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

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Induced representation

I'm doing the problem section of the induced representations chapter by Steinberg, and I'm having problems with the following one: Let $G$ be a group and $H$ subgroup. Given a representation ...
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Projective representaions of $(\mathbb{Z}/3\mathbb{Z})^2$

I have a very short question: is there a faithful projective representaion $\rho: \mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}\to {\rm PGL}(4,\mathbb R)$? Thanks!
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Adjoint representation of the isotropie group of a homogeneous space

I have difficulties seeing why is the following true: Let $G$ be a lie group and $H$ a closed subgroup, with $\tilde{g}$ and $\tilde{h}$ their lie algebras. The adjoint action of $g\in G$ is given by ...
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A 4x4 matrix representation of SU(3)?

Is it possible to find a representation of the infinitesimal generators of the special unitary group SU(3) that contains 4 by 4 matrices, by say taking a Kronecker product of its irreducible ...
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Schur's Lemma: Is the isormorphism between two irreducible spaces unique?

Suppose $V_1 \neq V_2$ are two irreducible representations of the finite group G. Then Schur's Lemma says that any G-invariant map between them is either 0 or an Isormorphism. I understand that if ...
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Permutation modules and their vector space dimensions

I'm given a field $k$, a finite group $G$ and a set $S$ which $G$ acts on transitively. I'm then told to consider the permutation module $M = kS$. My first problem is understanding what the ...
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Exponentials of Representations of Lie Algebras

Assume G is a lie group and g is its lie algebra. Consider a representation of G : D:G->End(V). Then there is a corresponding representation of g : d:g->End(V). My question is, when you can express ...
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Krull-Schmidt theorem and internally cancellable modules?

According to this lecture notes (in Lemma2.1) the statement $Ae\simeq Ae^{\prime}\to A(1-e)\simeq A(1-e^{\prime})$ is true for finite dimensional algebras by using Krull-Schmidt theorem. Can anyone ...
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Representation is reducible

Suppose $V$ is a representation of a finite group $G$ over a field $k$ of characteristic $0$, and suppose dim$V=3$ and $\wedge ^2V$ is reducible. Then $V$ is reducible. I was trying to do it by ...
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39 views

irreducible representation contained in regular rep

Why is every irreducible representation contained in the regular representation? Suppose $W$ is a irreducible representation. ( i.e. a vector space over $\mathbb{C}$ which $G$ acts on with no ...
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23 views

Representation Theory Symmetric Group Book?

I'm looking for a nice book that discusses the representation theory of the symmetric group. My background is an introductory class in representation theory.
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Are the orbits of a connected Lie group acting on a vector space always embedded manifolds?

Setting: We have a connected Lie group $G$ and a smooth map $G \to GL(V)$, where $V$ is a finite-dimensional vector space. Are the orbits of $G$ on $V$ embedded submanifolds? More precisely, if one ...
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Let $A''=\operatorname{End}_{A'}(V)=\operatorname{End}_{\operatorname{End}_A(V)}(V)$. Show that $A''$ is a $k$-algebra.

Let $A$ be a $k$-algebra for a field $k$. And let $V$ be a representation of $A$. Define $A''=\operatorname{End}_{A'}(V)=\operatorname{End}_{\operatorname{End}_A(V)}(V)$. Show that $A''$ is a ...
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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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Under what conditions can I expect the restriction of scalars functor to preserve tensor products

Suppose I have the canonical injection $i:H\hookrightarrow G$. Evidently I can induce the map on modules which restricts scalars from $\mathbf{Z}[G]$ to $\mathbf{Z}[H]$; that is, ...
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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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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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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 $V$. ...
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Nondegenerate representation

By the definition, we say a representation $(\pi,H)$ is nondegenerate if $cl[\pi(A)H ]= H$. Below I have two theorem, the first from Conway's Functional analysis and the second from Takesaki's ...
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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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Decomposition of a representation of $\mathfrak{S}_4$

I know that upon deocomposition we obtain two 3-dimensional irreducible representations, the first one is obtained by restricting the representation $ M: \mathfrak{S}_4 \rightarrow Gl_\mathbb{C}(4)$ ...
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(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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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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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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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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27 views

Using function parameters as representation

I was wondering if there is some field of mathematics which analyzes situations where you use function partners as representations, e.g. for classification or regression. For example, let's say I ...
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A question about the article 'You can't hear the shape of a drum'

I have read the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
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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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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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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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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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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 ...
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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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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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Kac's question 'Can one hear the shape of a drum' and Sunada method, a clarification

I'm reading the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
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31 views

The irreducible representation of $S_n$ of degree $n-1$ [duplicate]

So I understand it's not hard to show that the standard $(n-1)$-dimensional irreducible representation of $S_n$ is the only irreducible representation of $S_n$ of degree $n-1$ using characters/Young ...
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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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sub-$G$-representations

"So let $G$ be a finite group, $H$ a proper, nontrivial normal subgroup of $G$. For any representation $\rho: G \to \text{GL}(V)$ define the $H$-invariants of $V$ as $$V^H := \{v \in V \text{ }|\text{ ...
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$G$-representations, $W \otimes V^* \to \text{Hom}(V,W)$

Let $V$ and $W$ be finite-dimensional vector spaces. I know how to construct an explicit isomorphism of vector spaces $W \otimes V^* \to \text{Hom}(V,W)$ and show that it's an isomorphism. But if I ...
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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 ...
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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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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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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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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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Dirac Group Representations

(I am asking this question in the physics stackexchange too. I hope it is not problematic for me to ask same questions at two different places to get wider perspective!) I am currently taking a ...