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

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Exercise 2.13, I. Martin Isaacs' Character Theory

I am trying to solve the exercise 2.13 in Isaacs' Character Theory Book. However I met some difficulties, let me sketch out what I am thinking so that you may tell me a hint. The problem 2.13 is ...
3
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Help with Fulton and Harris question 6.16

The Pieri formula gives a decomposition $$\textrm{Sym}^d V \otimes \textrm{Sym}^d V = \bigoplus \mathbb{S}_{(d+a, d-a)}V,$$ the sum over $0\leq a \leq d$. The left-hand side decomposes into a ...
4
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3answers
260 views

Unitary representation of $SO(3)$

Definition: $\mathcal{H}$ be a Hilbert space and $U(\mathcal{H})$ denote the unitary operators on it, If Unitary representation of a matrix lie group $G$ is just a homomorphism $\Pi:G\rightarrow ...
4
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186 views

Representation of Lie algebra of $\textrm{SU}(2)$

$V_m=$Homogeneous polynomials in complex variable with total degree $m$, Let $U\in SU(2)$ is just a linear map on $\mathbb{C}^2$, Define a Linear Transformation $\Pi_m:V_m\rightarrow V_m$ given by ...
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1answer
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Some representation of $SU(2)$

$V_m=$Homogeneous polynomials in complex variable with total degree $m$, could any one tell me how is that Linear Transformation look like explicitly?And would you please tell me how this is an ...
3
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67 views

Question about the global dimension of End$_A(M)$, whereupon $M$ is a generator-cogenerator for $A$

Let $A$ be a finite-dimensional Algebra over a fixed field $k$. Let $M$ be a generator-cogenerator for $A$, that means that all proj. indecomposable $A$-modules and all inj. indecomposable $A$-modules ...
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1answer
224 views

Hom and Tensor Product of Linear Maps

I am reading Claudio Procesi's book on Lie groups and on page 105 there is something I don't understand. Let $U,V,W$ be vector spaces. Let us consider the product space $\hom(V,W) \times \hom(U,V) $ ...
3
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76 views

correspondence between finite dimensional complex representation

I would like to understand the following fact, shall need help, Thank you. " There is a one- to- one correspondence between the finite dimensional complex representation $\Pi$ of $SU(3)$ and finite ...
5
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1answer
73 views

Representations of $\pi_1M$ and Heegaard Splittings

I am reading Floer's Instanton-Invariant paper, and am stuck on a sentence. To set the stage: Consider a closed connected oriented 3-manifold $M$ and the nonabelian group $SU_2$. Denote the ...
4
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262 views

Group representations $\pi$ and $\rho$ are irreducible, but their outer tensor product $\pi\times\rho$ is not

Let $\mathbb{H}$ be the four-dimensional real vector space of quaternions, G multiplicative group $\mathbb{H}\backslash\{0\}$ and H multiplicative group $\mathbb{C}\backslash \{0\}$. Let $\pi$ be a ...
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2answers
89 views

Minimal Right Ideals in $\Bbb{C}[G]$

Let $G$ be a finite group. Consider the group algebra $\Bbb{C}[G]$ as a right module over itself. By Maschke's Theorem, $\Bbb{C}[G]$ is semisimple. How can we identify all the minimal right ideals of ...
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109 views

Decomposition of $\text{Sym}^{k+6}V$

Let $V$ be the standard representation of $S_3$. Show that $\text{Sym}^{k+6}V\cong \text{Sym}^kV\oplus \mathbb{k} S_3$. This is exercise 1.12 from Fulton and Harris. There's a solution here that ...
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1answer
145 views

Definition of differential of Adjoint representation of Lie Group

Let $g$ be an element of Lie Group $G$, and $\gamma(t) : \mathbb{R} \rightarrow G$ be a path in $G$ such that $\gamma(0) = e$, the identity element of $G$. Denote the tangent space at $e$ as $T_eG$, ...
4
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128 views

Showing a representation is irreducible by showing that a degenerate subspace has codimension one.

Throughout $\phi$ be a continuous character from a locally compact abelian group $G$ to the circle. I'm trying to understand this implication. Basically we want to show that a certain representation ...
3
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Stuck in proof of combinatorial identity - Fulton and Harris A.39

I'm trying exercise $A.39$ in Fulton and Harris. They suggest to first prove the formula $$|x_j^{l_i}| \prod_{j=1}^k(1-x_j)^{-1} = \sum |x_j^{m_i}| \hspace{1in} (\ast)$$ where the sum on the right ...
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Weyl character formula for locally compact Lie groups.

