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

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Multiplicity of a dual simple module in the dual module?

Let $A$ be a finite dimensional $k$ algebra. Let $S$ be a simple left $A$-module and $M$ be any left $A$ module. Then my first question is that is it true $$[M:S]=[M^*:S^*]$$ where ...
1
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1answer
22 views

Is a virtual vector bundle the same as a vectorial bundle?

What is a virtual vector bundle? Is a virtual vector bundle the same as a vectorial bundle? The current entry in nLab states the following: "In one class of models for K-theory – generalized ...
5
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1answer
20 views

Are the weights of an irreducible representation of a simple Lie algebra in a single Weyl orbit?

When we consider the weights of an irrep of a simple Lie algebra, are they always in a single orbit under the Weyl group of the Lie algebra, or do they form a set of disjoint orbits? If they form ...
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0answers
84 views
+50

How to construct $\operatorname{End}(V_{\pi})$ using a representation $\pi$

Let $(\pi, V)$ be a representation of the group $G$. To make the setting as general as possible, I will not put any restrictions on $\pi, V$, and $G$ from the beginning. By the very definition, for ...
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0answers
13 views

unique part of induced representation.

In the book about representation by Harris and Fulton . It proofs a proposition(3.17), where H is a subgroup of G: It only proofs existence at glance, why is it unique?
4
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2answers
60 views

A question in representations theory

My question is about irreducible representations of groups over the field $\mathbb{Q}$. Let $G$ be a cyclic or an abelian group. I want to check that under what conditions we have a ...
1
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1answer
32 views

Eigenspace of finite abelian group

Let $\rho: G\to {\rm GL}_n(\mathbb{C})$ be faithfull representation of finite abelian group $G$ and $V$ is the eigenspace of some $g\in G$. Is it true that $V$ is also eigenspace for all $G$ (that ...
5
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4answers
296 views

Polynomials as module over symmetric polynomials

Consider ring of polynomials $\mathbb{k} [x_1, x_2, \ldots , x_n]$ as a ring over symmetric polynomials $\Lambda_{\mathbb{k}}$ Is $\mathbb{k} [x_1, x_2, \ldots , x_n]$ free $\Lambda_{\mathbb{k}}$ ...
5
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1answer
41 views

A problem similar to Maschke's theorem

My question is about a problem that its assumptions are like Maschke's theorem in some ways. Let $\mathbb{F}$ be a field that $char \mathbb{F}$ doesn't divide $|G:H|<\infty$ and $M$ be an ...
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1answer
30 views

The Weyl group of $\widehat{\mathfrak{sl}}_2$.

On page 5 of this paper, example 3.1, it is said that the Weyl group of $\widehat{\mathfrak{sl}}_2$ is $$ W= \langle s_1, s_2 \mid s_1^2 = s_2^2 = 1 \rangle. $$ Why the Weyl group of ...
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0answers
42 views

How to compute $Ext_A^{1}(S_1, S_2)$ and $Ext_A^{1}(S_2, S_1)$?

Let $A = kQ/\rho $, $Q$ is the quiver \begin{align} 1 \overset{a}{\underset{a^*}{\rightleftarrows}} 2 \end{align} $\rho$ is the relation $a a^* - a^* a = 0$. Question: compute $Ext_A^{1}(S_1, S_2)$. ...
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0answers
20 views

Unitary dual of $\mathrm{Sp}_4(\mathbb{R})$

We know the unitary dual of $GL_n(\mathbb{R})$, unitary dual of $SU(2,2)$, how about $\mathrm{Sp}_4(\mathbb{R})$? Is there any known result? If so, can anyone provide me any references? Thanks!
2
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1answer
31 views

Decomposing $\mathfrak{sl}_3(\mathbb{C})$

There is a pretty standard exercise on $\mathfrak{sl}_2 (\mathbb{C}$) representations that consists in decomposing the representation given by $\mathfrak{sl}_3(\mathbb{C})$ via $\operatorname{ad}$, ...
3
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1answer
96 views

Reference for a proof of a projective representation of $A_6$

I want to understand the proof of There is a projective representation of $A_6 \hookrightarrow PSU(3).$ I am looking for a reference, but could not find. Suggestions are welcome. EDIT: We ...
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0answers
156 views

Representation theory of the general linear group over a finite prime field

The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely classified and well-understood via Schur-Weyl duality, the algebraic Peter-Weyl theorem and the entire ...
2
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0answers
28 views

Normal basis theorem

Let $K$ be a finite Galois extension of, say, $\mathbb{Q}$. Then is known(and called normal basis theorem) that if i view $K$ as a representation of $Gal(K/\mathbb{Q})$ over $\mathbb{Q}$ it is ...
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0answers
12 views

