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.
10,318 questions
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Good Textbook or Relevant Literatures for Learning Graph Embedding
I'm interested in the following problem in statistical characteristics of graph embedding, and it seems to fall between traditional graph theory and Graph neural networks.
I looked up:
William L. ...
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50
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Confusion regarding quivers with potential and cluster tilted algebras
I was looking at P. Plamondon's paper titled "GENERIC BASES FOR CLUSTER ALGEBRAS FROM THE CLUSTER CATEGORY". I'm confused about the calculation in Example 4.3. The example starts with a ...
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Cohomology of smooth representations of profinite groups
Let $G$ be a profinite group. A discrete $G$-module is an Abelian group $A$ with discrete topology and continuous $G$ action $G\to\mathrm{Aut}_\text{Grp}(A)$. We know the group cohomology $H^n(G,-)$ ...
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Equivalence of categories between quiver representations and module for infinite quiver
Every time I look for resources, authors assume quivers to be finite. I’m sure this question has been answered somewhere, but I cannot find it. I am reading Assem’s book on the representation theory ...
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Existence of irrational eigenvalues of a sum of representation matrices
Let $G$ be a finite non-abelian group and $H$ be a non-normal subgroup of $G$. It can be shown that there exists a non-linear irreducible unitary representation $(\rho,V)$ of $G$ such that the matrix $...
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Abelianity of Toral Lie subalgebras
I'm a bit confused about the abelianity of the Toral lie subalgebras and the related discussion reported here.
Preliminarily I set a notation and recall some results:
Let $\mathfrak{g}$ be Lie ...
- 705
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Computing the action of GL(2) on the Fourier-transformed principal series representation
Below is the setting,
Let $F$ be a non-Archimedean local field, $G=\text{GL}_2(F)$, and let $\chi_1, \chi_2:F^\times \to \mathbb{C}$ be characters. Define a character on the Borel subgroup $B$ of $G$ ...
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Lie algebra of the unipotent radical
I am not very versed in linear algebraic groups, and I am a bit confused about the unipotent radical.
Let $G$ be a linear algebraic group over, say, ${\mathbb C}$ with Lie algebra ${\mathfrak g}$. Let ...
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Do the Riemann tensors lie in $\text{Young}(13,24)$?
In the Wikipedia article on the Riemann curvature tensor, it is stated that:
“The algebraic symmetries are also equivalent to saying that $R$ belongs to the image of the Young symmetrizer ...
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How to prove that $\operatorname{rad}({\mathfrak g}) = [{\mathfrak g}, {\mathfrak g}]^\perp$?
Let ${\mathfrak g}$ be a finite dimensional Lie algebra over a subfield $k\subset {\mathbb C}$. Let us denote the derived ideal
$$
{\mathfrak g}'=[{\mathfrak g},{\mathfrak g}].
$$
It seems well known ...
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conjugate Young diagram means conjugate representation
It is well known that if you flip the rows and columnns of a Young diagram, you get a conjuated representation of the original representation. But I cannot find a proof of it in the literature.
Here, '...
- 1,168
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Proof of theorem of Burnside ( Anyone who have the Steinberg's Representation theory of finite groups book ? )
I am reading the proof of Theorem 9.1.10 in the Benjamiin Steinberg's book, representation theory of finite groups and stuck at some statement.
First let me arrange associated preliminaries
...
- 4,056
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Example of the nilradical sequence for a Lie algebra that does not split as in the Levi decomposition
Let ${\mathfrak g}$ be a finite-dimensional Lie algebra over, say, ${\mathbb R}$, and ${\mathfrak r}$ its radical, i.e. maximal solvable ideal, and ${\mathfrak n}$ its nilradical, i.e. maximal ...
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Is there exist a decompositions for the space of functions on manifolds with group action?
Suppose that there is a group action on a manifold, $G\times M\to M$. Does the space of functions on $M$ decompose into the direct sum of irreducible representations of $G$?
I am familiar with the (...
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Literature on finite-dimensional subrepresentations in $\text{End}(H)$ (including unbounded operators)
I have a question about the representation theory of semisimple Lie groups, motivated by concepts from quantum mechanics.
Let $G$ be a semisimple Lie group and let $(\pi, H)$ be a unitary ...
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What's the exact statement of minimal model Virasoro algebra has finitely many simple modules?
I've been encountered arguments like when $c=1-6\frac{(p-p^{\prime})^2}{pp^{\prime}},\,p,p^{\prime}\in \mathbb{Z}^{+}$, then Virasoro algebra has finitely many simple modules.
But we have the ...
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Motivations of Artin's theorem and Brauer's Theorem on Characters
I've been reading Serre's Linear Representations of Finite Groups (GTM 42), but I’m a bit confused about the motivation and applications of two classical theorems:
Artin’s Theorem: Each character of $...
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85
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Complete reducibility of infinite-dimensional representations
We have Maschke’s theorem stated in the following form (I am working over a field $k$ of characteristic $0$):
Theorem. Let $G$ be a finite group (or a compact Lie group), and let
$V$ be a (possibly ...
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Benjamin Steinberg, Representation theory of finite groups, proof of Theorem 6.3.9
I am reading the Benjamin Steinberg, Representation theory of finite groups, proof of Theorem 6.3.9 and stuck at some statement.
First, let me arrange associated preliminaries.
Corollary 4.1.10. Let $...
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Determining the composition factors of induced epresentations of semi-direct products
Let $H$ be a subgroup of a finite group $G$ and let $\phi:\mathbb{Z}/2\to \text{Aut}(G)$ such that $\phi(1)(H)=H$. I will write $G^+:=G\rtimes_\phi \mathbb{Z}/2$ and $H^+:=H\rtimes_\phi \mathbb{Z}/2$,...
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Reference request : Point of view for representations of the symmetric group
I am doing my thesis studying representations of the symmetric group $\mathfrak{S}_n$. More precisely, I consider cyclic representations. I want to consider these representations as subrepresentations ...
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Infinite dimensional representation of $SL_2(\mathbb{C})$
This question is a simple example of this post.
$\DeclareMathOperator\SL{SL}$Let $G=\SL_2(\mathbb{C})$.
Let $u(a)=(\begin{smallmatrix} 1&a\\0&1 \end{smallmatrix})$, $h(t)=(\begin{smallmatrix} ...
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Sufficient condition for weight to be a weight of irreducible standard cyclic module $V(\lambda)$
I am trying to prove the following proposition from Humphreys. Here $V(\lambda)$ is the unique irreducible standard cyclic module of weight $\lambda$.
If $\lambda$ is a dominant weight, then the set $...
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Two actions of quotients on classifying stack
Let
$$
1 \to H \to G \to Q \to 1
$$
be an exact sequence of commutative algebraic groups.
The following diagram
$$\begin{array}
BBH & {\longrightarrow} & BG \\
\downarrow & & \...
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Nearby cycles, Vanishing cycles and Gluing: Constant sheaf
Let $i: Z \hookrightarrow X$ be a closed embedding and $j: U \hookrightarrow X$ be the inclusion of the complementary open subset. It is known that a vanishing cycles gluing datum, i.e. $(\mathcal{F}, ...
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