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

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Artin Algebra Representation Chapter Resource Request

I am working through chapter 10 of Artin's Algebra 2ed which introduces Group Representations. However, I've found that the approach to introducing groups is unlike the usual method used by more ...
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How to rigorously show tensor identities using symmetry arguments?

I am wondering how to rigorously show tensor identities such as the following. Let $n$ denote the radial unit vector in $D$ dimensions. Then $\langle n_i n_j \rangle = \frac 1 D \delta_{ij}$ and ...
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Decomposing some representations as a direct sum of irreducibles

I'm taking $V$ to be the standard representation of $S^3$. I'm looking for the decomposition of the following representations as a direct sum of irreducibles. (a) $V\bigotimes V$ (b) $Sym^2$ $V$ (c) ...
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69 views

Inner product doesn't matter for Schur orthogonality?

I'm reading Knapp's Basic Algebra, specifically the section about Schur orthogonality relations. Given a representation $R: G \to \text{End}(V)$, he defines $V_R$ to be the vector space of matrix ...
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cohomology of general linear group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let $\mathrm{GL}_n(\mathbb{Z}_2)$ be the group consisting of all $n\times n$ matrices with entries in $\mathbb{Z}_2$ with non-zero determinant. What is the ...
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32 views

$\overline\pi$ vs. $\check\pi$

Let $\pi$ be the automorphic representation of ${\rm GL}_2$ or $B^\times$ ($B$ an indefinite quaternion algebra over $\Bbb Q$) of central character $\varepsilon$ attached to some holomorphic newform ...
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29 views

problem book suggetion

I'm taking a course on basic representation theory, where upto midsem we're supposed to learn upto $5$ th chapter of serre. Can one please suggest some book which consists of nice problems ? I've ...
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46 views

Is there such a notion of “expansion” in groups?

Given a subset of elements of a finite group $G$, I would like it to be such that the set of all distinct words (as elements of $G$) that can be formed from this set is exponentially large in the size ...
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Irreducible representation and rank one projetion

Let $A$ be a C*-algebra with a nonzero minimal projection $e$. a - Show that if $\{\pi, H\}$ is an irreducible representation of $A$ such that $\pi(e) \neq 0$, then $\pi(e)$ is a projection of rank ...
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25 views

Filling in the last two rows of a character table?

I have a hopefully simple question about character tables. If I know all but two rows of a character table, and the character table is sufficiently large (say, at least 4 rows), do the orthonormality ...
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55 views

Simplifying a direct sum $\mathbf{3}\oplus\mathbf{3}\oplus\mathbf{2}$ etc

In particle physics, one often uses the dimensionality of the irrep to label the irrep (apparently this is not a very good idea since the dimension does not unambiguously determine the rep.). What are ...
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36 views

Relation between Poisson bracket and commutator.

In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b \in \mathbb{C}_q[T].$ It seems that we have $$ [a, b]=(q-1)\{a,b\}+o((q-1)^2). $$ ...
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29 views

What is the root structure of the Diffeomorphism Group?

Being a physicist, I think it'd be cool to have Coxeter plane projections of the root systems of the symmetry groups associated with the fundamental forces hanging on my walls (example for E8: ...
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25 views

Showing that a subrepresentation is isomorphic to the trivial representation

I'm considering $V$ to be the regular representation of a group $G$ and $W$ to be the 1-dimensional subspace of $V$ generated by the element $x=\sum_{s\in G} e_s$. I'm trying to show that $W$ is a ...
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21 views

When are all Gorenstein projective also pure-injective?

For an artin algebra of finite global dimension, each Gorenstein projective module is projective then is pure-injective. Are there any other examples having this property? That is, all Gorenstein ...
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50 views

Simple question on the unitary representation.

Let $\pi_1,\pi_2,\pi_3$ are all irreducible, unitary representation of some algebraic group G. Then is it ture that $$Hom_{G}(\pi_1,\pi_2 \otimes \pi_3) \simeq Hom_{G}(\pi_1 \otimes ...
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36 views

Irreducible Representations of Nilpotent Lie algebras

By Lie's theorem all irreducible representations of a solvable Lie algebra over $\mathbb C$ are one dimensional. What are all irreducible representations of a nilpotent Lie algebra ?
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80 views

Constructing irreducible representations of quaternion group over $\mathbb{Q}$

I am a beginner in studying the representation theory and I am doing some exercises in this field. So this is not a homework. My question is about constructing all irreducible representations of ...
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39 views

Matrix representations of free groups?

What is the general form of faithful matrix representations of free groups? How about for the simple case of $F_2$?
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Infinite dimensional representation such that every subrepresentation is reducible

Let $V$ be a nonzero finite dimensional representation of an algebra $A$. a) Show that it has an irreducible subrepresentation. b) Show by example that this does not always hold for ...
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45 views

Can we say that $A$ is a complement for a group $G$?

