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

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Why is the Fourier transform of a non-Abelian finite group the weighted superposition over all irreps?

I am going through the lecture note of Andrew Childs on Nonabelian Fourier analysis. I would like to quote from the note: My question: Why does it have to be weighted superposition and not equal ...
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39 views

Degree one irreducible representations

In section 2.5 of his Linear representations of finite groups (I have the french copy), Serre gives an example of determination of the character table of a group $G$. The group $G$ is taken to be $S_3$...
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Action of $sl(2,\mathbb{C})$ on Dual of Polynomials does not Exponentiate

Let $V$ be the space of holomorphic polynomial functions in two complex variables $\xi,\eta$ and let $V^\ast$ be its dual space with subspace $W$ of linear functionals of the form $Df(1,0)$ where $D$ ...
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31 views

What is the use of a right-module?

It seems that only the left-module provides a representation of a group. So what is the use of a right-module?
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72 views

Exterior power of irreducible representation

I am new to representation theory. Suppose that $G$ is a finite group with an irreducible representation over a (real or complex) vector space $V$. In my application, $G$ is a symmetric group and the ...
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46 views

Degrees of irreducible characters of an extension of $A_5$ by an elementary abelian 5- group

I'm reading a recent paper of G. Navarro, The set of character degrees of a finite group does not determine its solvability, in which he construct two finite groups $H$ and $G$ with $cd(G)=cd(H)$ such ...
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129 views

What are some Group representation of the rubik's cube group?

The Rubik's cube corresponds to valid sequences of moves of the Rubik's cube. What are some group representations of this group (with respect to finite dimensional vector spaces on finite fields)? ...
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Assume a homomorphism of groups gives a full and faithful functor on reps. Was it surjective?

Let $\phi: H \to G$ be a finite group homomorphism. Then there is a functor on representations $\operatorname{Rep}(\phi): \operatorname{Rep}(G) \to \operatorname{Rep}(H)$ given by precomposition with $...
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72 views

Degrees of Irreducible Characters of $GL(n,q)$

I know that in an old paper, R. Steinberg computed the irreducible (complex) characters of groups $GL_n(q)$ and $PGL_n(q)$ where $n \in$ {3, 4}. I want to know is there any known method for computing ...
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Is there an intuitive way of seeing why there are only finitely many irreducible representations?

Let $G$ be a finite group. A basic result in representation theory is that up to $\mathbb{C}[G]$-module isomorphism, there are only finitely many irreducible representations of $G$ over $\mathbb{C}$. ...
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34 views

Undecidability and the representation theory of $K<X,Y>$

The question comes from the problem here: http://mathoverflow.net/questions/73940/are-wild-problems-related-to-undecidable-ones It has already been proven that the representation theory of $K<X,Y&...
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64 views

Definition of the category of group representations

One usually considers the category of complex linear group representations for a fixed group $G$. It is defined as the category whose objects are group morphisms $G \rightarrow GL(V)$ where $V$ is a ...
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27 views

Young tableaux of partition $3+1+1$ for the conjugacy classes of $S_5$

I just computed the Young tableaux of partition $3+1+1$ for the conjugacy classes of $S_5$. It would be nice if anyone could confirm it's correctness. Thanks.
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1answer
75 views

Does the supposed to exist functor considered in Langlands program bear a peculiar name?

I'm trying to figure out what a very rough sketch of the Langlands program could be. From what I (think I) understand, objects called reductive algebraic groups together with related so-called ...
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63 views

A Lie group that has an immersion in $\mathrm{GL}(n,\Bbb R)$ but no embedding?

Question: Is there a Lie group $G$ that admits a smooth immersion $$i:G\longrightarrow\mathrm{GL}(n,\Bbb R)$$ for some $n\in\Bbb N$, but no smooth embedding $$j:G\longrightarrow\mathrm{GL}(m,\...
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1answer
50 views

Is there a faithful linear representation of the additive group of integers?

