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

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8
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1answer
417 views

Osp, USp, SU(,) and PSU

I would be glad if someone can give me some (hopefully easy to understand!) references for learning about these groups Osp, USp and PSU and their representations. I run into these mostly while ...
4
votes
1answer
130 views

How to compute the Gel'fand Models for a (quantum) Lie Algebra

Given a lie algebra $g$, how does one approach finding the Gel'fand models? For clarity, by this I mean $\bigoplus_{\lambda\in P^+}V(\lambda)$ where $P^+$ are the dominant weights, and ...
3
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1answer
121 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 ...
1
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1answer
627 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 $\mathbb{R}$?
5
votes
2answers
143 views

When are the principal series of GL_2(F) square integrable?

Let $F$ be a local field, $G=GL_2(F)$, $\chi_1,\chi_2$ quasicharacters, and $B(\chi_1,\chi_2)$ be the principal series (we are assuming $\chi_1,\chi_2$ to be such that the representation is ...
1
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1answer
214 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
votes
1answer
223 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
232 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? ...
20
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4answers
1k views

Irreducible representations of Poincaré group

I am looking for any reference on Wigner's classification of irreducible representations of the Poincaré group. I know the classification, but is there any reference where the representations ...
6
votes
1answer
483 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 ...
1
vote
1answer
75 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
225 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 ...
1
vote
1answer
276 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 ...
4
votes
1answer
1k views

Representation theory of $SO(n)$

This is probably not a very ethical question to ask but I need to have a fast introduction to a range of concepts about the representation theory of the $SO(n)$ and I would be happy to see some online ...
1
vote
2answers
114 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 ...
2
votes
0answers
47 views

notation question $Aut_{Modk}(A)$

What is $Aut_{Modk}(A)$ if A is a finite generated algebra, k is a field? Is it a group of automorphisms of A as a vector space over k or what?
1
vote
1answer
445 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) ...
19
votes
1answer
795 views

Why is a general formula for Kostka numbers “unlikely” to exist?

In reference to Stanley's Enumerative Combinatorics Vol. 2: right after he has defined Kostka numbers (section 7.10), he mentions that it is unlikely that a general formula for $K_{\lambda\mu}$ ...
3
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0answers
148 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
218 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 ...
6
votes
1answer
514 views

Group representations over p-adic vector spaces

Recently I have found a need to learn more about p-adic group representations over a p-adic vector space. Generally, this motivates a study of representations $\left( V, \rho \right)$ for some group ...
3
votes
2answers
263 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 ...
3
votes
1answer
309 views

Line bundles, line bundles on a homogeneous space, and sections of line bundles

I have some difficulty in understanding the concepts: line bundles, line bundles on a homogeneous space, and sections of line bundles. These concepts are on page 140 (the first paragraph of section ...
0
votes
1answer
280 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 ...
2
votes
3answers
363 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 $d\rho_e: T_eG \to T_eH$?
4
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1answer
348 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
203 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 ...
15
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2answers
316 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 ...
5
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0answers
101 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 ...
1
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1answer
399 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
2answers
477 views

sl(2,C) and the harmonic oscillator

I've been studying the finite-dimensional representations of the lie algebra sl(2,C). I've read that these representations are related to the harmonic oscillator and the associated raising and ...
3
votes
1answer
444 views

Eigenvalues of representations

Let $\rho$ be a representation of $G$ on $V$. Why are its eigenvalues roots of unity?
5
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1answer
215 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 ...
4
votes
2answers
333 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 + ...
19
votes
4answers
546 views

How is $\operatorname{GL}(1,\mathbb{C})$ related to $\operatorname{GL}(2,\mathbb{R})$?

I am trying to get a grasp on what a representation is, and a professor gave me a simple example of representing the group $Z_{12}$ as the twelve roots of unity, or corresponding $2\times 2$ matrices. ...
0
votes
1answer
271 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.
5
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1answer
325 views

Connection between Hecke operators and Hecke algebras

Hecke operators are things that act on modular forms and give rise to a lot of interesting arithmetical results: http://en.wikipedia.org/wiki/Hecke_operator On the other hand on the wikpedia page ...
2
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1answer
247 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 ...
18
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3answers
1k views

Representation theory of the additive group of the rationals?

What do the finite-dimensional continuous complex representations of the additive group $\mathbb{Q}$ with the usual topology look like? With the discrete topology? Which representations are ...
1
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1answer
65 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
votes
1answer
503 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 $\mathrm{Hom}(V, W) \cong V^* \otimes W$?
1
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1answer
97 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 ...
89
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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 ...