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.
10,320 questions
0
votes
0
answers
31
views
Could someone recommend me some references for algebraic groups? [closed]
I'm currently taking a number theory course, specifically a representation of reductive p-adic groups. To work through examples, could someone recommend me books or lecture notes that explain ...
- 1
-4
votes
0
answers
87
views
Schur's lemma for $S_3$ discrete group [closed]
If we consider the discrete group $S_3$, for which we write the 3DF (unitary) matrix representation. One can reduce this representation to a sum of irreducible representations. This means, that one ...
- 367
1
vote
0
answers
159
views
Unitary representations of non-compact Lie Groups
I have often read the following statement:
Let $G$ be a connected, simple, non-compact Lie Group of dimension $n \geq 2$. Let $ρ: G \to U(H)$ be a unitary representation of $G$ on the Hilbert Space $H$...
- 102
0
votes
0
answers
23
views
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. ...
- 1
0
votes
1
answer
75
views
What exactly is the adjoint representation - confused by varying definitions [closed]
In class, we first only defined the adjoint representation as a matrix of structure constants. We proved everything only using this. I tried to review the class material with other resources but I'm ...
- 31
0
votes
0
answers
65
views
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 ...
- 847
2
votes
1
answer
88
views
Confusion about one of Cartan-Weyl Basis rules for Lie algebra
(forenote: forgive my bad notation and my inability to move away from the word eigenvector - I'm still trying to wrestle with these concepts very crudely, so I don't want to use words I'm not ...
- 31
0
votes
0
answers
42
views
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 $H^n(G,-)$ ...
- 649
0
votes
0
answers
57
views
Restriction of character tables to subgroups
Srinivasan's paper The Characters of the Finite Symplectic Group $\mathrm{Sp}_4(q)$ provides the character table of $Sp_4(q)$ for $q=p^e$ and $p$ an odd prime. It is not clear to me how I can use this ...
- 473
1
vote
0
answers
24
views
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 ...
- 41
0
votes
1
answer
60
views
Irreducibility of $\mathfrak{sl}(m_\beta, \mathbb{C})\ \oplus\ \mathfrak{sl}(m_\alpha, \mathbb{C})$ module
Fix integers $m_\alpha,m_\beta\ge1$. Consider the vector space$$
U_{\beta\alpha}\ :=\ Hom(\mathbb{C}^{m_\alpha}, \mathbb{C}^{m_\beta})\ \cong\ M_{m_\beta\times m_\alpha}(\mathbb{C}),$$
with the ...
- 143
4
votes
0
answers
160
views
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 $...
- 161
5
votes
0
answers
49
views
Hecke algebras over fields other than $\mathbb{C}$
I am trying to prove some statements about specific Gelfand pairs, when considering representations of some finite groups over field $k$ which can be of positive characteristic. For this I study some ...
- 994
0
votes
1
answer
54
views
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
-1
votes
1
answer
66
views
Distribution of roots in a matrix representation [closed]
I'm reading the first chapter of Humphreys's Lie algebra, and I have a vague question.
Consider the Lie algebra $\mathfrak{so}(8)=\{X\in M_{8\times 8}(\mathbb{C})\mid SX+X^TS=0 \}$, with $S=\begin{...
- 75