I was just wondering if there exists such a formula. Specifically I need to calculate characters of irreducible representations of GSp$(4,\mathbb{C})$. I know how to do it for the compact Lie group ...
2
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1answer
67 views

Modules of a group over different fields.

Let $G$ be the cyclic group of order 2 and let $V$ be the regular $\mathbb{F}G$-module of $G$. Determine the $\mathbb{F}G$-submodules of $V$ where $\mathbb{F}$ has each of zero or positive ...
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117 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
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38 views

Smallest dimensional irreps of semi-simple Lie algebras

I'm wondering if there is a reference that lists the first couple smallest dimensional irreducible representations of each semi-simple Lie algebra. I know these can be found using the Weyl dimension ...
0
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1answer
51 views

If $\rho: G \to GL(V)$ is a representation with sub-representation $\tau$, is $\tau^{\otimes n}$ a subrepresenation of $\rho^{\otimes n}$?

I'm working over an algebraically closed field of characteristic $p>0$ so I'm not assuming that $\tau$ is a direct summand of $\rho$. I think I can prove this by looking at the Kronecker product of ...
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212 views

Idempotent and Hermitian vectors in Group Algebra

Let $C$ be the field of complex number and $G$ a finite group, then define $C[G]$ be a vector space over $C$, with elemnts of $G$ as the basis. Then any element in $C[G]$ can be written as $\sum_{g ...
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506 views

Lie algebra action from Lie group action: coordinates

Here's the setup: I have $SL(2;\mathbb{C})$ acting on $V = \mathbb{C}[z,w] = \oplus_d V_d$, where $V_d$ is the homogeneous complex polynomials of degree $d$. The action is precomposition: ...
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1answer
542 views

What is a rational character?

Let $G$ be the group of $F$-points of a connected, reductive group over a $p$-adic field $F$. The unramified character of $G$ are $\chi\circ\psi$ where $\chi$ is an unramified character of ...
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336 views

Faithful representation implies group is cyclic

Suppose there is a faithful representation $\rho:G\to SL_2(\mathbb{R})$. Prove that $G$ is cyclic. I know there has to be something special about its representation being special (no pun ...
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Methods of Multilinear Algebra in Representation Theory

I have been interested in representation theory lately in particular on that of Lie algebras. Now I have noticed that one way of building representations is to take tensor/exterior/symmetric powers. I ...
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1answer
110 views

Action of $\mathfrak{sl}_2(\Bbb{C})$ on $\textrm{Sym}^2 V$

I am reading Fulton and Harris and on page 150, there is the following passage (in the second paragraph) that I don't understand: "Similarly, a basis for the symmetric square $W = ...
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2answers
166 views

questions about representation theory/structure theorem for finitely generated modules

I am reading a book in representation theory by James and Liebeck. They define an FG-module as: Let $V$ be a vector space over the field $F$ and let $G$ be a group. Then $V$ is an FG-module if a ...
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31 views

Help with understanding certain basic representation theoretic objects and notation

I heard a talk recently on number theoretic representation theory, in which the speaker suggested that we focus on the case $G=$GL$_2$, with $\rho$, the representations, being thought of as sym$^n$ ...
2
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1answer
108 views

Irreducibility of a 4 dimensional representation of $\mathfrak{sl}_2(\Bbb{C})$

Let $\mathfrak{g}$ be the complex Lie algebra $\mathfrak{sl}_3(\Bbb{C})$. Consider adjoint representation $\textrm{ad} : \mathfrak{sl}_3(\Bbb{C}) \to \textrm{gl}(\mathfrak{g})$. $\mathfrak{g}$ has ...
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1answer
78 views

Estimation of a polynomial

I'm currently reading the following paper: http://arxiv.org/abs/1209.0612 and got stuck on Proposition 3.1 (2). The claim translated to polynomials is the following: Assume $n\geq 3, c\geq 1, ...
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is there a better classification of a desirable algebra?

Consider a finite-dimensional, associative algebra presented as follows: $$\mathcal{A} = e_1 \mathbb{R}\oplus e_2 \mathbb{R} \oplus \cdots \oplus e_n \mathbb{R} $$ with multiplication $*: \mathcal{A} ...
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10answers
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What's a good place to learn Lie groups?

