# Questions tagged [representation-theory]

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

5,752 questions
22 views

### Point groups where the tensor square of two-dim. irreps (over $\mathcal{O}(3)$) does not contain a two-dim. irrep in its decomposition

Which are the point groups where the tensor square of a two dimensional irreducible representation does not decompose into a sum that contains a two-dimensional irreducible representation? For ...
22 views

### What is the meaning of $\rho \nu_{\rho}$?

I am reading the paper. On page 14, section 3.1, I am trying to understand the notation $\rho \nu_{\rho}$, where $\rho$ is a supercuspidal representation $GL_n(F)$, $F$ is a non-archimedean local ...
27 views

### Equivariant projective modules and skew group algebras

This is a question related to the two dimensional McKay correspondence. Let $R = \mathbb{C}[x,y]$, and $G$ a finite group acting on $R$. Recall that a $G$-equivariant $R$-module is an $R$-module with ...
17 views

### When are indecomposable projective modules finitely generated?

What conditions to you need to put on your ring to guarantee that the indecomposable projective modules are all finitely generated? Edit: I was hoping there was some general result for this. If your ...
19 views

50 views

48 views

39 views

### Simple formula for the dimension of weight spaces of Verma module?

Let $\mathfrak{g}$ be a simple Lie algebra (e.g. $\mathfrak{sl}_n$), and let $M_\lambda$ be the Verma module with highest weight $\lambda$. Is there a simple formula for $\dim (M_\lambda)_\mu$, where ...
23 views

### Invariant subspaces of the representation of $T(\sigma)[x_1,…,x_n] = [x_{\sigma^{-1}(1)},…,x_{\sigma^{-1}(n)}]$

Where $\sigma \in S_n$ and the representation is over the vector space $\mathbb{C}^n$ I'm trying to find as many invariant subspaces of this representation as possible. I don't know how to find these ...
16 views

### structure of subrepresentations of (infinite) sums of irr. representations

Let $G$ be a (locally compact) group and $( \pi_1 , V_{\pi_1} ) , ( \pi_2 , V_{\pi_2} ) , \ldots$ irreducible, unitary representations. I think, I can show, that for the (finite) direct sum ...
25 views

### Show that $\dim V^{G} = 1/|G| \sum_{g \in G} \chi_{T} (g).$

Let $T$ be a complex linear representation of the group $G$ in a space $V$ Show that $\dim V^{G} = 1/|G| \sum_{g \in G} \chi_{T} (g).$ If I know the definition of $V^G$ from this corollary: Which ...
21 views

### Representation of sigma complete Boolean algebras

In Terrence Tao's article 245B notes 4: The Stone and Loomis-Sikorski representation theorems he gives a proof that not each sigma-complete Boolean algebra can be realized as a $\sigma$-complete ...
28 views

24 views

### Calculate the characters of the left and right regular representationsof an arbitrary finite group.

Calculate the characters of the left and right regular representations of an arbitrary finite group. The answer of the question is given below: But I do not know why the character of the left and ...
29 views

### Computing the inner product $(\chi_{R},\chi_{R})$.

Let $\chi_{R}$ denote the character of the right regular representation $R$. Compute the inner product $(\chi_{R},\chi_{R})$ directly and also by decomposing $R$ into a sum of irreducible ...
14 views

### Decomposition of Induced and Restricted Modules

I am taking a course in representation theory, and as I missed many classes I am having a lot of trouble in dealing with the decomposition of induced and restricted representations. I have no idea how ...
31 views

### How to find the matrix generators of higher dimensional irreducible representations of $\operatorname{SU}(2)$?

Using the most general form of a $2\times 2$ unitary matrix $U$ of determinant $+1$ and using the formula $$T^a=-i\frac{\partial U}{\partial\theta^a}|_{\{\theta_a\}=0} ~{\rm with}~ a=1,2,3.$$ In this ...
21 views

### Characters of $(\mathbb{Z}/2\mathbb{Z})^m$

I just started to read the book 'Representation theory of finite groups' by B. Steinberg and I'm trying to solve Exercise 4.5., which discusses the characters of $(\mathbb{Z}/2\mathbb{Z})^m$. For a ...
24 views

### Schur Index for Quaternion Algebra

I learned form this question and this answer that Schur index in GAP can be found using LoadPackage("wedderga") the functions "SchurIndex". But I am working on the field $K=\mathbb Q (\sqrt{-39})$ ...
40 views

32 views

### direct sum of representation of product groups

Given two finite groups $G_1$ and $G_2$, and some representations $\rho_1: G_1 \to V_1$ and $\rho_2: G_2 \to V_2$, it seems the standard way to create a representation for $G_1 \times G_2$ is to use ...
14 views

### What is the alternative of an algebraic direct sum?

I'm working through some Lie theoretic representation theory, and the definition of a $(\mathfrak{g},K)$-module states that the representation $V$ decomposes into an algebraic direct sum of finite-...
24 views

### Difference between 2 questions on the alternating group $A_{4}$.

I have to answer 2 questions on the alternating group $A_{4}$: One of them asks me to find all irreducible complex representations of the group $A_{4}$ (and calculate their characters) while the ...
76 views

### A difficulty in understanding the definition of “Spaces of Matrix Elements.”

The definition is given below: If the definition of the Spaces of Matrix Elements is as given below: But I do not understand why: 1- Any linear combination of matrix elements can be expressed ...
In Carlson's Cohomology and representation theory, the author states Schanuel's lemma, and then derives a consequence that I cannot understand. They define, for a $kG$ module $M$, $\Omega (M)$ to be ...
### Construct a basis of the matrix elements in the space $\mathbb{C}[S_{3}]$
Construct a basis of the matrix elements in the space $\mathbb{C}[S_{3}]$. If the definition of the Spaces of Matrix Elements is as given below: And the answer of the question at the back of ...