Finding the 2D irreducible representation of quaternions $Q_8$ in the space of functions $f\colon Q_8 \rightarrow \mathbb{C}$

The space of functions $F=\{f\colon Q_8\rightarrow\mathbb{C} \}$ is 8 dimensional, since we can choose for each element of $Q_8$ an element in $\mathbb{C}$ to send it to. The action of $Q_8$ on this ...
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3answers
56 views

A question about the proof of a theorem in Representation theory of groups

My Question is about one part of the proof of theorem in the book "A Course in the Theory of Groups" by Derek J.S. Robinson. I highlight the part that my question is about. We know that if $G$ is a ...
2
votes
1answer
80 views

Tannaka reconstruction: reference request

What is a classical and perhaps even original reference for the following result, often called Tannaka reconstruction? Let $G$ be a group and $R$ be a commutative ring in which $0,1$ are the only ...
2
votes
1answer
23 views

Holonomy representation: is it actually a class of representations?

In D. Joyce's book "Riemannian Holonomy Groups and Calibrated Geometry" (2007) the author writes that if $\nabla$ is a connection on a vector bundle $E$ (over a connected base) with the fibre $\mathbb ...
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0answers
28 views

Cauchy Identity for a specialized product of Schur polynomials

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_d)$ be a partition, with $|\lambda|=n$. Let $\nu=\nu(\lambda):=(\lambda_1-1,\lambda_2,\cdots,\lambda_d).$ In other words, $\nu$ is obtained from ...
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0answers
19 views

Finite-dimensional, irreducible Representations of the Diffeomorphism Group $Diff(R^4)$

Is there any possible way to study the finite-dimensional, irreducible representations of $Diff(R^4)$ systematically? My interests stems from the fact, that the symmetry group of general relativity is ...
2
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0answers
22 views

Some questions on Langlands Classification of Irreducible Admissible Representation

I am trying to construct some representations using Langlands classification theorem. But I get confused and have some problems when constructing these representations..... i) In the classification ...
3
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0answers
16 views

Lusztig's $h$-function of a dihedral group

Following the notations in Hecke algebras with unequal parameters, let $(W,S,L)$ be a weighted Coxeter system, and $H$ be the corresponding Hecke algebra with $\{c_w |w \in W\}$ the Kazhdan-Lusztig ...
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0answers
40 views

Character as sum with regular representation

Suppose $G$ is a group and $\chi$ is a character of $G$ with $\chi(g_1)=\chi(g_2)$ for all non-identity $g_1,g_2 \in G$, and let $\chi_{reg}$ denote the regular representation character. I read that ...
2
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1answer
26 views

$Hom_G(\pi,\sigma)$ = $Hom_{\mathfrak{g}}(d\pi,d\sigma)$?

Let $G$ be a Lie group. Let $\mathfrak{g}$ be the corresponding Lie algebra. Let $(\pi,V)$ and $(\sigma, W)$ be representations of $G$, with corresponding differentials $d\pi$ and $d\sigma$, which are ...
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0answers
10 views

proof that Linear transformation is isometry

let ∑=set of all continous unitary representation and $ Ψ \in{ ∑}$ $π_Ψ: \frac{L^1(G)}{N_Ψ}→ B ( \oplus H_π) $ is definde by $$π_Ψ(f^0)=\oplus π(f) , π \in{ Ψ},f^0\in{\frac{L^1(G)}{N_Ψ} }$$ ...
1
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1answer
22 views

Existence of Irreducible Character s.t. $\chi(g) \neq 0, \chi(1) \neq 0 \text{ mod } |C(g)|$ for Elements in Conjugacy Class of Prime Order

Given a finite group $G$, and a non-identity representative $g$ in a conjugacy class of prime order $p$, I'm trying to show that some nontrivial irreducible character of $G$ must have $\chi(g) \neq 0$ ...
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1answer
32 views

Describing $GL(2,\mathbb{C})$ with generators and relations.

My question is : how can I describe $GL(2,\mathbb{C})$ with generators and relations ? I do not know how to start ? Thanks for your help in advance,
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1answer
18 views

Trivial representation from the row-shape Young diagram

For the Young diagram $\lambda$ which is the row with, say $d$ squares, i.e. $\lambda = (d)$, the corresponding Young symmetrizer is $c_\lambda = \sum\limits_{g\in\mathfrak S_d}g$ such that the ...
4
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1answer
126 views

How to prove that representations on $S^k(V), \bigwedge ^ k(V)$ are irreducible?

