Questions tagged [representation-theory]

Question 1

I want to compute the composition factor multiplicities of the Verma module $M(0)$ for the complex Lie algebra $\mathfrak{sl}_3$ using elementary methods. I already have some results: $[M(w. 0) : L(...

Question 2

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 $H$...

Question 3

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. ...

Question 4

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 ...

Question 5

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 ...

Question 6

(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 ...

Question 7

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 $H^n(G,-)$ ...

Question 8

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 ...

Question 9

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 ...

Question 10

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 ...

Question 11

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 $...

Question 12

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 ...

Question 13

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 ...

Question 14

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{...

Question 15

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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Questions tagged [representation-theory]

Computing Composition Factor Multiplicities of the Verma Module $M(0)$ for $\mathfrak{sl}_3$

Unitary representations of non-compact Lie Groups

Good Textbook or Relevant Literatures for Learning Graph Embedding

What exactly is the adjoint representation - confused by varying definitions [closed]

Confusion regarding quivers with potential and cluster tilted algebras

Confusion about one of Cartan-Weyl Basis rules for Lie algebra

Cohomology of smooth representations of profinite groups

Restriction of character tables to subgroups

Equivalence of categories between quiver representations and module for infinite quiver

Irreducibility of $\mathfrak{sl}(m_\beta, \mathbb{C})\ \oplus\ \mathfrak{sl}(m_\alpha, \mathbb{C})$ module

Existence of irrational eigenvalues of a sum of representation matrices

Hecke algebras over fields other than $\mathbb{C}$

Abelianity of Toral Lie subalgebras

Distribution of roots in a matrix representation [closed]

Computing the action of GL(2) on the Fourier-transformed principal series representation

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