Questions tagged [representation-theory]
For questions about representations or any of the tools used to classify and analyze them. A representation linearizes a group, ring, or other object by mapping it to some set of linear transformations. A common goal of representation theory is classifying all representations of some type. Representation theory is a broad field, so questions not including the word "representation" may be appropriate.
3,317
questions with no upvoted or accepted answers
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conjugacy class preserving deformation of symmetric group
It is a well-known fact in a group representation theory that all elements of the same conjugacy class possess the same trace (character) in any representation of a group. I deal specifically with ...
2
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91
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Lemma 8.1 of "The Local Langlands Conjecture for GL(2)"
Let $F$ be a non-Archimedean local field, and $N$ the subgroup of $G=GL_2(F)$ of the form $\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}$ with $x \in F$. Let $(\pi,V)$ be a smooth ...
2
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43
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Decomposition of the exterior algebra of a representation of $Sp(1)\times GL(n,\mathbb{H})$
Let $H:=\mathbb{C}^2$ be the standard representation of $Sp(1)$ and $E^\ast:=\mathbb{C}^{2n}$ be the standard dual representation of $GL(n,\mathbb{H})$. Then $H\otimes E^\ast$ is a representation of $...
2
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108
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Splitting of a topological vector space (TVS) into an (a) countable sum and (b) direct integral of subspaces
TVS = topological vector space. Any subspace of a TVS is a TVS in the induced-topology sense.
DEFINITION
For TVS spaces ${\mathbb{V}}_1\subset\mathbb V$, a TVS subspace ${\mathbb{V}}_2\subset\mathbb V$...
2
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37
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Representations of $O(2)$ and related problems
I'm currently studying Group Theory in order make a further application to physics and understand the math of some physical theories.
I know that $SO(2)$ literally is a special case for $O(2)$ and ...
2
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1
answer
141
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D-module matching with a Verma module
I am recently learning about the Beilinson-Bernstein localization theorem and I am working on examples matching D-modules with Lie algebra representations. I would like to ask which D-module ...
2
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333
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Why find Casimir elements?
Given a semi-simple Lie algebra $\mathfrak g$ the Killing form induces a canonical central element of $U(\mathfrak g)$. This is seen as a very important and useful thing.
Why is it useful to know ...
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61
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Residual Mod 3 Representation Attached to Elliptic Curve is Not Induced By a Certain Galois Group
In Cornell, Silverman, Stevens, "Modular Forms and Fermat's Last Theorem," Edixhoven, the author of the chapter on Serre's Conjecture, asserts a few times, beginning on page 234 of the text,...
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66
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Does the image of a unitary irrep span all unitary operators?
Consider an irreducible unitary representation $\rho: SE_3 \to U$ where $SE_3 = \mathbb{R}^3 \rtimes SO(3)$ is the (special) Euclidean group and $U$ is the (infinite dimensional) Lie group of unitary ...
2
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130
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Confusion with Proof of Schur's Lemma
I'm following a proof of Schur's lemma as presented in these notes on Representation Theory and Quantum Mechanics by Noah Miller. I'll walk through their presentation of the lemma and the proof, to ...
2
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100
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Explicit construction of polynomial representation of $GL(n,\mathbb{R})$
In order to understand representations of the general linear group $GL(n,K)$, with $K=\mathbb{R}$ or $K=\mathbb{C}$,
I'm looking for an explicit construction of the polynomial representation matrices
...
2
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33
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Bounded Representations on a Group vs. on the Group C*-algebra
I'm learning about representations of groups and their group C-algebras and I am trying to understand the relationship between bounded representations of a group and its group C-algebra.
Let $G$ be a ...
2
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28
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Classification of binary subgroups of $\text{SL}_2(\mathbb{C})$
I am currently working through the classification of finite subgroups of $\text{SL}_2(\mathbb{C})$ as done in Klaus Lamotke's book on "Regular Solids and Isolated Singularities" (pages 33 ff....
2
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63
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Spectrum of C* algebras
In the spectral theory of bounded operators on a Banach space $X$, the spectrum of a operator has three parts: the point spectrum, the continuous spectrum and the residue spectrum. And note the ...
2
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1
answer
74
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Left versus Right regular representations.
Let $G$ be a finite group. $G$ can bear the so-called regular representation. Let $\chi_g(h) \colon= \delta_{g,h} ~ {\mathrm{for}} ~ h \not= g$. Let $X \colon= {\mathrm{the\, vector\,space\,of\,...
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363
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2-cocycles in Lie group vs Lie algebra cohomology (context of projective reps)
I'm confused by the relationship between the cocycle condition in Lie algebras vs Lie groups.
For Lie groups, a 2-cocycle is defined (e.g. here) as a map $\Phi : G \times G \rightarrow \mathbb{F}$ ...
