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

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Are the orbits of a connected Lie group acting on a vector space always embedded manifolds?

Setting: We have a connected Lie group $G$ and a smooth map $G \to GL(V)$, where $V$ is a finite-dimensional vector space. Are the orbits of $G$ on $V$ embedded submanifolds? More precisely, if one ...
4
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26 views

Let $A''=\operatorname{End}_{A'}(V)=\operatorname{End}_{\operatorname{End}_A(V)}(V)$. Show that $A''$ is a $k$-algebra.

Let $A$ be a $k$-algebra for a field $k$. And let $V$ be a representation of $A$. Define $A''=\operatorname{End}_{A'}(V)=\operatorname{End}_{\operatorname{End}_A(V)}(V)$. Show that $A''$ is a ...
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1answer
28 views

What is the natural action of $\mathfrak{sl}(4,\Bbb{C})$ on $\wedge^2 \Bbb{C}^4$?

What is the natural action of $\mathfrak{sl}(4,\Bbb{C})$ on $\wedge^2 \Bbb{C}^4$? We know that $\wedge^2 \Bbb{C}^4$ is generated by $\{e_1 \wedge e_2, e_1 \wedge e_3, e_1 \wedge e_4, e_2 \wedge e_3, ...
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1answer
21 views

Under what conditions can I expect the restriction of scalars functor to preserve tensor products

Suppose I have the canonical injection $i:H\hookrightarrow G$. Evidently I can induce the map on modules which restricts scalars from $\mathbf{Z}[G]$ to $\mathbf{Z}[H]$; that is, ...
2
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1answer
18 views

Can we say anything about the unit of a $k$-algebra $A$ in terms of the unit $1\in k$?

Context: Being confused about new concepts and trying to make new distinction to better understand it. Let's say we have have associative $k$-algebra $A$. Where $k$ denotes a field. An algebra is a ...
1
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1answer
17 views

Let $M,N$ be isomorphic as $\mathbf{Z}[G]$-mods, are they isomorphic as $\mathbf{Z}[H]$-mods, where $H<G$

So I have recently been looking at the isomorphisms of $\mathbf{Z}[G]$-mods ($G$ finite), and noticed that a couple of my examples saw them isomorphic as $\mathbf{Z}[H]$-mods also, where $H$ is a ...
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30 views

Definition of Representation in terms of Group Action

The definition of a representation of a group $G$ over a vector space $V$ is a map $p: G \to GL(V)$. According to wikipedia, for finite groups an equivalent definition is an action of $G$ on $V$. ...
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20 views

Nondegenerate representation

By the definition, we say a representation $(\pi,H)$ is nondegenerate if $cl[\pi(A)H ]= H$. Below I have two theorem, the first from Conway's Functional analysis and the second from Takesaki's ...
2
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1answer
35 views

Cyclic representation on $L^2(\mu)$

Show that if $(X,\Omega,\mu)$ is a $\sigma-$ finite measure space and $H=L^2(\mu)$, then $\pi:L^\infty(\mu)\to B(H)$ defined by $\pi(\phi)=M_\phi$ is a cyclic representation and find all the cyclic ...
3
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1answer
50 views

(Tensor) Product of irreducible representations

Suppose that $T: G \rightarrow GL(U)$ and $ S: G \rightarrow GL(V)$ are two finite dimensional irreducible representations of some group $G$ . I consider the tensor product representation $ T^*S : G ...
2
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1answer
81 views

Character table of the non-abelian group of order 21

I'm working my way through the first Chapter of Fulton and Harris' Representation Theory and I'm trying exercise 3.26: There is a unique nonabelian group $G$ of order 21, which can be realized as ...
1
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1answer
57 views

Lie algebras and the Killing form.

The Killing form is defined by $K(x,y) = \text{tr}(\text{ad} x, \text{ad} y)$, right? In this lecture, we assume that $\{x_1, ... , x_n\}$ is a basis for $g$ and $\{y_1, ... ,y_n\}$ is a dual basis ...
1
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1answer
32 views

Proof that a group representation matrix is diagonalizable?

Suppose we have a finite group $G$ and and an $n$-dimensional vector space $V\cong \Bbb C^n$ over the field $\Bbb C$ of complex number. My professor said the other day that for every group element $g$ ...
2
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2answers
41 views

Every Irreducible Representation of $G \times H$ is tensor product of Irreducible Reps of $G$ and $H$?

