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

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On the structure of a vector bundle

Let $P \rightarrow X$ be a principal $G$-bundle, $\rho: G\rightarrow GL(V)$ and $\sigma: G\rightarrow GL(W)$ be two finite dimensional linear representations of $G$. Let $E=P\times_\rho V$ and ...
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31 views

Baily Borel Compactification: choice of boundary

In Borel/Ji " compactifications of symmetric and locally symmetric spaces " the Baily Borel compactification of a locally symmetric space is defined as $$\Pi\backslash(X\coprod_{\bf{P}}X_{P,h})$$ ...
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53 views

Formula for the number of inequivalent k-dimensional complex representations of an abelian group

Let G be an abelian group, $|G|=n$. Prove that the number of inequivalent k-dimentional complex representions of G is equal to the coefficient of $t^k$ in series $(1-t)^{-n}$. Find this coefficient. ...
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Highest weights of irreducible components of tensor product of irreducible sl(3)-module.

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows: For each weight $\mu$, let $L(\mu)$ be the irreducible ...
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41 views

Equation on the vertices of regular polyhedra

I found in this book, on page 6 that the equation on vertices of icosahedron inscribed in sphere considered as $\mathbb{CP}^1$ by means of stereographic projection is $xy(x^{10}+14x^5y^5-y^{10})=0$. ...
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60 views

Irreducible representations of nonabelian group generated by $3$ elements

My question is rather commonplace, but nevertheless I'd like to discribe irreducible representations of the so called Heisenberg group (I suppose this one is just a special case of Heisenberg group). ...
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1answer
44 views

The dimension of space of morhisms as the number of orbits

All groups are finite, all representations are over $\mathbb{C}$ (just in this context, of course), $G$ is a group, $K,H\subset G$ - its subgroups. By $\mathbb{C}$ we denote the trivial representation ...
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30 views

representations into $\mathfrak{sl}(n,\mathbb{C})$

in Borel/Ji "Compactification of symmetric and locally symmetric spaces" the standard Satake compactification is constructed and general Satake compactifications are realized via an embedding into the ...
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146 views

Irreducible representations of group of order $pq$

There is the problem to describe dimensions of irreducible representations of a group of order $pq$, where $p$ and $q$ a distinct primes. I am doing it as follows: Suppose $p>q$. Then by the Sylow ...
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62 views

SU(2) representations and differential equations in physics.

I studied that $SU(2)$ has a spin $j$ representation $U_j$on a homogeneous space of 2 variables with dimension $2j+1$. Now I am trying to understand the following sentences. Suppose $\phi: ...
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Space of morhisms of representations, its dimension in special case

The symmetric group $S_n$ acts linearly on $\mathbb{C}^n$, hence it brings up to the representation in $\Lambda^m\mathbb{C}^n$. The goal is to evaluate the dimension of morphisms ...
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Decomposition of representation of symmetric group

Let $V$, $\dim V=n-1$ be the standard representation of the symmetric group $S_n$ and let $V'= \langle x_1,x_2,\ldots,x_n \rangle$ be its natural representation. Then ( see. Fulton, Harris, 4.19) ...
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182 views

Transitivity of representation induction

Let $K\subset H\subset G$ be some triple of finite groups and $T: K\longrightarrow \mathrm{GL}(V)$ - some representation f $K$. We are to prove the transitivity of induction: $Ind_K^G(V)\simeq ...
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70 views

Each irreducible representation is a subrepresentation of induced one

I'm learning what irreducible representation is and need some examples. One of them is as follows: Let $G$ be any group and $H$ - it abelian subgroup. How to prove that each irreducible representation ...
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47 views

how to deduce that finite group $G$ has at most $|G|$ characters

Let $G$ be a finite abelian group. A character of $G$ is a group homomorphism $\chi: G\longrightarrow \mathbb{C}^{\times}$. I have proved by induction that for distinct $\chi_1,\cdots,\chi_r$, they ...
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If a group has a faithful reducible two-dimensional representation then its commutant is abelian.

