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

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2
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1answer
68 views

Question concerning Morita equivalence and an algebra over a field which is not algebraically closed

I would like to know, whether there are a quiver $Q$ and an admissible ideal $I$ such that the quiver algebra $\mathbb{F}_3Q/I$ and the group algebra $\mathbb{F}_3 (C_3\times C_3)$ are Morita ...
10
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6answers
2k views

The use of conjugacy class and centralizer?

This is more or less for a conceptual and better-understanding question in group theory and in representation theory: (1) Why are conjugacy class and centralizer important concepts in the group / ...
0
votes
1answer
220 views

Exponents of a semisimple Lie algebra

I'd like to compute the exponents of a semisimple complex Lie algebra $\mathfrak{g}$. According to http://math.berkeley.edu/~theojf/LieQuantumGroups.pdf proposition 8.1.2.18, this amounts to ...
3
votes
1answer
82 views

Building invariants of non-fundamental $SU(2)$

Suppose you have two objects, $ \phi _i $ and $ \psi _j $ that form representations of $ SU(2) $. With both fields in the fundamental representation, I believe there is one invariant, \begin{equation} ...
1
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1answer
34 views

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, ...
1
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1answer
385 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 ...
2
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0answers
58 views

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 $G=...
1
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1answer
96 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 ...
3
votes
0answers
80 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 $a_i'=[\rho_1(\...
2
votes
1answer
63 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 ...
1
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2answers
281 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 ...
3
votes
2answers
152 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 ...
1
vote
0answers
111 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" ...
6
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1answer
1k 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 ...
1
vote
1answer
62 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 either$...
0
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1answer
37 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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0answers
52 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 $\mathbb{R}/\mathbb{...
4
votes
1answer
109 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 = \mathbb{C}G(1-...
2
votes
2answers
47 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 ...
3
votes
2answers
101 views

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 ...
3
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1answer
168 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?
4
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1answer
144 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 $X=Mat_n(\...
3
votes
1answer
69 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 $n$-...
1
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1answer
234 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 ...
0
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1answer
32 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 ...
1
vote
1answer
108 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 \...
0
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1answer
104 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 ...
6
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0answers
95 views

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 Zariski-...
0
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0answers
180 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 ...
10
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0answers
79 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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0answers
50 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) ...
1
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1answer
66 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 $\lambda=(\lambda_1\ge\cdots\ge\...
3
votes
1answer
827 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 ...
2
votes
1answer
79 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 \...
7
votes
1answer
155 views

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 ...
8
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3answers
646 views

Do all Groups have a representation?

I know that many kind of groups can be represented by matrices; for example: rotation groups can be represented by matrices. Especially all elements of rotation groups can be represented by ...
10
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3answers
1k 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 $...
1
vote
1answer
113 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 $\...
0
votes
1answer
38 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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0answers
177 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 ...
5
votes
1answer
81 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 ...
3
votes
1answer
67 views

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 ...
5
votes
1answer
506 views

Jordan-Holder theorem for modules?

Let $A$ be a finite dimensional algebra over some field $k$. I think from the 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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vote
1answer
131 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 ...
7
votes
2answers
189 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 $\mathrm U_n(\mathbb C)$?...
3
votes
1answer
99 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 ...
2
votes
1answer
52 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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vote
2answers
110 views

On the graph of induction-restriction for group-subgroup representations

Let $G$ be a finite group, and $H$ a subgroup. Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ and $H$ (up to isom.). Consider the graph $\mathcal{G}(...
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1answer
93 views

Representation of $SL(3,\mathbb Z)$

I read the paper "Real-analytic actions of lattices", it says that any representation of any finite-index subgroup of $SL(3,\mathbb Z)$ into $GL(2,\mathbb R)$ has finite image, so how to prove it? ...
0
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1answer
116 views

Determinant of the irreducible characters table of a finite abelian group

Let $G$ be an abelian group, $|G|=n<\infty$. Let $\Phi$ be an irrep of G. Find the modul of determinant of the characters table. Ther is answer in my book. It is $n^{n/2}.$ If G is cyclic then ...