It's an easy task to prove with character theory that if $V_1$ and $V_2$ are irreducible representations of $G_1$ and $G_2$ respectively, then $V_1 \otimes V_2$ is an irreducible representation of ...
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27 views

Using function parameters as representation

I was wondering if there is some field of mathematics which analyzes situations where you use function partners as representations, e.g. for classification or regression. For example, let's say I ...
5
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204 views

A question about the article 'You can't hear the shape of a drum'

I have read the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
3
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34 views

Unipotent representations of SL(2,R) by quantization

I'm a PhD student in mathematical physics and I happen to need some elements of Kirillov's "orbit method" for producing representations of Lie groups. I'm familiar with symplectic geometry, geometric ...
2
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0answers
22 views

Finding expansions for elements of a group algebra

Suppose $G$ is a finite group of order $\left|G\right|$, with an associated group algebra $K[G] = \left\{\sum_{g \in G} a_g g\right\}$ over, say, the complex numbers. Suppose we represent $G$, and by ...
2
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2answers
133 views

Precise definition of affine, smooth, and irreducible

A book which I'm reading now says that "the Drinfeld curve $$ \mathbf{Y} = \{\, (x, y) \in \mathbf{A}^2(\mathbb{F}) \mid xy^q - yx^q = 1 \,\}$$ is affine, smooth, and irreducible." Here $p$ is an odd ...
1
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1answer
49 views

Endomorphism ring of indecomposable representations

Let $Q$ be the quiver given by an $n\times n$ grid where every square commutes and let $F:Q\to {\rm vec_k}$ be an indecomposable (finitely dimensional) representation of $Q$. I am interested in ...
2
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0answers
15 views

Visualisation of representations and their decomposition into irreps

A question in a Representation Theory midterm got me thinking, and made me realise I didn't really understand irreps. The question was on the subject of reps of $S_4$, and went: An obvious ...
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1answer
24 views

Matrices of subrepresentations and quotient representations.

Suppose that $V$ is a $5$ dimensional representation (with generators $\{y_1, ... , y_5\}$ of the lie group $\mathcal{g}$, with the lie algebra homomorphism $\rho: \mathcal{g} \rightarrow ...
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19 views

Subrepresentations of finite dimensional semisimple representations of an algebra

I'm following the notes by Prof. Etingof, linked here, and am stuck on a detail from Prop. 2.2, on page 23. To briefly recap what is in the notes, we have a finite dimensional, semisimple ...
2
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1answer
66 views

Does this representation have a name?

Let $G$ be a group acting on a set $X$. Let $F(X) = \{f: X \to \mathbb C\}$ be the set of complex valued functions on $X$. This is a complex vector space. Then $G$ acts on $F(X)$ linearly via the ...
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0answers
50 views

Kac's question 'Can one hear the shape of a drum' and Sunada method, a clarification

I'm reading the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
1
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0answers
33 views

The irreducible representation of $S_n$ of degree $n-1$ [duplicate]

So I understand it's not hard to show that the standard $(n-1)$-dimensional irreducible representation of $S_n$ is the only irreducible representation of $S_n$ of degree $n-1$ using characters/Young ...
0
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1answer
35 views

A Representation of $C(X)$ is a positive map.

I quote this excerpt from Conway: "A representation $\rho:C(X) \rightarrow \mathcal{B(\mathcal{H}})$ is a $\ast$-homomorphism with $\rho(1)=1$. Also, $\|\rho\|=1$. If $f\in C(X)_+$, then $f=g^2$ ...
3
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1answer
34 views

sub-$G$-representations

"So let $G$ be a finite group, $H$ a proper, nontrivial normal subgroup of $G$. For any representation $\rho: G \to \text{GL}(V)$ define the $H$-invariants of $V$ as $$V^H := \{v \in V \text{ }|\text{ ...
4
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1answer
63 views

$G$-representations, $W \otimes V^* \to \text{Hom}(V,W)$

Let $V$ and $W$ be finite-dimensional vector spaces. I know how to construct an explicit isomorphism of vector spaces $W \otimes V^* \to \text{Hom}(V,W)$ and show that it's an isomorphism. But if I ...
2
votes
1answer
57 views

Different definitions of Casimir element

I read about the Casimir element just recently and I found it a bit difficult to wrap my mind around the definition(s). In fact, I have seen two different definitions. For concreteness, let ...
2
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41 views

Why do Ad(K) orbits in the $-1$ eigenspace of a Cartan decomposition intersect the Weyl chamber once?

Let $G$ be a semisimple Lie group and let $\frak p\oplus t$ be a Cartan decomposition of $\frak g$ and $K$ the connected subgroup with Lie algebra $\frak t$. Choose a maximal abelian subalgebra ...
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36 views

Let $V=V_{1}⊕ ⋯ ⊕ V_{n}$ be semisimple. $U$ irreducible. Show that $\dim_{k} (Hom_A(U,V)) $ is equal to the number of $V_i$ equivalent to $U$.

$\DeclareMathOperator{\End}{End} \DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\Irr}{Irr}$ Definition. An $A$-module $V$ is ...
0
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1answer
31 views

Contains the representation with multiplicity n

In a problem I'm asked to prove that a representation contains the trivial representation with multiplicity $n$. I'm a little confused. What exactly does "contain" mean and "multiplicity"? Does it ...
0
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1answer
34 views

Is a representation of a $k$-algebra a $k$-vector space?