A group $G$ has a faithful reducible two-dimensional representation. Prove that commutant of the group $G'$ is Abelian. I think to so. Commutant $G'\triangleleft G$. Let $\rho$ is the faithful ...
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Doubt about Proposition 2.39 in Dana Williams' crossed product book

You can see the proposition in a google books preview here. First and foremost, my question is: Question: Am I correct to interpret Proposition 2.39 as setting up a bijective correspondence ...
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47 views

Find the number of inequivalent two-dimensional complex representations of the group $Z_4$

Find the number of inequivalent two-dimensional complex representations of the group $Z_4$ Any hints will be greatly appreciated. Thank you all
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Irreducible representations and abeliean subgroups

There is a theorem in representation theory which is surprising to me: the dimension of irreducible (complex) representation of finite group is not greater that $(G:H)$ - an index of abelian subgroup ...
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71 views

Irreducible and regular representation

I was working on representations of $S_3$ and thought about the following problems: The symmetric group $S_4$ acts on $\mathbb{C}^{4}$ by permuting coordinates. Decompose this representation into ...
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109 views

Action of the Weyl group on the symmetric algebra $ S\mathfrak{h} $

Let $\mathfrak{g}$ be a complex semi-simple Lie algebra. Let $\mathfrak{h}$ be a cartan subalgebra. Let $ \Delta $ be the resulting root system. Denote by $ V $ the real span of the roots. Let $ ...
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111 views

The decomposition of the exterior of the symmetric square over Lie algebra sl(3)

I am studying the representation theory of finite dimensional modules over the simple Lie algebra $\operatorname{sl}(3)$. I know some basics facts about the decomposition of some construction of ...
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96 views

Maschke's theorem and the problem of the irreducible representation

Need to prove the following statement Let $\rho_k:<a>_n\rightarrow GL_2(R)$ is representation. $\rho_k(a)= \left( \begin{array}{cc} \cos {\frac{2 \pi k}{n}} & -\sin{\frac{2 \pi k}{n}} \\ ...
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on the simple group $M_{11}$

As we know, the simple group $M_{11}$ is a important group,it has order $7920$, how can we prove the simple group of order $7920$ is isomorphic to $M_{11}$ ?
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Representation theory and character proof

I have come across the following statement in various sources without any proof. Apparently they say the proof is trivial. However, I don't see the triviality in this case: $\rho_1,\cdots , \rho_r$ ...
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204 views

Representation theory and direct sum

I came across the following theorem in one of the online notes regarding representation theory which I thought should have a simple proof. I am trying to prove it using basic linear algebra tools: ...
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Infinite dimensional reps of the rotation group

$\mathbf{Background:}$ The following is paraphrased from ``Representations of the rotation and Lorentz groups and their applications,'' by Gel'fand. Consider a finite-dimensional representation $T: ...
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63 views

how can I show $H^1(g , Hom_C(g,M))=0$?

For a simple Lie algebra $g$ and a finite dimensional vector space $M$ with a trivial $g-$action, how can I show $H^1(g , Hom_C(g,M))=0$?
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53 views

An existence of exponential function for a Lie algebra.

Let $G$ be a Lie group (given by a matrix). Let $\frak g$ be its Lie algebra. I would like to know if the following is true. "Let $X$ be a matrix in $\frak g$. Then $\gamma(t)=\exp(tX)$ is a curve ...
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1answer
49 views

Representation in Banach space and norms 'induced' by representation

By $G$ we denote some compact group, $X$ stands for some Banach space. Suppose $\pi\colon G\longrightarrow \mathrm{GL}(X)$ to be some representation in $X$. I'm trying to prove that there is an ...
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57 views

Questions about Haar integral.

Questions about Haar integral. Let $B$ be the subgroup of $GL_2 (\mathbb{R})$ defined as $$ B =\{ \left( \begin{matrix} 1 & b \\ 0 & c \end{matrix} \right), b, c \in \mathbb{R}, c \neq 0 ...
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Questions about Haar integral for the group $GL_2(\mathbb{R})$.

