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

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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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61 views

Irreducible representations of groups of order $pq$: induction from normal subgroups

Consider a group $G$ of order $pq$ ($p$ and $q$ are distinct primes and also $p<q$). It is easy to show that the dimension of each irreducible representation of $G$ is $1$ or $p$. Also, it can be ...
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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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98 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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52 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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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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52 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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43 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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2answers
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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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44 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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60 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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1answer
95 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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66 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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79 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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137 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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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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46 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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44 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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52 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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77 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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74 views

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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60 views

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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1answer
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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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27 views

Understanding analytic construction of induced representation

I'd like to get some intuition for analytic construction of induced representations as described on Wikipedia. Algebraic construction also described there is much more intuitive and clear to me, but ...
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1answer
57 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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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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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 G ...
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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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107 views

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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50 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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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 ...
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What are the “irreducible factors” in an algebra?

What are the irreducible factors in an algebra? In the book "Spin Geometry" by Lawson and Michelsohn, page 35, line 1, there occurs "take irreducible factors of...", but I don't know what irreducible ...
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100 views

If every quotient by normal subgroup is abelian, then the irreducible representations are injective

The following is Problem 6 of January 2006 algebra qualifying exam from University of Maryland. See here for the problems. Let $G$ be a finite group. Suppose that for each normal subgroup $K\neq ...
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A corollary to the Wedderburn-Artin theorem.

Suppose we proved the Wedderburn-Artin theorem, i.e. we have the fact that if S is a semisimple algebra over a field $F$, then $$ A \cong M_{n_1} (D_1) \times ... \times M_{n_k} (D_k), $$ where ...
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Role of the discrete subgroups of Lie groups

This is a question I don't believe is too vague to admit a sensible answer: In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What ...