Given a $\mathbb{C}$ vector space $V$, let $GL(V)$ act on $\bigotimes^k(V)$ via: $GL(V) \times \bigotimes^kV \to \bigotimes^k(V), \ (A,v_1\otimes...\otimes v_k)\mapsto Av_1\otimes...\otimes ...
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0answers
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Exercise 2.4.1 Automorphic Forms and Representations, Daniel Bump

I am working my way through D. Bump's Automorphic Forms and Representations. I was trying my hand at this problem. Problem: Let $K$ be a compact subgroup of $GL(n,\mathbb{C})$ and let $(\pi, H)$ be a ...
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0answers
20 views

Trivial representation in tensor square

Taken from another question in this website. I am not sure why the following statement is true. Suppose $G$ is a group and $V$ an irreducible representation of $G$. One has that ...
2
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1answer
44 views

Request a reference in group theory

Although the book "A Course in the Theory of Groups" by Derek J.S. Robinson is an excellent up-to-date introduction to the theory of groups and covers various branches of group theory, it is hard for ...
4
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1answer
72 views

Question about isotypical components

Consider $V=\bigotimes^3(\mathbb{C}^2)$ as a $\mathfrak{S}_3$ representation. One of its isotypical component is $S^3(\mathbb{C}^2)$, which is a linear subspace of symmetric tensors of ...
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91 views

Clifford algebra - Gamma matrices

Let's say we have $\gamma^{a}$ matrices $(a=1,2,...,D)$. They satisfy the following condition $$\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\delta^{ab}I^{N\times N}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ ...
2
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2answers
36 views

Semisimple modules and the radical

I don't need a proof, but can someone tell me whether it is true that for all $A$-modules $V$ we have that $V/\text{rad}V $ is semisimple, where we define $\text{rad} V$ as the intersection of all ...
5
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2answers
78 views

About integral binary quadratic forms fixed by $\operatorname{GL_2(\mathbb Z)}$ matrices of order $3$

I am reading this paper of Manjul Bhargava and Ariel Shnidman, and I want to prove this claim, which appear at the first paragraph of Theorem $14$: Up to $\operatorname{SL_2}(\mathbb Z)$ ...
3
votes
2answers
68 views

Quaternion–Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $ \mathbf{R}^3 $ by conjugation. I've recently encountered spinors and notice similarities in their definitions ...
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1answer
29 views

Induced representation

I'm doing the problem section of the induced representations chapter by Steinberg, and I'm having problems with the following one: Let $G$ be a group and $H$ subgroup. Given a representation ...
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1answer
37 views

Projective representaions of $(\mathbb{Z}/3\mathbb{Z})^2$

I have a very short question: is there a faithful projective representaion $\rho: \mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}\to {\rm PGL}(4,\mathbb R)$? Thanks!
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0answers
26 views

Adjoint representation of the isotropie group of a homogeneous space

I have difficulties seeing why is the following true: Let $G$ be a lie group and $H$ a closed subgroup, with $\tilde{g}$ and $\tilde{h}$ their lie algebras. The adjoint action of $g\in G$ is given by ...
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1answer
57 views

A 4x4 matrix representation of SU(3)?

Is it possible to find a representation of the infinitesimal generators of the special unitary group SU(3) that contains 4 by 4 matrices, by say taking a Kronecker product of its irreducible ...
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1answer
41 views

Schur's Lemma: Is the isormorphism between two irreducible spaces unique?

Suppose $V_1 \neq V_2$ are two irreducible representations of the finite group G. Then Schur's Lemma says that any G-invariant map between them is either 0 or an Isormorphism. I understand that if ...
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23 views

Permutation modules and their vector space dimensions

I'm given a field $k$, a finite group $G$ and a set $S$ which $G$ acts on transitively. I'm then told to consider the permutation module $M = kS$. My first problem is understanding what the ...
3
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1answer
38 views

Exponentials of Representations of Lie Algebras

Assume G is a lie group and g is its lie algebra. Consider a representation of G : D:G->End(V). Then there is a corresponding representation of g : d:g->End(V). My question is, when you can express ...
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1answer
36 views

Krull-Schmidt theorem and internally cancellable modules?

According to this lecture notes (in Lemma2.1) the statement $Ae\simeq Ae^{\prime}\to A(1-e)\simeq A(1-e^{\prime})$ is true for finite dimensional algebras by using Krull-Schmidt theorem. Can anyone ...
3
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1answer
46 views

Representation is reducible

Suppose $V$ is a representation of a finite group $G$ over a field $k$ of characteristic $0$, and suppose dim$V=3$ and $\wedge ^2V$ is reducible. Then $V$ is reducible. I was trying to do it by ...
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1answer
41 views

irreducible representation contained in regular rep

Why is every irreducible representation contained in the regular representation? Suppose $W$ is a irreducible representation. ( i.e. a vector space over $\mathbb{C}$ which $G$ acts on with no ...
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0answers
28 views

Representation Theory Symmetric Group Book?

I'm looking for a nice book that discusses the representation theory of the symmetric group. My background is an introductory class in representation theory.