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

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15
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1answer
126 views

Decomposing $V_1^{\otimes n}$, $\text{Sym}^2V_n$ into irreducibles, formula for all $n$?

$``$Let $G = \text{SU}(2)$, and let $V_n$ be the space of homogeneous degree $n$ polynomials in $\mathbb{C}[x, y]$. Decompose $V_1^{\otimes n}$, $\text{Sym}^2V_n$ into irreducibles.$"$ For ...
7
votes
4answers
90 views

A trigonometric integral identity

How can we prove the following identity? $$ \int_{0}^{2\pi}\cos^{n}\left(\,\theta\,\right) \sin\left(\,\left[\,n + 1\,\right]\theta\,\right)\sin\left(\,\theta\,\right) \,{\rm d}\theta ={\pi \over ...
4
votes
1answer
61 views

Flattening Young Tableaux

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k)$ be a partition with $|\lambda|=n$ and $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_k$. For any Standard Young Tableaux (SYT) $T$ of shape ...
10
votes
2answers
59 views

$SU(2)$ acting by conjugation, decomposition into irreducibles

I am attempting past exam questions of the Cambridge Math Tripos. I know how to do the first few parts, which involves giving the irreducible representations of $U(1)$ and $SU(2)$. But I am not sure ...
0
votes
0answers
38 views

Jacobson radical of a particular group algebra

I am studying for my Algebra test and I got stuck in this question: Determine the Jacobson radical of the group algebra $F[G]$, where $F$ is a field of characteristic $p>0$ and $G$ is the cyclic ...
0
votes
1answer
49 views

Decomposition into irreducibles of representations of semisimple Lie groups.

Let $G$ be a connected semisimple Lie group and $\mathfrak{g}$ it's Lie algebra. Then $\mathfrak{g}$ is semisimple. Let $V$ be a finite dimensional representation of $G$. Viewing $V$ as a ...
1
vote
1answer
22 views

Irreducible representation

I know that correspondence every pure state on a C*-algebra $A$, there is an irreducible representation of $A$. Also we have the following theorem: Let $A$ be a C*-algebras and $(\pi,H)$ be an ...
2
votes
1answer
16 views

Restriction of an irreducible representation

Let $A$ be a C*-algebra and $\pi:A \to B(H)$ be a irreducible representation. Could we claim $\pi_{|B}$ is an irreducible representation if $B$ is a C*-subalgebra of $A$ ?
0
votes
2answers
40 views

construction of an injective representation of $C_0(X)$

Let X be a locally compact noncompact Hausdorff space and consider the C$^*$-Algebra $C_0(X)$ of continuous functions vanishing at infinity. I want to construct an injective *-represenatation of ...
6
votes
0answers
78 views

Duals of representations of affine group schemes, in particular $\mathrm{GL}_n$

Duals of representations of affine group schemes Let $R$ be a commutative ring. If $G$ is a group and $V$ is a dualizable i.e. finitely generated projective $R$-module on which $G$ acts, then it is ...
3
votes
0answers
13 views

How can the generators of subalgebra $\mathfrak g^{\sigma}$ of $\sigma$-stable elements be expressed through generators of Lie algebra $\mathfrak g$?

Let $\mathfrak g$ be the semisimple Lie algebra of type $D_{4}$. Let $\sigma$ be the 3-rd order automorphism of $\mathfrak g$ induced by the triality of $D_{4}$: $$ ...
0
votes
1answer
19 views

How to find the multiplicity of weight in a Verma module?

In particular, let $\mathfrak g$ be the semisimple Lie algebra of type $A_{2}$ et let $\alpha,\beta$ be its simple roots. How can the multiplicity of weight $-2\alpha -3\beta$ be calculated in the ...
1
vote
0answers
22 views

Infinite dimensional representation such that every subrepresentation is reducible

Let $V$ be a nonzero finite dimensional representation of an algebra $A$. a) Show that it has an irreducible subrepresentation. b) Show by example that this does not always hold for ...
0
votes
0answers
7 views

The proportionality constant between the Casimir and the identity.

By Schur's Lemma, in any irreducible representation of a Lie algebra, the Casimir operator $J$ is proportional to the identity $Id$. How can we see that $J=j(j+1)Id$ for some natural number $j$ and ...
1
vote
1answer
32 views

character of irreducible representations of odd-ordered groups

I want to prove that if $G$ is a group and the order of $G$ is odd, and $\chi$ is a real-valued irreducible character of $G$, then $\chi$ must be the trivial representation, $\chi = \epsilon$. So ...
0
votes
1answer
29 views

Difference between to Tensor products with regards to modules

What would be the difference between $$ \otimes_B $$ and $$ \otimes $$ both in the following context and in general? Let A be a ring with $$ B \subset A $$ and M a B-Module. We can construct the ...
3
votes
0answers
31 views

Irreducible representations of $S_n$ [duplicate]

I want to prove that $S_n$ has an irreducible representation of dimension $n-1$. Intuitively, I know that the $\forall n$, the trivial representation is irreducible, and there should be some ...
3
votes
0answers
23 views

On Schur map and tableaux

My post refers to Jerzy Weyman's "Cohomology of vector bundles and syzygies" pag. 37. Let $R$ be a ring and $E$ a free $R$-module of rank $n$. Let ${e_1, \cdots, e_n}$ a basis of $E$. Let us consider ...
1
vote
2answers
34 views

If $V$ is a $\mathbb CG$-module then we may take $\rho(g)$ as a diagonal matrix?