Let $A$ be a frobenius complement for a $G$ i.e. $A$ act on $G$ by automorphism s.t. $C_A(g)=e$ for all nonidentity $g$. Now, Action of $A$ can be linearly extended so that $A$ act on $F[G]$. As a ...
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Reference request: classification of simple Lie groups and simple real Lie algebras

I am trying to understand the classification of simple Lie groups and the theory of highest weights for semisimple Lie groups by first understanding the case for complex Lie algebras, then relating to ...
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36 views

How to tackle a research journal - level course in Lie Theory and Representation Theory?

I am taking a course in Lie Theory and Theory of Representations this year, where starting from the second lecture, Lie Theory is heavily bundled with Theory of Representations. It is pretty much a ...
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42 views

Character of a Representation

Suppose that we have a representation $V$ of a group $G= SU(2)$ . Is it true that if $ \chi_V \cdot c \neq 0$ for some non-zero constant c, then the trivial representation must be one of the ...
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39 views

Schur-Weyl Duality - references

I'm trying to understand the Schur-Weyl duality. Unfortunately the lecture notes I have don't describe it very detailed. Any good references?
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49 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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23 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!
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31 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 ...
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47 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 ...
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37 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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34 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 ...
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38 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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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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52 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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22 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 ...
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multiplicities of irreducible representations

Let $G$ be a finite group and $G'$ be a subgroup. Let $\rho$ be a one-dimensional group of $G'$. Then define $\psi$ to be the induced action of $\rho$ - $\psi:= Ind_{G'}^G \rho$ Is there any general ...
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30 views

Spin Representations and Galois correspondence?

I have a vague question regarding the Spin representations. Is there a "quick" way of seeing that $Spin(2n)$ has exactly two irreducible representations which do not factor through $SO(2n)$, and one ...
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45 views

Matrix representation and permutation matrices

In order to find the matrix representation associated to a permutation representation I identify each permutation with the corrisponding matrix representation. How can I prove that these matrices ...
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limit of regular hyperbolic integrals is a unipotent integral (GL2 calculation)

In developing a simple trace formula for $G$=GL$_2$ over a number field $F$ one encounters the following identity of local integrals: $$\int_{G_v}f_v(g^{-1}\begin{pmatrix}1 & 1\\ 0 & ...
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44 views

Questions about the bracket

In the map $\phi : L \mapsto \mathfrak {U}(L) $, where $ L $ is a lie algebra and $\mathfrak {U} $ is a universal enveloping algebra of $ L $. (1) Is the following relation true? If $[xy]=z$ in $ L ...
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20 views

P-adic Lie groups - Representation theory

I am quite familiar with the Representation Theory for locally compact groups and nilpotent Lie groups. I want to start with the study of $p$-adic Lie groups representation theory, in particular ...
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25 views

How do I map Df(w) to it's [lie] group/algebra representation?

E.G. For $p,w\in(\mathbb{R}^3,+,\times_\vartheta)$ with $(\mathbb{R}^3,+)$ a vector space and with $p=(r,s,t)$, $w=(x,y,z)$ where we have $p\times_\vartheta ...
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33 views

regular representation of algebras

Let suppose we have universal enveloping algebra, what is the meaning of the notion of the right regular representation of that? How can we determine the right regular representation of universal ...
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44 views

Intuition behind PBW

The PBW theorem states: $\omega:\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras. Where $\mathfrak {S} $ is the symmetric tensor algebra of a Lie algebra $ L $. Where $\mathfrak ...
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50 views

Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and let $H$ be ...
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Representation of a group, and finite index subspaces

Been working on this for a while and haven't gotten anywhere. I would really appreciate some hints. Let $G$ be a group, not necessarily finite. $V=\mathbb C [G] $ a vector space with basis $(e_g, ...
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Why does $\rho_{\mathbf{Z}^t\otimes P}=\rho_R$ imply the isomorphsim of $\mathbf{Z}^t\otimes P\cong R$?

so I have happened upon a thesis regarding the calculations of various $\mathbf{Z}D_6$ modules and their isomorphisms and came across a technique which is bothering me. Let me give an example. Let ...
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24 views

Examples of the local Langlands correspondence

I'm trying to compile some examples of the local Langlands correspondence, with the aim of motivating the statement and also just giving some concrete traction on how it works. I would especially like ...
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80 views

matrix of the dual representation: inverse of the transpose

I have a doubt concerning the dual representation. Can someone check that what I wrote is correct please? Let $A: V \longrightarrow V$ be linear, the dual map $A^T : V^* \longrightarrow V^*$ is ...
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84 views

Prove the Weyl's complete reducibility Theorem on finite-dimensional $\mathfrak{g}-modules$ by Kostant's $\mathfrak{n}$-cohomology result

I've met an exercise in Kumar's book ("Kac-Moody Groups, their Flag Varieties and Representation Theory", Chapter III, page 89, Ex. 3.2. E, (1) & (2)). But I have no idea about its proof. Any ...