2
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164
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The radical of indecomposable modules
Let $R$ be a finite dimensional algebra and $V$ a R-module. Denote the radical of $V$ by $Rad(V)$ which is defined to be the intersection of all maximal submodules of $V$. The socle of $V$ is the sum ...
2
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91
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Definition of Schubert cells.
I found two versions of definition of Schubert cells in two papers. In the paper, page 45, line 7 of Section 4.1, the Schubert cells are defined as
$$
{X}_{w}^{\circ} = \{ \mathcal{F} \in \mathcal{Fl}...
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112
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Degrees of irreducible constituents of induced characters
Let be $G$ a finite solvable group and $M$ a minimal normal subgroup of $G$. The group $M$ is an elementary abelian $q$-group, for a prime $q$ that we assume to be odd. Suppose that $1_M \neq\lambda \...
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Presentation of a group from a paper of Charles Ford
The question is very small - I am trying to understand an example of a group, but its presentation looks incomplete.
In a paper, Charles Ford (1970) mentions following example of a group.
Let $G$ be ...
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359
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Reference request: Representation Theory of Indefinite Orthogonal Lie Group and its Lie Algebra
In most textbooks, I have seen material on representation theory of the orthogonal group and its Lie algebra, and at most of the Lorentz group. However, I haven't seen any references talking about the ...
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107
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Exercise 4.1 in Humphreys' BGG category O theory, about contravariant form.
The following modules are all over a finite dimensional simple complex Lie algebra $\mathfrak{g}$.
Let $M$ be a nonzero submodule of a Verma module $M(\lambda)$. If $M$ has a nondegenerate ...
2
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49
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A representation problem of the universal enveloping algebra of a graded nilpotent Lie algebra
Assume we work over $\mathbb C$.
Let $\mathfrak n$ be a finitely dimensional nilpotent Lie algebra. Assume there is an $\mathbb N_+$-grading on $\mathfrak n$, i.e. there exists integers $0<w_1<\...
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74
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Trying to understand a proof on representation of $C^*$-algebra
I am trying to understand a proof of Proposition 3.1.2 from Bohle - $K$-theory
for
Ternary Structures. Let me explain the notations below.
Let $V$ be a TRO (a closed subspace of $B(H,K)$ which is ...
2
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133
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Complete reducibility, in linear algebra and in topology
Consider a representation $A(G)$ of a group $G$ in a vector space $\mathbb V$.
The following two definitions of the complete reducibility are equivalent.
Definition 1
A representation $A$ is called ...
2
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76
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Is there a nice explicit description of Verma modules when $\mathfrak{g} = \mathfrak{sl}_3(\mathbb{C})$?
For reference, I've read through and generally understand the general construction at a high level. I've also read and understood Verma modules for $\mathfrak{sl}_2(\mathbb{C})$, but I can't link my ...
2
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77
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Prove that a matrix is symplectic?
Let $\omega$ be the skew-symmetric bilinear form on $R^{2n}$ given by
$$\omega(x,y)=\sum_{j=1}^{n}(x_{j}y_{n+j}-x_{n+j}y_{j})$$
Let $\Omega$ be the $2n \times 2n$ matrix
\begin{pmatrix}
0 & I_{n}\\...
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72
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A technical detail in the definition of a complementary representation.
DEFINITION
Suppose a representation $A(G)$ of a group $G$ is acting in a linear vector space $\,\mathbb{V}$ with an inner product and, generally speaking, with some topology.
Consider a subspace ${\...
2
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156
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Examples of applications of Beilinson-Bernstein localization?
I'm a sheaf theorist who knows a little representation theory but is not familiar with it. According to my memory, I was once told that one virtue of the Beilinson-Bernstein localization theorem is ...
2
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136
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What is the eigenvalue of the Casimir element of $su(n)$ in the defining representation?
I am currently in the process of learning some representation theory, and I am trying to derive a form for the Casimir element $C_n$ of $\mathfrak{su}(n)$ given some fixed basis. To verify if my ...
2
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121
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LCH group representation: SOT $\iff$ norm continuity?
Let $G$ be a locally compact Hausdorff group, $H$ a complex Hilbert space and $\pi : G \to \Bbb{B}(H)^\times$ a group homomorphism from $G$ to the group of invertible bounded linear operators on $H$.
...
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173
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Duality functor is equivalence of category
I am reading a book on Frobenius algebra in which I read all the following definitions:
Given a linear map $f:V\to W$ the dual map is defined by $f^{*}:W^{*}\to V^{*}$( as, $h\to f\circ h$)
Let there ...
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48
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references on irreducible $(\mathfrak{g},K)$-modules for small rank groups
Classification of irreducible $(\mathfrak{g},K)$-modules for $SL_2(\mathbb{R})$ and $SL_2(\mathbb{\mathbb{C}})$ can be found in many standard textbooks, like Wallach's "real reductive groups"...
