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

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What is the natural action of $\mathfrak{sl}(4,\Bbb{C})$ on $\wedge^2 \Bbb{C}^4$?

What is the natural action of $\mathfrak{sl}(4,\Bbb{C})$ on $\wedge^2 \Bbb{C}^4$? We know that $\wedge^2 \Bbb{C}^4$ is generated by $\{e_1 \wedge e_2, e_1 \wedge e_3, e_1 \wedge e_4, e_2 \wedge e_3, ...
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Can we say anything about the unit of a $k$-algebra $A$ in terms of the unit $1\in k$?

Context: Being confused about new concepts and trying to make new distinction to better understand it. Let's say we have have associative $k$-algebra $A$. Where $k$ denotes a field. An algebra is a $...
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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 ...
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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 ...
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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'...
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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 ...
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225 views

Rank of an action and definition of an orbital

Let $G$ be a group acting on a set $X$. In group theory sometimes it is helpful to consider the action of $G$ on $X\times X$; a good example is perhaps finding the dimension of $\operatorname{Hom}(\...
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Application of Hilbert's basis theorem in representation theory

In Smalo: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand: Two orders are defined on the set ...
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Theorem about the subgroup of a Lie group fixed by an involution

When trying to do Lie-theoretic calculations on Lie groups (finding the Bruhat decomposition, etc.) I've often come across expositions that seem to be implicitly using a result something like the ...
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352 views

Character table of the non-abelian group of order 21

I'm working my way through the first Chapter of Fulton and Harris' Representation Theory and I'm trying exercise 3.26: There is a unique nonabelian group $G$ of order 21, which can be realized as ...
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524 views

(Tensor) Product of irreducible representations

Suppose that $T: G \rightarrow GL(U)$ and $ S: G \rightarrow GL(V)$ are two finite dimensional irreducible representations of some group $G$ . I consider the tensor product representation $ T^*S : G ...
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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 $g$ ...
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Every Irreducible Representation of $G \times H$ is tensor product of Irreducible Reps of $G$ and $H$?

It's an easy task to prove with character theory that if $V_1$ and $V_2$ are irreducible representations of $G_1$ and $G_2$ respectively, then $V_1 \otimes V_2$ is an irreducible representation of $...
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79 views

Endomorphism ring of indecomposable representations

Let $Q$ be the quiver given by an $n\times n$ grid where every square commutes and let $F:Q\to {\rm vec_k}$ be an indecomposable (finitely dimensional) representation of $Q$. I am interested in ...
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64 views

Unipotent representations of SL(2,R) by quantization

I'm a PhD student in mathematical physics and I happen to need some elements of Kirillov's "orbit method" for producing representations of Lie groups. I'm familiar with symplectic geometry, geometric ...
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27 views

Finding expansions for elements of a group algebra

Suppose $G$ is a finite group of order $\left|G\right|$, with an associated group algebra $K[G] = \left\{\sum_{g \in G} a_g g\right\}$ over, say, the complex numbers. Suppose we represent $G$, and by ...
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Precise definition of affine, smooth, and irreducible

A book which I'm reading now says that "the Drinfeld curve $$ \mathbf{Y} = \{\, (x, y) \in \mathbf{A}^2(\mathbb{F}) \mid xy^q - yx^q = 1 \,\}$$ is affine, smooth, and irreducible." Here $p$ is an odd ...
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316 views

Example of a simple module which does not occur in the regular module?

Let $K$ be a field and $A$ be a $K$-algebra. I know, if $A$ is artinain algebra, then by Krull-Schmidt Theorem $A$ , as a left regular module, can be written as a direct sum of indecomposable $A$-...
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Visualisation of representations and their decomposition into irreps

A question in a Representation Theory midterm got me thinking, and made me realise I didn't really understand irreps. The question was on the subject of reps of $S_4$, and went: An obvious ...
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113 views

How to show that a local finite dimensional algebra is basic from definition?

A basis algebra is an algebra $A$ such that $e_i A \not\simeq e_j A$ for any $i \neq j$, where $e_1, \ldots, e_n$ is a complete set of primitive orthogonal idempotents. A local algebra is an algebra ...
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Matrices of subrepresentations and quotient representations.