Ok so I read the following article the other day: http://www.aimath.org/E8/ and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought "oh it must have something to ...
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3answers
139 views

Character of $S_3$

I am trying to learn about the characters of a group but I think I am missing something. Consider $S_3$. This has three elements which fix one thing, two elements which fix nothing and one element ...
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1answer
351 views

Proof of the second part of Schur's lemma

Background According to PlanetMath Schur's lemma is this: Let $G$ be a finite group and let $V$ and $W$ be irreducible $G$-modules. Then, every $G$-module homomorphism $\,f: V \to W$ is either ...
3
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2answers
274 views

Distinguishing the two irreducible representations of odd-dimensional complex Clifford-Algebras

The complex Clifford algebra $A$ of a complex, non-degenerate quadratic space $(V,q)$ of odd dimension $2k+1$ admits up to isomorphism exactly two non-trivial, irreducible and finite-dimensional ...
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Best books on Representation theory

What are some of the best books on Representation theory for a beginner? I would prefer a book which gives motivation behind definitions and theory.
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53 views

“Two ways” to reduce a module

Let $M$ be a module over a principal ideal domain $R$ and $\mathfrak{m}$ a maximal ideal of $R$ with residue field $R/\mathfrak{m}=k$ of characteristic $p$. Under what circumstances are the modules ...
3
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1answer
148 views

Simple algebra over a field of char zero

Let $A$ be an associative finite dimensional simple algebra over a field $K$ of charicteristic zero. Prove that $A$ can not have a basis consists of nilpotent elements. $Remark$: The statement is ...
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1answer
165 views

Finite dimensional algebra with a nil basis is nilpotent

Prove that a finite-dimensional algebra $A$ over a field $K$ of characteristic zero having a basis consisting of nilpotent elements $\{e_1,...,e_n\}$ is nilpotent. My approach: Let $m_i$ be the ...
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1answer
66 views

representation of associative algebra is faithful iff it's an isomorphism?

Let $A$ be an associative algebra over a field $K$ and let $\rho:A \to \operatorname{End}_K(V)$ be a representation of $A$. Is it true that if $\rho$ is faithful then it's also an isomorphism? Note: ...
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Generating function for characters of representations

One example of such a generating function that I know how to derive is for $SU(2)$, $\frac{1}{(1-tx)(1-\frac{t}{x})}$. The coefficient of $t^n$ in the above function is the character in the $n+1$ ...
10
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1answer
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Do all representations of finite groups have one-dimensional subrepresentations?

Let V be a representation of a finite group G, and $v\in V$ - a nonzero vector. Put $$u = \sum_{g\in G} gv.$$ Then for any $g\in G$ we have $gu = u$ and therefore $<u>$ is a subrepresentation of ...
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Some irreducible character separates elements in different conjugacy classes

Let $x$ and $y$ be elements that are not conjugate in $G$. Then there is some irreducible character $\chi$ such that $\chi(x) \not = \chi(y)$. Clearly the "irreducible" part isn't important, ...
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What is the relationship between Mackey's theorem in character theory and Mackey's theorem in transfer theory?

Here are the statements of the two theorems. The first statement I took from a paper I have been reading, but I believe can also be found in Isaacs' Character Theory of Finite Groups as an exercise. ...
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1answer
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conjugacy classes in representation theory

I have a question on conjugacy classes in this post, especially to this sentence: "if $g$ is a rotation of order $5$, then it is $K$-conjugate to each of $g^{13},g^{13^2},g^{13^3},g^{13^4}=g$". ...
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How can the trivial representation not be in some $R\otimes R$?

Where do I foul up? In (e.g.) $SU(3)=A_2$ the tensor product $(1,0)\otimes (1,0)=(2,0)+(0,1)$. No $(0,0)$. But on the other hand still $(1,0)\otimes (0,0)=(1,0)$. This immediately begs ...
3
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1answer
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Eigenvalue of an element

what does it mean, if you say that $x$ is an eigenvalue of an element $g \in G$, where $G$ is a group? I know this definition just for matrices, not for elements.
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1answer
237 views

Connection between ramification in number fields and Clifford theory

Consider algebraic number fields $\mathbb{Q} \subseteq K \subseteq L$ with rings of integers $\mathbb{Z}\subseteq \mathcal{O}_K \subseteq \mathcal{O}_L$. If $0 \neq \mathfrak{p} \trianglelefteq ...
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How to expand a representation

If there is a finite group with a normal subgroup and a representation of this subgroup over a finite field. How can one expand this representation to a representation of the whole group? Are there ...
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A non-continuous p-adic representation

I am looking for an example of a non-continuous homomorphism $$G \to GL_r(\mathbb C_p)$$ from a profinite (topologically finitely generated) group $G$, where $\mathbb C_p$ is the completion of an ...