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79
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Restate Galois representation from category theory
Given an arbitrary category $C$, a representation of $G$ that is a category with a single object in $C$ is a functor from $G$ to $C$. A group representation is a representation of $G$ in the category ...
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44
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Restriction on $n$ for ${Spin(n)\times Spin(n+2)} \subset Spin(2n+2) $ to be true
We know that
$$SO(n) \times SO(m)\subset SO(n+m) \tag{1}$$
$$\frac{Spin(n)\times Spin(m)}{{\mathbf{Z}/2}}\subset Spin(n+m) \tag{2}$$
are both true.
Eq (1) is true for the obvious reason. Eq (2) can be ...
2
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answers
221
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Representations of $GL_n(F_q)$ over a finite field
If $F_q$ denotes a finite field of characteristic $p,$ then I want to learn about the representations of $G = GL_n(F_q)$ over a finite field $K$ such that $char(K) \ \nmid |G|.$ Any reference ...
2
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What are the prerequisites for studying Automorphic Representation Theory?
I am interesting in getting into Automorphic Representation Theory but am unsure where to start. It seems like there are a lot of prerequisites. What are the main areas (above abstract algebra) that ...
2
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129
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Representation of Lie algebra $SE\left(2\right)$
When I read the paper Universal approximations of invariant maps by neural networks of Dmitry Yarotsky, it happens on page 36 that he used some concepts about the representation of Lie algebra of the ...
2
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67
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Tensor products of irreducible representations of $GL_{2}(\mathbb{F}_{q})$
Throughout the post $G = GL_{2}(\mathbb{F}_{q})$ where $q$ is a prime power with the prime not being 2.
Let $V_{1}$ and $V_{2}$ be cuspidal representations of $G$ over $\mathbb{C}$. I can understand ...
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103
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Explicit 45 Lie algebra generators as rank-16 matrix spinor representations of $𝑆𝑝𝑖𝑛(10)$
A simple Lie group $𝑆𝑝𝑖𝑛(10)$ has a spinor representations of 16 dimensions, which is distinct from the vector representation of 10 dimensions (coming from standard vector representation of SO(10))...
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140
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Show irreducibility of polynomial representation of $SU(2)$
I am looking at this exercise for 2 days and honestly cannot make any progress, so I really appreciate any help.
Let $V_n$ ⊆ $C[x_1, x_2]$ be the space of all homogeneous polynomials of degree
n. Let $...
2
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answers
102
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Trying to understand the proof of the main theorem in Tannakian categories
I am trying to understand Lemma 2.13 in Deligne and Milne's Tannakian Categories
Lemma: Let $C$ be a $k-$linear abelian category and let $\omega:C\to\text{Vec}_k$ be a $k-$linear exact faithful ...
2
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188
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Branching rule for $S_n$ proof by James
Apologies for my English in advanced..
The following is a part from James' proof for the branching rule on the symmetric group:
It can be found in "The Representation Theory of the Symmetric ...
2
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93
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Gabriel's theorem
Need some help with Gabriel's theorem, doing the part "if a graph is of finite type, it must be a Dynkin graph" like this:
Let $\vec{Q}$ be of finite type. All representations is a direct ...
2
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64
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A Quick Question Concerning Iwasawa's Paper "On Galois Groups of Local Fields"
I am reading Iwasawa's paper "On Galois Groups of Local Fields", because I want to understand more about $U^{(1)}$ as a Galois module. He started his paper with the following data:
$q=p^{f_0}...
2
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52
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Operators acting on the space of solutions of a system of ODEs
A system of the hypergeometric equation (HGE), associated with a set $(\alpha, \beta, \gamma)$ of parameters is defined/obtained in the following way. Let $\Psi$ be a solution of the hypergeometric ...
2
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168
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What is the relationship between the representations of a lie algebra and its complexification?
When reading Brian Hall's Lie Groups, Lie Algebras, and Representations, I found this statement confusing
Since SU(2) is simply connected, Theorem $5.6$ will tell us that the representations of $\...
2
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144
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Difference between dual representation and the adjoint
The dual or contragradient representation from a vector space $V$ on $V^*$ (the dual vector space of $V$) is defined as the linear operator
$$
(\pi^{-1})^T(g): V^*\rightarrow V^*,
$$
where I ...
2
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77
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Infinite, finitely generated linear group has indices of subgroups divisible by infinitely many primes
Let $G$ be an infinite finitely generated group and suppose that $G$ is linear, i.e., that $G \leq \operatorname{GL}_n(K)$ for some field $K$.
Are there infinitely many primes $p$ such that $p$ ...
2
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31
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Irreducible complex characters of $G/K$ induced by characters of $G$ for representations $\varrho$ with $K \leqslant \ker \varrho$.
Let $G$ be a finite group, and $K \lhd G$. I want to prove that there is a bijection between the set of characters of $G$ that correspond to representations $\varrho$ of $G$ with $K \leqslant \ker \...