In other words, does there (constructively) exist a faithful representation $\phi : \mathbb{Z} \rightarrow GL_{n}(\mathbb{C}) $?
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Why do we care about two subgroups being conjugate?

In classifications of the subgroups of a given group, results are often stated up to conjugacy. I would like to know why this is. More generally, I don't understand why "conjugacy" is an equivalence ...
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Flag varieties from the representation of a solveable Lie algebra

I've been reading Lie Algebras, and I've come across this problem: "Let $\mathfrak{g}$ be a solveable Lie Algebra over $\mathbb{R}$. $V$ a vector space over $\mathbb{R}$, and $\rho$ a representation ...
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35 views

Radical of a Lie algebra bracket itself

Let $\mathfrak{g}$ be a Lie Algebra over $k$, $\mathfrak{n}$ its radical. Why is $[\mathfrak{n},\mathfrak{g}]$ the smallest of its ideals $\mathfrak{a}$ such that $\mathfrak{g}/\mathfrak{a}$ is ...
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38 views

Nilpotent Lie Algebra with determinant 0 [duplicate]

If I have a nilpotent Lie Algebra $\mathfrak{g}$ and a representation $\rho(X)$ in a vector space $V$ such that $det \rho(X) = 0 $ for all $X \in \mathfrak{g}$, then how do I show that there is a ...
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37 views

Characters of modules of associative algebras

Let $F$ be a field, and let $A$ be a finite-dimensional associative $F$-algebra with multiplicative identity 1. If $M$ is a finite-dimensional module of $A$, define the character $\chi_M:A\rightarrow ...
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Why does a skew-linear form on kG determine a triangular structure on k[G]?

I'm trying to understand braidings on finite group representations. They are the same as quasitriangular structures on the group algebra $k[G]$. The original reference seems to be http://arxiv.org/pdf/...
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27 views

Irreducible invariant tensor spaces of the general linear group $GL_n(\mathbb{C})$

I am learning the representation theory of the general linear group $GL_n(\mathbb{C})$. As far as I understand, the way to decompose the $\nu$-fold tensor product space $ V^{\otimes \nu}=V \otimes V ...
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1answer
57 views

Finite dimensional algebras with finite global dimension.

Let $A$ be a finite dimensional $k$-algebra, $k$ is a field, with a finite global dimension. I wonder if that implies $A$ is tame or finite type? or more generally is there a relation between these ...
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1answer
34 views

Representation of indefinite Kac-Moody algebras

The Kac-Moody algebras are divided in three very distinct classes: finite-dimensional, affine and indefinite type. For the first class the finite-dimensional representation theory is very known. For ...
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42 views

Surjective quadratic mapping

Are there any known values of $n$ for which there exists a surjective quadratic mapping $Q:\mathbb{R}^n \rightarrow \mathbb{R}^n$ with non-trivial zeroes?
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1answer
51 views

Question about Schur's lemma and irreducible representations of $S_n$

Schur's lemma says that if $M,N$ are two irreducible representations of a group $G$, then either $Hom_G(M,N)=0$ if $M,N$ are not isomorphic, or every $\varphi\in Hom_G(M,N) $ is invertible if they are ...
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2answers
60 views

A question on the decomposition of the group algebra $\mathbb{C}[G]$ of a finite group $G$

I am am very confused about a fundamental result in representation theory of finite groups. Please let me first introduce the setting. Let $G$ be a finite group. The group algebra $\mathbb{C}[G]$ is ...
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18 views

Help with a stage in Peter-Weyl proof: that “matrix entry” functions separate points

Edit The question as originally phrased was clumsy. What I really need is the simplest proof, or reference, anyone can rustle up of this: "for $G$ a compact Lie group, and $g$ and $h$ distinct ...
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21 views

Induced representation via diagonal embedding

I've been working on a couple of problems from Fulton-Harris' Representation theory book. In particular, for 6.11, we want to prove that $\mathbb{S}_v(V\otimes W)=\oplus C_{v\mu\lambda} (\mathbb{S}_\...
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1answer
47 views