If $G$ is a group and $\mathbb K$ is a field let $\mathbb KG$ be the usual group ring. We know a representation $\rho:G\longrightarrow GL(V)$, where $V$ is a $\mathbb K$-vetor space, is the same as a ...
3
votes
1answer
49 views

Invariants of $O(2) \times O(2)$ under simultaneous conjugation

Let $G= \textrm{O}(2)$ be the group of orthogonal $2 \times 2$ matrices over $\mathbb{C}$. $G$ acts on $G \times G$ by conjugation: $g \cdot (a,b) :=(g a g^{T}, g b g^T)$. This induces an action on ...
0
votes
0answers
14 views

Conjugation of $A_5$ by $S_5$ and effects on irreducible representations of $A_5$

I want to ask q5 of the following https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2010-2011/repex2.pdf The group $S_5$ acts on $A_5$ by conjugation. How does this action act on the ...
0
votes
0answers
26 views

permutation representation of Symmetric group

The symmetric group $S_n$ acts on $\mathbb{C}^n$ by permuting the coordinates. Decompose this representation explicitly into irreducible representations. For the action $\sigma (a_1,\cdots, a_n)= ...
7
votes
2answers
82 views

exponential function, lie group homomorphism

Let $f: \mathbb{R} \to \mathbb{C}^*$ be a continuous map satisfying for all $x, y \in \mathbb{R}$: $f(x + y) = f(x)f(y)$. $f(x) = 1$ for all $t = 2\pi n, n \in \mathbb{Z}$. Show that there exists ...
2
votes
0answers
22 views

Faithful representation of a finite group over reals [duplicate]

http://mathoverflow.net/questions/77325/finite-groups-with-faithful-real-two-dimensional-representation Suppose $\pi :G \rightarrow SL_2(\mathbb{R})$ is a faithful representation of a finite group ...
0
votes
0answers
26 views

Equations in group representation theory

I am reading this book by Margenau and Murphy: "The Mathematics of Physics and Chemistry". I think that there is a typo: In the 2nd edition, in the chapter on Group Theory, page 568, in equation ...
1
vote
0answers
43 views

Can we say that $A$ is a complement for a group $G$?

Let $A$ be a frobenius complement for a $G$ i.e. $A$ act on $G$ by automorphism s.t. $C_A(g)=e$ for all nonidentity $g$. Now, Action of $A$ can be linearly extended so that $A$ act on $F[G]$. As a ...
0
votes
1answer
34 views

Character of dual Representation?

Let $G$ be a finite group and consider the group ring $\mathbb C[G]$. If $M$ is a $\mathbb C[G]$-module consider the dual representation in $M^*=\operatorname{Hom}(M, \mathbb C)$ given by $$(g\cdot ...
4
votes
1answer
49 views

A converse of schur's lemma

Suppose $\rho: G \rightarrow GL(V)$ is a representation. and if $T: V \rightarrow V$ is a linear operator such that $T\circ \rho_g= \rho_g\circ T$ for all $g\in G$ implies $T=k\cdot Id$ for some ...
2
votes
0answers
32 views

If $G$ is a finite non-trivial group of odd order, it has an irreducible representation not realisable over the reals.

$\textbf{Lemma }$If $V$ is a representation of a finite group $G$, then $V$ is of real type if and only if $V$ is the complexification of a representation $V_{\mathbb{R}}$ over the field of real ...
1
vote
1answer
61 views

Is quotient under $S_4$ action on “cube” representation a flat morphism?

Consider a three-dimensional irreducible representation $V$ of $S_4$, corresponding to symmetries of cube. Let $p$ be canonical projection $p: V \rightarrow V/S_4$. My question: is $p$ flat? I want ...
1
vote
1answer
37 views

Relationship between simple group and character table

Prove that a group $G$ is not simple iff $\chi (g)=\chi (1)$ for some nontrivial character $\chi$ and some $g\not= 1$. I have no idea how to do this, please help, thanks.
1
vote
1answer
17 views

The Weyl group and eigenspaces

Let $V$ be a representation of the Weyl group. For any reflection $\sigma_{\alpha}$ (where $\alpha$ is a root), we know that $V$ has two eigenspaces with eigenvalues $1$ and $-1$. The ...
1
vote
1answer
31 views

Is the map from representation ring to class functions a isomorphism?