Is a representation $V$ of an $k$-algebra $A$ a $k$-vector space ? I've been studying representation theory for some weeks, but sometimes I get a little bit confused about all the different ...
2
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0answers
44 views

Remark 3.1.3 from Introduction to Representation Theory from Pavel Etingof

$\DeclareMathOperator{Hom}{Hom}$I'm trying to prove the following proposition (remark 3.1.3 from Introduction to Representation Theory from Pavel Etingof). Proposition. Any semisimple representation ...
1
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1answer
49 views

Basis of vector space invariant under group action (of symmetric group)

Suppose I have a finite-dimensional real vector space $X$ and a finite group $G$ that acts faithfully on X. The task is to find a $G$-invariant basis of $X$. This means the set of basis vectors is ...
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0answers
22 views

Representations and mutually singular measures

I'm finding some difficulties with an exercise from Conway and I ask for some help in understanding it: "Let X be a compact space and let $\{\mu_n\}$ be a sequence of measures in X. For each $n$ let ...
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0answers
16 views

Dirac Group Representations

(I am asking this question in the physics stackexchange too. I hope it is not problematic for me to ask same questions at two different places to get wider perspective!) I am currently taking a ...
3
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0answers
26 views

Finite-dimensional unitary representations of $SL_n(\mathbb{R})$

In Proposition 2.6.4 of his book Automorphic Forms and Representations, Bump is trying to prove that $SL_n(\mathbb{R})$ has no non-trivial finite-dimensional unitary representations. His argument is ...
2
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1answer
85 views

Rank of an action and definition of an orbital

Let $G$ be a group acting on a set $X$. In group theory sometimes it is helpful to consider the action of $G$ on $X\times X$; a good example is perhaps finding the dimension of ...
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1answer
30 views

Let $A=k[x]$ and $V=k[x]/((x-λ)^n)$. Find a filtration $V=V_0 ⊃ V_1 ⊃ \dots⊃ V_n=0$ such that the subsequent quotients $V_{i-1}/V_i$ are irreducible.

Let $A$ be the algebra $A=k[x]$ and let $V$ be the representation $V=k[x]/((x-\lambda)^n)$ for some $\lambda \in k$ and $n\in\Bbb N$. Find a filtration $V=V_0 \supset V_1 \supset \dots \supset ...
0
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1answer
30 views

Let $A=k[x]$ and let $V=k[x]/\big((x-λ )^{n} \big)$ for some $λ \in k$ and $n\in \Bbb{N}$. Then $V$ is indecomposable.

Theorem. Let $A=k[x]$ and let $V=k[x]/\big((x-\lambda )^{n} \big)$ be a representation of $A$ for some $\lambda \in k$ and $n\in \Bbb{N}$. Then $V$ is indecomposable. This is a theorem in my book. ...
0
votes
1answer
30 views

A smooth non-stably trivial smooth vector bundle

This may well be just a look-up, but do you have an example of a non-stably trivial smooth vector bundle? If it has a presentation as the vector bundle associated to the representation of some ...
3
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0answers
55 views

Characters of a finite group

Recently, I have been studying about Character Theory of Finite Groups, mostly from "Groups and Representations" by J. Alperin & R. Bell. In the aforementioned textbook, the characters of a finite ...
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47 views

Covering Spaces in Representation Theory.

I'm reading the paper "Covering Spaces in Representation Theory" of K. Bogartz and P. Gabriel. Now I'm in section 2, proposition 2.3, on the first three lines concludes that the functor $l \mapsto ...
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74 views

Theorem about the subgroup of a Lie group fixed by an involution

When trying to do Lie-theoretic calculations on Lie groups (finding the Bruhat decomposition, etc.) I've often come across expositions that seem to be implicitly using a result something like the ...
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32 views

Question about the equivalence of two linear representations.

I would like to know if this approach is correct. I have two distinct permutation representations and I have to prove that the associated linear representations are equivalent. In order to do this I ...
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0answers
8 views

$\overline{\phi}: G/H \to GL(V)$ irreducible representation then $\phi= \overline{\phi}\circ \pi :G\to GL(V)$ it's irreducible

Let $H\trianglelefteq G$ be a normal subgroup of $G$ and let $\pi: G\to G/H$ be the canonical projection. Suppose that $\overline{\phi}: G/H \to GL(V)$ it's an irreducible representation. Define the ...
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26 views

Infinite Cyclic group representation

I am trying to learn Group representation and have a basic question regarding infinite cyclic groups. I am trying to find a representation of infinite cyclic group in $GL_n(\mathbb{C})$ and ...
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15 views

Restriction of a Specht module to the alternating group

Let $n\in\mathbf{N}$ and denote by $S_n$ the symmetric group on $n$ letters. For $\lambda\vdash n$ a partition of $n$ the Specht module $S^\lambda$ defines an irreducible representation. What ...