Questions about Haar integral for the group $GL_2(\mathbb{R})$. How to show that a Haar integral for the group $GL_2 (\mathbb{R})$ is given by $$ I(f ) = \int_{\mathbb{R}} \int_{\mathbb{R}} ...
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125 views

Coxeter numbers for semisimple and reductive algebraic groups

I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on ...
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Group ring of galois group [duplicate]

Suppose $E/F$ is Galois extension. What is it known about structure of $F[Gal(E/F)]$? I've learned only one fact in this direction - existence of normal basis in $E/F.$ But it's not truly about ...
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How does one decompose the regular representation of the group

How does one decompose the regular representation of the group $<a>_2*<b>_3$ in the direct sum of 1-dimensional representations I know what is regular representation of the group (G ...
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How to decompose representations into irreducible ones?

I have some trouble understanding the decomposition of representations into irreducible ones. For example, take $G = S_3$, the symmetric group. Then $G$ has three irreducible representations, namely, ...
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Questions about real-valued measure on a vector space.

I am reading the lecture notes on representation theory. I have some difficulty in solving Exercise 1.8 on page 4. Let $K$ be a non-archimedian local field and $v : K \to Z \cup \{\infty \}$ a ...
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A representation of $SU(2)$ is self dual

Let $SU(2)$ be a set of $2 \times 2$ unitary matrices over $\mathbb{C}$ with determinant $1$. Let $H_j$ be a $2j+1$ dimensional vector space with basis $x^ay^b$ with $a+b=2j$. A representation $U_j$ ...
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Character tables of finite groups in positive characteristic - not the modular case

Let $G$ be a finite group and let $F$ be an algebraically closed field of characteristic $p$ with $p \nmid |G|$. So, the group algebra $F[G]$ is semisimple. What are the techniques to compute the ...
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Induced representations

I am studying representation theory and the induced representations are one thing that I really can't ''grok''. I was reading Wikipedia article and in the beginning it says: (...) induced ...
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1answer
93 views

Is there a Peter-Weyl theorem for the quasi-invariant measure on a homogeneous space of a compact semisimple group?

Let $H \hookrightarrow G$ be an inclusion of semisimple, compact Lie groups. There is a measure on the homogeneous coset space $G/H$ by pulling back the Haar measure on $G$ via the projection $G ...
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75 views

Regular representation

I am stuck at the following question and dont know where to begin: Let $\rho $ be the permutation representation associated to the operation of $D_3$ of order 6 on itself by conjugation. Decompose ...
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144 views

Representation theory

I am trying to study representation theory from the book Algebra by Artin. I came across the following problem which seemed interesting: Prove that the linear operator $T=\sum_{g\in C} \rho_{g}$ is ...
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1answer
83 views

Schur functors as spaces of “flag tensors”?

Consider the following construction: for a vector space $V$, define $W \subseteq \bigwedge^2 V \otimes V$ by $W = \langle\ \alpha \otimes v : v \in \text{Span}(\alpha) \ \rangle$, that is, $W$ is ...
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Representation theory and characters

I have been studying representation theory for 6 months now. I came across the following question in a graduate course example sheet. Let $\chi$ be the character of a representation $\rho$ of ...
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Are Lie algebras $u_n$ and $su_n$ simple?

I think, that $u_n$ isn't simple, because, for example, any matrix $(\begin{matrix} ia & 0 \\ 0 & ia \end{matrix}) \in Z(u_n)$, and hence $u_n$ has non-trivial ideal. But i don't know ...
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Is there any groups $G$ with the property $(*_d)$?

Let $G$ be a finite group of even order has only one non-principal irreducible character $\chi$ of degree $d$, $d\in \mathbb{N}$, with the following property (we name it $(*_d)$): $(*_d)$: There ...
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55 views

Show that we have an algebra homomrohpsim

I need to show that we have an algebra homomorphism $\phi: M_n(K)\otimes_KA \simeq M_n(A)$ Where A is a K-algebra and K is some field. I suspect it's really easy but I don't know what to do. Is ...
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Semisimplicity of restriction: Representation theory

Let $G$ be a group, $E$ be a vector space over field $K$ and $\rho : G \rightarrow \operatorname{GL}(E)$ a semisimple $K$-representation of $G$. Let $H \lhd G$ be a finite-index normal subgroup of ...
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1answer
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About a representation of matrix algebra

Let $\rho: M_n(\mathbb R)\rightarrow M_{2n}(\mathbb R)$ be an algebra morphism, i.e., a $2n$-dimensional real representation of the matrix algebra $M_n(\mathbb R)$. Then what can we say about the ...