I have a questions from representation theory. ($G$ is finite group) Fulton and Harris in "Representation Theory. A first Course" write that: the character defines a map $$\chi : R(G) \to ...
1
vote
1answer
42 views

What is the center of the Valentiner group $\mathcal{V}=\langle I, Q \rangle$?

(Please refer to this question first: Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$? ) I want to understand the center of the Valentiner group: $$\mathcal{V}=\langle I, Q ...
2
votes
1answer
35 views

character group of finite abelian group and induced homorphism

This is ex 5.7 of chapter 10 of artin's algebra (2nd edition) Suppose $\varphi:G \rightarrow G'$ is a homomorphism of abelian groups. Define an induced homomorphism $\hat{\varphi}" \hat{G'} ...
3
votes
1answer
73 views

Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$?

My question is the one in the title. If you want to understand the context of the problem, please read further. I reduced a problem to proving the question. Background is: Valentiner group ...
1
vote
1answer
27 views

Is the center of a compact Lie algebra precisely the set of vectors on which the Killing form is zero?

Suppose a Lie algebra $\frak{g}$ has a killing form, $B$, which is negative semidefinite. Suppose $B(X,X)=0$ for some $X\in \frak{g}$. Is $X$ necessarily in the center of $\frak{g}$?
0
votes
0answers
29 views

Partial generalisation to Whitehead's second Lemma

Let $k$ be an algebraically closed field of characteristic $0$ and let $\mathfrak{g}$ be a finite dimensional semisimple $k$-Lie algebra. By Whitehead's second Lemma, we know that $H^{2}(\mathfrak{g}, ...
2
votes
0answers
30 views

action of $SU(2)$

Let $L=\mathbb C[u,v]$ (the $\mathbb C$-algebra of polynomials over two commuting variables $u,v$. For each non negative integer $n$ let $L_n$ be the linear subspace of homogeneous polynomials of ...
2
votes
4answers
136 views

Alternate proof of Schur orthogonality relations

I am trying to find an alternate proof for Schur orthogonality relations along the following lines. Let $G$ be a finite group, with irreducible representations $V_1$, $V_2$, $\cdots$, $V_d$. Let $V$ ...
1
vote
0answers
28 views

Irreducible representations of $S_3$

I am trying to do a problem from artin's algebra 2nd ed (Chapter 10, Exercise 2.3) but having trouble: Let $(\rho , V)$ be a representation of the symmetric group $S_3$. Let $x=(123), y=(12)$ be the ...
1
vote
1answer
32 views

How does one define weights for a semisimple Lie group?

For compact Lie groups one considers a maximal torus to define the weight space decomposition of a representation. For a complex semisimple Lie algebra one considers a Cartan subalgebra. How does ...
3
votes
1answer
57 views

An irreducible character of degree prime affords a faithful representation.

This is a long question, but hopefully someone can give me a suggestion, as I've been hitting my head against the wall... Take a non-abelian group $P$, of order $p^3$, $p$ prime. We've already ...
0
votes
2answers
23 views

Do elementary row operations give a similar matrix transformation?

So we define two matrices $A,B$ to be similar if there exists an invertible square matrix $P$ such that $AP=PB$. I was wondering if $A,B$ are related via elementary row operations (say, they are ...
1
vote
0answers
22 views

Reference request: classification of simple Lie groups and simple real Lie algebras

I am trying to understand the classification of simple Lie groups and the theory of highest weights for semisimple Lie groups by first understanding the case for complex Lie algebras, then relating to ...
1
vote
1answer
22 views

Character of the algebra $\mathbb{C}[G]$ as $G \times G $-module

Let $G$ be a finite group. We can define an action of $G\times G$ on the group algebra $\mathbb{C}[G]$ in the following way: If $x \in \mathbb{C}[G]$ then $(g,h)\cdot x=gxh^{-1}$. Now, what about ...
4
votes
0answers
40 views

Convolution of matrix coefficients is also a matrix coefficients

I have a question about the convolution of matrix coefficients as follows: Let $G$ be a compact Lie group. A Map $f:G\rightarrow \mathbb{C}$ is called a matrix coefficient if there is a finite ...
0
votes
1answer
21 views

Why if we have representation $\rho$ of finite group then $\rho(g)$ is diagonalisable matrix?

Why if we have representation $\rho:G \to GL(V)$ of finite group $G$ then $\rho(g)$ is diagonalisable matrix? I read that it's because $x^{o(g)} -1$ splits, but I don't understand how this fact is ...
4
votes
1answer
42 views

Decompose the following representation of $A_5$ in irreducible representations

Denote the space of functions on the set of faces of the icosahedron by $V_f$, this is a 20-dimensional representation of $A_5$ which acts here on as the group of rotational symmetries. Here follows ...
1
vote
1answer
26 views

Dimension of subspace stabilized by group and principle character

Let $\phi: G \rightarrow GL(V)$ be a representation with character $\chi$. Let $W$ be the subspace $\{v \in V : \phi(g).v = v$ for all $g \in G \}$ of $V$. Prove that $\dim W = \langle \chi, ...