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

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9
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1answer
333 views

Field of definition of representations of symmetric groups

Can one show in an elementary way, without recourse to Young tableaux etc., that the complex representations of symmetric groups are realisable over $\mathbb{R}$? It is easy to show that they are all ...
6
votes
2answers
4k views

What is the meaning of an “irreducible representation”?

What does it mean to talk about the "irreducible representatives of SO(3)"? I'm struggling to understand the concept of irreducible representations. Could someone give a concrete example for someone ...
0
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1answer
63 views

A compact group with a $G$-finite function

Suppose we have a compact group $G$ with continuous $f$ on $G$ that is also $G$-finite. I am told that then, out of all the irreducible representations $\pi$ of $G$ we must have $\pi(f)\neq 0$ only ...
2
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1answer
576 views

Irreducible representations and simple lie algebras

Could you give me a hint how to prove the following? A representation $R$ of a complex Lie algebra $\mathfrak{g}$ is irreducible iff the image $R(\mathfrak{g})$ is a simple Lie algebra.
1
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1answer
315 views

Irreducible representation decomposition

Any faithful finite group representation can be written as a sum of irreducible representations $\rho = \oplus_{i} a_i \rho_i$ such that $Ker(\rho)=0=\bigcap_i Ker(\rho_i)$ - is this sufficient to ...
0
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1answer
122 views

Sum of non-principal representation over the group is zero?

I am stuck on an exercise. (a) Let $\mathfrak{X}$ be an irreducible $F$-representation of $G$ over an arbitrary field. Show that $\sum_{g \in G} \mathfrak{X}(g) = 0$ unless $\mathfrak{X}$ is the ...
4
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2answers
174 views

Character theory questions

I am following the text by Isaacs on character theory and I have a few questions. From p. 10, it seems like an reducible representation is one whose matrix at each group element can be written in a ...
1
vote
1answer
297 views

the volume of 3-sphere

My question is why the volume of 3-sphere that lies between a rotation angle of $\theta$ and $\theta+d\theta$ is $$2\pi \sin^2(\theta/2) d\theta$$
1
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1answer
417 views

Dimension of irreducible representations of finite p-groups

Let $G$ be a finite $p$-group. Is it true that every irreducible representation of $G$ over an algebraically closed field of characteristic zero ($\mathbb{C}$, for example) must have dimension a ...
3
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1answer
115 views

What is the dimension of a representation

What is meant by dimension of a representation in the following excersize: "Prove that any irreducible representation of an abelian group has the dimension of 1"? I looked at the solution, and it ...
7
votes
1answer
237 views

What is a description for the following number theoretic object?

The title couldn't quite contain the question, so I didn't attempt to make it precise. I should note that this is the third or fourth question I've asked these past two days about problems I've been ...
6
votes
3answers
322 views

Basic Representation Theory

I'm reading a paper that uses representation theory, and I'm stuck on something simple. Say $\Delta$ is an abelian group, and $\hat{\Delta}$ its group of irreducible characters. Say $M$ is a ...
2
votes
2answers
318 views

An 'opposite' or 'dual' group?

Let $(G, \cdot)$ be a group. Define $(G, *)$ as a group with the same underlying set and an operation $$a * b := b \cdot a.$$ What do you call such a group? What is the usual notation for it? I tried ...
4
votes
1answer
166 views

Questions about p-adic representations

In a paper I'm currently reading, they have the following situation: $k$ is some number field that doesn't have a primitive $p^{th}$ root of unity, and $k(\zeta_p)$ a field above it with Galois group ...
1
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1answer
104 views

Question about notation / terminology

I'm given the following in a homework question: Let $G$ be a group and $k$ an algebraically closed field. (a) Show that the action of $G \times G$ on $C_k (G)$ defined by $$ (g_1, g_2) \varphi (x) ...
1
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2answers
299 views

What are the differences between classical Yang-Baxter Equation and quantum Yang-Baxter Equation?

what are the differences between classical Yang-Baxter Equation and quantum Yang-Baxter Equation? Thank you very much.
2
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2answers
581 views

Dimension of Hom(U, V)

all. I am reading a paper and I could not understand the dimension of the vector space $Hom_{\Gamma}(\rho_i\otimes \rho, \rho_j)$. The paper is http://pages.uoregon.edu/njp/Lusztig92.pdf, section 1.1, ...
5
votes
1answer
355 views