Suppose that $V$ is a $5$ dimensional representation (with generators $\{y_1, ... , y_5\}$ of the lie group $\mathcal{g}$, with the lie algebra homomorphism $\rho: \mathcal{g} \rightarrow \mathcal{gl}(...
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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 ...
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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 ...
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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 ...
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176 views

Different definitions of Casimir element

I read about the Casimir element just recently and I found it a bit difficult to wrap my mind around the definition(s). In fact, I have seen two different definitions. For concreteness, let $\mathfrak{...
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49 views

A Representation of $C(X)$ is a positive map.

I quote this excerpt from Conway: "A representation $\rho:C(X) \rightarrow \mathcal{B(\mathcal{H}})$ is a $\ast$-homomorphism with $\rho(1)=1$. Also, $\|\rho\|=1$. If $f\in C(X)_+$, then $f=g^2$ ...
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56 views

Why do Ad(K) orbits in the $-1$ eigenspace of a Cartan decomposition intersect the Weyl chamber once?

Let $G$ be a semisimple Lie group and let $\frak p\oplus t$ be a Cartan decomposition of $\frak g$ and $K$ the connected subgroup with Lie algebra $\frak t$. Choose a maximal abelian subalgebra $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Representations and mutually singular measures

I'm finding some difficulties with an exercise from Conway and I ask for some help in understanding it: "Let X be a compact space and let $\{\mu_n\}$ be a sequence of measures in X. For each $n$ let $...
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Let $A=k[x]$ and $V=k[x]/((x-λ)^n)$. Find a filtration $V=V_0 ⊃ V_1 ⊃ \dots⊃ V_n=0$ such that the subsequent quotients $V_{i-1}/V_i$ are irreducible.

Let $A$ be the algebra $A=k[x]$ and let $V$ be the representation $V=k[x]/((x-\lambda)^n)$ for some $\lambda \in k$ and $n\in\Bbb N$. Find a filtration $V=V_0 \supset V_1 \supset \dots \supset ...
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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. ...
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Covering Spaces in Representation Theory.

I'm reading the paper "Covering Spaces in Representation Theory" of K. Bogartz and P. Gabriel. Now I'm in section 2, proposition 2.3, on the first three lines concludes that the functor $l \mapsto ...
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1answer
47 views

A smooth non-stably trivial smooth vector bundle

This may well be just a look-up, but do you have an example of a non-stably trivial smooth vector bundle? If it has a presentation as the vector bundle associated to the representation of some (...
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Characters of a finite group

Recently, I have been studying about Character Theory of Finite Groups, mostly from "Groups and Representations" by J. Alperin & R. Bell. In the aforementioned textbook, the characters of a finite ...
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Show that $k[x]/(x^{2})$ is an indecomposable (1), but not irreducible (2) $k[x]$-module.

Exercise: Show that $k[x]/(x^{2})$ is an indecomposable (1), but not irreducible (2) $k[x]$-module. I'm not sure about all different kind of modules, but this is a question of a book about ...
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Restriction of a Specht module to the alternating group

Let $n\in\mathbf{N}$ and denote by $S_n$ the symmetric group on $n$ letters. For $\lambda\vdash n$ a partition of $n$ the Specht module $S^\lambda$ defines an irreducible representation. What happens,...
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Tensors furnish representations of the group

I'm bad at english, so what exactly does it mean in simple english that Tensors furnish representations of the group?
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multiplicities of irreducible representations

Let $G$ be a finite group and $G'$ be a subgroup. Let $\rho$ be a one-dimensional group of $G'$. Then define $\psi$ to be the induced action of $\rho$ - $\psi:= Ind_{G'}^G \rho$ Is there any general ...
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Is there anywhere some explicit Bruhat decompositions are written down?

Question in title: most places I see Bruhat decompositions treated they're only briefly mentioned and no examples are given. Also, I calculated the following regarding the Bruhat decomposition of $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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limit of regular hyperbolic integrals is a unipotent integral (GL2 calculation)

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{...
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A question about positive forms on involutive algebras.

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^\...