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

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Some questions on Langlands Classification of Irreducible Admissible Representation

I am trying to construct some representations using Langlands classification theorem. But I get confused and have some problems when constructing these representations..... i) In the classification ...
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Lusztig's $h$-function of a dihedral group

Following the notations in Hecke algebras with unequal parameters, let $(W,S,L)$ be a weighted Coxeter system, and $H$ be the corresponding Hecke algebra with $\{c_w |w \in W\}$ the Kazhdan-Lusztig ...
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51 views

Character as sum with regular representation

Suppose $G$ is a group and $\chi$ is a character of $G$ with $\chi(g_1)=\chi(g_2)$ for all non-identity $g_1,g_2 \in G$, and let $\chi_{reg}$ denote the regular representation character. I read that ...
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31 views

$Hom_G(\pi,\sigma)$ = $Hom_{\mathfrak{g}}(d\pi,d\sigma)$?

Let $G$ be a Lie group. Let $\mathfrak{g}$ be the corresponding Lie algebra. Let $(\pi,V)$ and $(\sigma, W)$ be representations of $G$, with corresponding differentials $d\pi$ and $d\sigma$, which are ...
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proof that Linear transformation is isometry

let ∑=set of all continous unitary representation and $ Ψ \in{ ∑}$ $π_Ψ: \frac{L^1(G)}{N_Ψ}→ B ( \oplus H_π) $ is definde by $$π_Ψ(f^0)=\oplus π(f) , π \in{ Ψ},f^0\in{\frac{L^1(G)}{N_Ψ} }$$ ...
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1answer
35 views

Existence of Irreducible Character s.t. $\chi(g) \neq 0, \chi(1) \neq 0 \text{ mod } |C(g)|$ for Elements in Conjugacy Class of Prime Order

Given a finite group $G$, and a non-identity representative $g$ in a conjugacy class of prime order $p$, I'm trying to show that some nontrivial irreducible character of $G$ must have $\chi(g) \neq 0$ ...
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38 views

Describing $GL(2,\mathbb{C})$ with generators and relations.

My question is : how can I describe $GL(2,\mathbb{C})$ with generators and relations ? I do not know how to start ? Thanks for your help in advance,
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30 views

Trivial representation from the row-shape Young diagram

For the Young diagram $\lambda$ which is the row with, say $d$ squares, i.e. $\lambda = (d)$, the corresponding Young symmetrizer is $c_\lambda = \sum\limits_{g\in\mathfrak S_d}g$ such that the ...
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1answer
135 views

How to prove that representations on $S^k(V), \bigwedge ^ k(V)$ are irreducible?

Given a $\mathbb{C}$ vector space $V$, let $GL(V)$ act on $\bigotimes^k(V)$ via: $GL(V) \times \bigotimes^kV \to \bigotimes^k(V), \ (A,v_1\otimes...\otimes v_k)\mapsto Av_1\otimes...\otimes ...
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33 views

Trivial representation in tensor square

Taken from another question in this website. I am not sure why the following statement is true. Suppose $G$ is a group and $V$ an irreducible representation of $G$. One has that ...
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1answer
78 views

Request a reference in group theory

Although the book "A Course in the Theory of Groups" by Derek J.S. Robinson is an excellent up-to-date introduction to the theory of groups and covers various branches of group theory, it is hard for ...
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117 views

Question about isotypical components

Consider $V=\bigotimes^3(\mathbb{C}^2)$ as a $\mathfrak{S}_3$ representation. One of its isotypical component is $S^3(\mathbb{C}^2)$, which is a linear subspace of symmetric tensors of ...
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159 views

Clifford algebra - Gamma matrices

Let's say we have $\gamma^{a}$ matrices $(a=1,2,...,D)$. They satisfy the following condition $$\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\delta^{ab}I^{N\times N}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ ...
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2answers
47 views

Semisimple modules and the radical

I don't need a proof, but can someone tell me whether it is true that for all $A$-modules $V$ we have that $V/\text{rad}V $ is semisimple, where we define $\text{rad} V$ as the intersection of all ...
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86 views

About integral binary quadratic forms fixed by $\operatorname{GL_2(\mathbb Z)}$ matrices of order $3$

I am reading this paper of Manjul Bhargava and Ariel Shnidman, and I want to prove this claim, which appear at the first paragraph of Theorem $14$: Up to $\operatorname{SL_2}(\mathbb Z)$ ...
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2answers
149 views

Quaternion–Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $ \mathbf{R}^3 $ by conjugation. I've recently encountered spinors and notice similarities in their definitions ...
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1answer
45 views

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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1answer
39 views

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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38 views

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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1answer
98 views

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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1answer
46 views

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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35 views

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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1answer
47 views

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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1answer
60 views

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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1answer
54 views

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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1answer
45 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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76 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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54 views

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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1answer
32 views

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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1answer
39 views

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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1answer
20 views

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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1answer
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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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38 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 $V$. ...
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1answer
47 views

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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1answer
43 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 ...
3
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1answer
145 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 ...
3
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1answer
159 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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72 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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1answer
124 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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2answers
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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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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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142 views

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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1answer
67 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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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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1answer
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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 ...