Drinfeld Center

Let $\mathscr{C}$ be a strict monoidal category. I will denote the product of $\mathscr{C}$ by $\otimes$. The Drinfeld center $\mathscr{Z(C)}$ of $\mathscr{C}$ is the category with object $(X,\phi)$ ...
4
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2answers
363 views

Defining representations; representations and semidirect products

Let $G$ be a group, $X$ a set. Defining an action $G\times X \to X$ is the same as defining a group morphism $\rho: G\to Sym(X)$, through the formula $g\cdot - = \rho(g)$. The morphism $\rho$ is ...
3
votes
1answer
119 views

a couple of canonical maps in representation theory

Today I came across two possible canonical maps in the context of representation theory from reading stackexchange and a friend's email. However, I don't really know what they are (and have thus far ...
79
votes
8answers
5k views

Importance of Representation Theory

Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I ...
1
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1answer
528 views

representation of quaternion group over $\mathbb{C}$ and $\mathbb{R}$

The quaternion group of order 8 has an irreducible two dimensional representation over $\mathbb{C}$ but how does one show that this representation cannot be defined over $\mathbb{R}$?
2
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3answers
337 views

How to differentiate a homomorphism between two Lie groups

Let $G$ and $H$ be two Lie groups and $\rho: G \to H$ be a homomorphism. How to differentiate $\rho$ to obtain a Lie algebra homomorphism $d\rho_e: T_eG \to T_eH$?
1
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1answer
200 views

Consequence of Schur's lemma

I am having trouble understanding part of a lecture in class. Let $G$ be a group, $\phi$ a representation of $G$ into $GL_n(C)$, and let $E^{ij}$ denote the matrix with 1 in the $i$th row and $j$th ...
4
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1answer
220 views

Symmetric matrices as module over the skewsymmetric ones

Let $\mathfrak {so}_{n}$ denote the skew-symmetric complex $n \times n$-matrices and let $M$ denote the symmetric $n \times n$-matrices of trace 0. As I understand, $M$ is a module over $\mathfrak ...
0
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1answer
214 views

Eigenspaces of a representation

Let $g$ a Lie algebra and $V$ a finite-dimensional irreducible $g$-module, then each generalized eigenspace of $V$ is actually an eigenspace? If not, what is a condition to guarantee this fact? ...
1
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1answer
359 views

Complexifying representations

Let me try to split the question in a few parts, I would like to understand the claim that all non-degenerate bilinear symmetric forms are equivalent over the complex while for the reals they can be ...
6
votes
1answer
434 views

Representations of a non-compact group are labeled by its maximal compact subgroup?

I don't have much of any awareness about the representation theory of non-compact Lie groups but I bumped into it for my work. Is there some idea that the representations of a non-compact group are ...
15
votes
2answers
285 views

Proving finite dimensionality of modular forms using representation theory?

It is well known how to use algebraic geometry (differentials, divisors, and Riemann-Roch) in order to prove the finite dimensionality of the vector space of modular forms of some fixed weight and ...
1
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1answer
73 views

semisimplicity for subgroup

let $G$ be a group (can assume finite) acting on a vector space $V$, and assume that the action is semisimple. Let $H$ a normal subgroup of $G$. Why also the action of $H$ on $V$ is semisimple too?
3
votes
1answer
209 views

Why is the identity representation of $SL(2,k)$ isomorphic to its dual representation?

By identity representation I mean the representation sending each element of $SL(2,k)$ to itself. Is there a simple way to see this isomorphism? I feel like I am missing something incredibly basic ...
0
votes
1answer
260 views

Representations of $S_3$

In these Karen Smith's notes on representation of finite groups, on page 50 the irreducible representations of $S_3$ are found. If $\sigma=(1 2 3)$ and $V$ is a complex representation of $S_3$, I ...
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2answers
113 views

An equality about characters of representations

This is an equality that I am gleaning out of some papers that I have been reading. I am not sure I am reading it right. Hopefully people will correct it. Let $U$ be a group element and $R$ be a ...
1
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1answer
417 views

Representations of U(n) using bosons and fermions

I would be glad if someone can help me understand the argument in the first paragraph of page 4 of this paper. Especially I don't understand their first sentence, "Using N bosons (fermions) ...
2
votes
0answers
132 views

