under this representation i.e ...
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17 views

$U(1)$ generators of $SU(2)$

I wanna get $U(1)$ out of $SU(2)$. I know for example that this can be done using the diagonal Pauli matrix, but I wonder if there are more $U(1)$-s in $SU(2)$. So, which are the all the ways in ...
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28 views

invariants of a representation over a local ring from the residual representation

Let $(R, \mathfrak m)$ be a local ring (not necessarily an integral domain) and $T$ be a free $R$-module of finite rank $n\geq 2$. Let $\rho: G \to \mathrm{Aut}_{R\text{-linear}}(T)$ be a ...
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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 ...
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54 views

Dummit and Foote on Galois and Representation Theory?

At some point, I'd like to learn both Galois Theory and Representation Theory. I currently know a lot of Group Theory and Linear Algebra, as well as some Ring Theory. I was thinking of reading ...
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11 views

Properties of characters that remain true for infinite compact groups

Which properties of irreducible characters for finite groups still hold for infinite (compact) groups? In particular, is it still true that the irreducible characters form a basis for the space of ...
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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 ...
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36 views

Matrix coefficients of representations of finite groups

In finite-dimensional complex representations of finite groups, I would like to understand what I can learn by looking at a single matrix coefficient. In particular, I would like to look at "diagonal" ...
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19 views

Finite-dimensional commutant of unitary representation

Suppose that we are given a unitary representation $\rho\colon G\to\mathcal{U}(\mathcal{H})$ of some group $G$, that moreover satisfies $$\dim\rho(G)'=n<\infty,$$ i.e., the space of intertwining ...
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126 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 ...
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1answer
26 views

Given certain set of symmetries of a tensor, how do you associate the corresponding young tableaux

I have a particular problem, the following. $T^{a_1 \dots a_p;b_1 \dots b_p}$ is a tensor with the following symmetries. 1) $a_i$'s and $b_i$'s are completely antisymmetric, ie restricted to ...
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11 views

A canonical map Aut$_{\mathsf{Lie}_R}(\mathfrak{n} \rtimes_\pi \mathfrak{g}) \to$ Aut$_{\mathsf{Lie}_R}(\mathfrak{n})$

Let $\mathfrak{n}$, $\mathfrak{g} \in \mathsf{Lie}_R$ be two Lie algebras over a commutative ring $R$, s.t. $\mathfrak{g}$ acts on $\mathfrak{n}$ as a derivation: $\pi:\mathfrak{g} \to ...
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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 ?
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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 ...
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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 = ...
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2answers
33 views

A question about Lie group homomorphisms

Suppose I have a Lie group $G$ and a Lie homomorphism $ \phi : G \rightarrow GL_n(\mathbb{R})$. Can $ \phi $ be viewed as some sort of representation of $G$? Can anyone make this rigorous for me ...
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Why do we want *unitary* representations of locally compact groups into $B(H)$?

This is related to a previous question of mine but I have a more philosophical issue with the material. Everywhere I have looked for representations of locally compact groups into $B(H)$, everyone ...
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35 views

Generalized Schur-Weyl Duality

Schur-Weyl duality relates representations of the symmetric group to representations of $GL(n)$. Is there a generalization to arbitrary reductive groups?
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54 views

Exercise from Etingof's notes on Representation Theory

I am reading through these notes of Etingof on Representation theory and I am stuck with one exercise (it's problem 4.69 in the notes). The problem is the following. Consider the space ...
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40 views

$\mathrm{GL}_n$-representation theory question or a Tale of Two Determinants

The irreducible representations of $\mathrm{GL}_n(\mathbb C)$ are indexed by partitions $\lambda$. These representations are denoted by $\mathbb S_{\lambda}(V)$, where $V$ is the standard ...
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1answer
37 views

Schur's Lemma and the Center of a Group

If I have a group $G$ and a complex irreducible representation $g:G\rightarrow GL_n(\mathbb{C})$. I am trying to use schur's lemma to show that for $x\in Z(G)$ we have that $g(z)=\lambda_z I_n$. Now ...
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1answer
25 views

Dimensions of the conjugacy classes of $S_3$ in $\Bbb{C}S_3$.

