# Tagged Questions

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

33 views

24 views

### Let $M,N$ be isomorphic as $\mathbf{Z}[G]$-mods, are they isomorphic as $\mathbf{Z}[H]$-mods, where $H<G$

So I have recently been looking at the isomorphisms of $\mathbf{Z}[G]$-mods ($G$ finite), and noticed that a couple of my examples saw them isomorphic as $\mathbf{Z}[H]$-mods also, where $H$ is a ...
147 views

### Lie algebras and the Killing form.

The Killing form is defined by $K(x,y) = \text{tr}(\text{ad} x, \text{ad} y)$, right? In this lecture, we assume that $\{x_1, ... , x_n\}$ is a basis for $g$ and $\{y_1, ... ,y_n\}$ is a dual basis ...
55 views

### Definition of Representation in terms of Group Action

The definition of a representation of a group $G$ over a vector space $V$ is a map $p: G \to GL(V)$. According to wikipedia, for finite groups an equivalent definition is an action of $G$ on $V$. I'...
57 views

### Cyclic representation on $L^2(\mu)$

Show that if $(X,\Omega,\mu)$ is a $\sigma-$ finite measure space and $H=L^2(\mu)$, then $\pi:L^\infty(\mu)\to B(H)$ defined by $\pi(\phi)=M_\phi$ is a cyclic representation and find all the cyclic ...
225 views

322 views

### Proof that a group representation matrix is diagonalizable?

Suppose we have a finite group $G$ and and an $n$-dimensional vector space $V\cong \Bbb C^n$ over the field $\Bbb C$ of complex number. My professor said the other day that for every group element $g$ ...
85 views

62 views

### Subrepresentations of finite dimensional semisimple representations of an algebra

I'm following the notes by Prof. Etingof, linked here, and am stuck on a detail from Prop. 2.2, on page 23. To briefly recap what is in the notes, we have a finite dimensional, semisimple ...
73 views

### Does this representation have a name?

Let $G$ be a group acting on a set $X$. Let $F(X) = \{f: X \to \mathbb C\}$ be the set of complex valued functions on $X$. This is a complex vector space. Then $G$ acts on $F(X)$ linearly via the ...
104 views

### Proof in Serre/Fulton's rep. theory of Artin-Wedderburn for $\mathbb C[G]$

I have figured out a proof myself for the following theorem, but in both Serre's "Linear Representation of Finite Groups" and Fulton's "Representation Theory" books, I don't understand their comments ...
176 views

69 views

### Let $V=V_{1}⊕ ⋯ ⊕ V_{n}$ be semisimple. $U$ irreducible. Show that $\dim_{k} (Hom_A(U,V))$ is equal to the number of $V_i$ equivalent to $U$.

$\DeclareMathOperator{\End}{End} \DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\Irr}{Irr}$ Definition. An $A$-module $V$ is ...
41 views

### Contains the representation with multiplicity n

In a problem I'm asked to prove that a representation contains the trivial representation with multiplicity $n$. I'm a little confused. What exactly does "contain" mean and "multiplicity"? Does it ...
43 views

### Is a representation of a $k$-algebra a $k$-vector space?

Is a representation $V$ of an $k$-algebra $A$ a $k$-vector space ? I've been studying representation theory for some weeks, but sometimes I get a little bit confused about all the different ...
93 views

### Remark 3.1.3 from Introduction to Representation Theory from Pavel Etingof

$\DeclareMathOperator{Hom}{Hom}$I'm trying to prove the following proposition (remark 3.1.3 from Introduction to Representation Theory from Pavel Etingof). Proposition. Any semisimple representation ...
156 views

### Basis of vector space invariant under group action (of symmetric group)

Suppose I have a finite-dimensional real vector space $X$ and a finite group $G$ that acts faithfully on X. The task is to find a $G$-invariant basis of $X$. This means the set of basis vectors is ...
46 views

38 views

### Let $A=k[x]$ and let $V=k[x]/\big((x-λ )^{n} \big)$ for some $λ \in k$ and $n\in \Bbb{N}$. Then $V$ is indecomposable.

Theorem. Let $A=k[x]$ and let $V=k[x]/\big((x-\lambda )^{n} \big)$ be a representation of $A$ for some $\lambda \in k$ and $n\in \Bbb{N}$. Then $V$ is indecomposable. This is a theorem in my book. ...
65 views

55 views

### Tate's thesis - continuous map from a local field to circle group

I am currently reading Decomposition of Unitary Representations defined by a discrete subgroups of nilpotent groups, by C.C. Moore. It is metioned that if $\mathbb{K}$ is a $p$-adic field in his ...
34 views

### Spin Representations and Galois correspondence?

I have a vague question regarding the Spin representations. Is there a "quick" way of seeing that $Spin(2n)$ has exactly two irreducible representations which do not factor through $SO(2n)$, and one ...
63 views

### Every representation of a finite group is reducible?

I somehow "proved" that every representation of a finite group is reducible. While I'm fairly sure the error is something silly, I can't seem to place it. Could someone please help me figure out what ...
75 views

### Matrix representation and permutation matrices

In order to find the matrix representation associated to a permutation representation I identify each permutation with the corrisponding matrix representation. How can I prove that these matrices ...
38 views

### How does Fulton and Harris establish that the differential of a group hom respects ad?

Fulton and Harris, Representation Theory, Section 8.1 (pages 104 - 107 in my copy) is concerned with showing that group homomorphisms $\rho : G \to H$, where $G$ is connected, are completely ...
In developing a simple trace formula for $G$=GL$_2$ over a number field $F$ one encounters the following identity of local integrals: \int_{G_v}f_v(g^{-1}\begin{pmatrix}1 & 1\\ 0 & 1\end{...
A linear form $f$ on an involutive algebra $A$ is said to be positive if $f(x^\ast x)\geq 0$ for every $x$ in $A$. To be useful, this definition requires that is not always possible to write \$-(x^\...