Reducibility of a faithful representation of a subgroup of a centreless group

I've been stuck on this homework question for a couple of days. Any hints would be much appreciated. Let $G$ be a finite group and $\rho : G \to \mathrm{GL}(V)$ be a faithful representation, with $V$ ...
4
votes
1answer
188 views

The primitive spectrum of a unital ring

I'm trying to investigate the notion of primitive spectrum and its so-called Jacobson or hull-kernel topology, but I can only find references which define it for C*-algebras: see the Wikipedia page ...
3
votes
2answers
247 views

what are good references for learning about vector bundles and their sheaves of sections?

I am a beginner in representation theory and algebraic geometry, so that references giving clear explanations of things like the tautological line bundle on $\mathbb P^n$, its dual, and the associated ...
0
votes
1answer
235 views

questions about coroot

I am reading the lecture notes of geometric representation theory: http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf. I have a question on coroot. In general, if we have a root $\alpha$, then the ...
4
votes
1answer
340 views

What is the Isomorphism between the adjoint representation of SU(2) and its representation on rank 2 symmetric tensors?

Let su(2) be the Lie algebra of SU(2), thought of as a representation of SU(2) by conjugation. Let S be the rank-2 symmetric tensors over the complex numbers, acted on by SU(2) in the obvious way. ...
4
votes
1answer
200 views

what is the usual topology on a vector space?

I do not understand the topology of a Lie group clearly. Let $G$ be a Lie group and $T_eG$ be its tangent space at the identity $e \in G$. Why $Aut(T_eG)$ is an open subset of the vector space of ...
5
votes
1answer
203 views

Parabolic subalgebra

Let $R$ a root system and $\Delta$ be a simple system of roots of a Lie algebra $\mathfrak g$, $\Delta'\subset \Delta$ and $R(\Delta')=R\cap \mathbb Z(\Delta')$. Define ...
5
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0answers
96 views

How to prove “The maps factoring through an injective object are precisely the null-homotopic maps”

Thanks for your attention, I'm an undergraduate. I'm reading the book of Dieter Happel, Triangulated categories in the representation theory of finite dimensional algebras, I cannot prove this ...
3
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1answer
388 views

Eigenvalues of representations

Let $\rho$ be a representation of $G$ on $V$. Why are its eigenvalues roots of unity?
4
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2answers
318 views

Representation is irreducible

Let $\rho : S_n \to \mathrm{GL}(\mathbb{C}^n)$, where $\rho(\sigma)(x_1, \ldots, x_n) = (x_{\sigma^{-1}(1)}, \ldots, x_{\sigma^{-1}(n)})$. How can you prove that $W = \{ (x_1, \ldots, x_n) : x_1 + ...
0
votes
1answer
261 views

Induced representation

Let H be a subgroup of G and V the trivial representation of C[H]. Prove that we have an isomorphism of C[G]-modules between $\mathrm{Ind}_{H}^{G}(V)$ and C[G/H]. Thanks for your help.
2
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1answer
241 views

torsion representation

Let $\mathbb{Z}_p$ be te ring of p-adic integers and let $T$,$T'$ be two free $\mathbb{Z}_p$-module with a continuous action of $G_{\mathbb{Q}_p}$ (the absolute Galois group of $\mathbb{Q}_p$). It is ...
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1answer
64 views

How to define the action of U(G) in this situation?

The usual action of $fg$ on $u⊗v$, where $f,g$ are elements in the Universal Enveloping Algebra $U(G)$ of a Lie algebra $G$ and $u,v$ are elements of a representation $V$ of $G$, is given by ...
3
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1answer
422 views

Representation over Hom(V, W)

Given representations $\rho_1 : G \to \mathrm{GL}(V)$ and $\rho_2 : G \to \mathrm{GL}(W)$, how can we define explicitly the representation of $G$ over $\mathrm{Hom}(V, W) \cong V^* \otimes W$?
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1answer
89 views

No linear relation between matrix coefficients of all the irreducible repn of finite group

Fix a finite group $G$, and look at all its irreducible representations/$\mathbb{C}$. It is said in Serre's book that "there cannot be any $\mathbb{C}$-linear relation between the matrix coefficients ...