Since the conjugacy classes of $S_3$ are $\{1\}$, $\{(1 2), (1 3), (2 3)\}$, and $\{(1 2 3), (1 3 2)\}$, I would think that they have dimensions 1, 3, and 2; respctively (because they are the basis ...
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1answer
42 views

Unitary representations of locally compact topological groups

Known: In the representation theory of finite groups (on finite dimensional vector spaces of course), given a finite group $G$ and a representation $\rho$, we can construct an inner product $\langle ...
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1answer
40 views

An example of finding the irreducible representations of G over C

I have an example in my lecture notes but I really don't understand what it is doing at each step so I was wondering if someone could help me work through it so that I can then do the problem sheet ...
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Closed orbits of complete flags in $\mathbb{C}^n$

Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the ...
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44 views

Definition of a splitting field of a finite group

This is a basic question from the journal 'Mathematische Zeitschrift' 208 (1991) page 243. Let $K/F$ be a finite Galois extension of number fields and $G={\rm Gal}(K/F)$. Also let $L$ be any number ...
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27 views

Reciprocity for branching rules of $\mathrm{GL}_n(\mathbb C)$

[Separated from another question] If I have information about the restriction of representations of the general linear group, can I make any statements about the induction (by Frobenius reciprocity)? ...
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30 views

How do non-semisimple modules over $\mathbb C\mathrm{GL}_n(\mathbb C)$ look like?

[Separated from another question] Can you give an example of a non-semisimple module over $\mathbb C\mathrm{GL}_n(\mathbb C)$? (Preferably one without direct summands, i.e. an indecomposable module) ...
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33 views

Orthogonal invariants of an irredubile GL-representation

Let $n\in 2\mathbb Z$ be an even number. Let $G=\operatorname{GL}_n(\mathbb{C})$ and $V_\lambda$ the irreducible complex $G$-module corresponding to the partition ...
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6 views

Augmented Positive Definite G-Invartiant Hermitain Form

We know that we can create a G-invariant positive definite Hermitian form on V by picking a arbitrary positive definite Hermitian form $\{ , \}$ and applying the averaging process: $$<v,w> = ...
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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 ...
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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 ...
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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 ...
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535 views

Why are we interested in irreducible representation but not faithful representation?

I am reading some materials of representation theory (of a group). The motivation of representation theory is to represent (by a homomorphism $h: G \to GL(V)$, from the group $G$ to a vector space ...
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1answer
32 views

Generalization of Schur's lemma

I would like to proof a generalization of Schur's lemma for representations. (Schur's lemma) (cfr. Jean-Pierre Serre, Linear representations of finite groups) Let $\rho^1$: G $ \to $ GL($V_1$) and ...
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1answer
29 views

I need a reference defining Representation Theorem

I am doing my research and need a reference in which Representation Theorem is defined. Albeit not on the web, somewhere valid in researches. The research is in the field of Continuum Mechanics. ...
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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 ...
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68 views

Construct a rational matrix $A$ s.t. $A^m = I$

Let $K$ be a field of either $\mathbb{C}$, $\mathbb{R}$ or $\mathbb{Q}$, Let $V$ be a $n$ dimensional vector space over $K$. I want to construct a matrix $A \in GL(V)$ s.t. $A^m = I$ for some $m$ and ...
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Inducing highest weight modules

I have a question regarding highest-weight modules: Let be $\mathfrak{g}$ a Lie algebra, $\mathfrak{b}$ a Borel subalgebra, $\mathfrak{h}$ a Cartan subalgebra and $U(\mathfrak{g})$ its universal ...
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52 views

Jordan-Holder theorem for modules?

Let $A$ be a finite dimensional algebra over some field $ k$. I think from Jordan-Holder Theorem, one might be able to claim that every simple $A-$module occurs in the series (by this I mean it is ...
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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 ...
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64 views

Correspondence of representation theory between $\mathrm{GL}_n(\mathbb C)$ and $\mathrm U_n(\mathbb C)$

If I know something about the representation theory of the general linear group $\mathrm{GL}_n(\mathbb C)$, what can I say about the representation theory of the unitary group $\mathrm U_n(\mathbb ...
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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 ...
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32 views

For z in Z(G) show that there exists $\lambda_z$ such that $z.v=\lambda_z v $ for all v in V

Let $V$ be an irreducible $\mathbb CG$ module. We define $Z(G)$ to be the centre of $G$. For $z\in Z(G)$ show that there exists $\lambda_z\in\mathbb C$ such that $z\cdot v=\lambda_z\cdot v$ for all $v ...
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Suppose that V is a 2 D FG module and that there exists g,h in G, v in V such that (gh).v is not equal to (hg).v.

Show that V is irreducible. I think I have possibly proved this by contradiction, but I just wanted to make sure my answer is thorough enough. The question gives the hint to use Maschke's theorem: Let ...