Show that character vanishes on specific element $g$ if for $H \le G$ we have $[\chi_H, 1_H] = 0$ and all elements of $Hg$ are conjugate in $G$

Let $G$ be a finite group and $H \le G$ with $g \in G$ such that all elements of the coset $Hg$ are conjugate in $G$. Let $\chi$ be a $\mathbb C$-character of $G$ such that $[\chi_H, 1_H] = 0$. Show ...
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33 views

The Jacobson radical of an algebra contains every maximal $A$-submodule of an $A$-module

Let $A$ be an algebra over the field $F$. For an $A$-module define $\mathcal A(V) = \{ a \in A \mid Va = 0 \}$, the annihilator of $V$. Denote by $\mathcal M(A)$ a set which contains one isomorphic ...
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1answer
24 views

Representation of the full ring of $n\times n$ matrices

I am now reading the book 'representations of groups' by Boerner. On page 72, he states the theorem: Every representation of $M_n$ is completely reducible; every irreducible representation is ...
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If $A$ is a semisimple algebra, and $M_1 \ncong M_2$ as irreducible $A$-modules, why we have that every ideal of $M_1(A)$ is an ideal of $A$

Let $F$ be a field and $A$ be an $F$-vector space which is also a ring with $1$. Suppose for all $c \in F$ and $x,y \in A$ we have $$ (cx)y = c(xy) = x(cy) $$ Then $A$ is called an $F$-algebra. If ...
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For a semisimple algebra and two $M$- and $W$-homogeneous parts for $M \ncong W$, why we have $M(A)W(A) = 0$.

Let $F$ be a field and $A$ be an $F$-vector space which is also a ring with $1$. Suppose for all $c \in F$ and $x,y \in A$ we have $$ (cx)y = c(xy) = x(cy) $$ Then $A$ is called an $F$-algebra. If ...
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1answer
20 views

Problem with $SL(2)$ isometric action on a compact homogeneous space

Let $G=SL(2,\mathbb{R})$, fix any left-invariant Riemannian metric $g$ on $G$. Let $\Gamma$ be a cocompact discrete subgroup of $G$ and $X=G/\Gamma$. Because $\Gamma$ acts by isometries $g$ descends ...
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101 views

Symmetric power of tautological representation of $U(n)$

Let $S^kV$ be the $k$-th symmetric power of tautological representation of $U(n)$ how to see that it's irreducible? I'm trying to do it using weight, but with no benefits..
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Why there is an isomorphism $D(soc^{i}M) \cong DM/rad^{i}DM$

I am reading the book Elements of the Representation Theory of Associative Algebras, volume 1, by Assem et al. On page 162, it is written $D(soc^{i}M)\cong DM/rad^{i}DM$, where $DM=\mathrm{Hom}(M,k)$. ...
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1answer
55 views

About a Corollary of Yoneda's Lemma

I am reading Assem-Simson-Skowronski's book "Elements of The Representation Theory of Associative Algebras". I do not understand a Corollary 6.2, (IV. 6.2, Functorial Aproach to almost split). It says ...
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29 views

Reference request: bounded derived categories and their Auslander-Reiten quivers

I have some knowledge of Auslander-Reiten theory, tilting theory, derived categories and triangulated categories though I still find most proofs using derived categories in "Tilting Theory and Cluster ...
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Prove that the augmentation ideal in the group ring $\mathbb{Z}/p\mathbb{Z}G$ is a nilpotent ideal ($p$ is a prime, $G$ is a $p$-group)

Let $p$ be a prime and let $G$ be a finite group of order a power of $p$ (i.e., a $p$-group). Prove that the augmentation ideal in the group ring $\mathbb{Z}/p\mathbb{Z}G$ (to be read as $\left( \...
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Is there a general formula for the following Lie algebra quantity?

Consider the generators of $SO(n)$, written as $M_{\mu\nu} = - M_{\nu\mu}$ and they satisfy $$ \left[ M_{\mu\nu} , M_{\rho\sigma} \right] = i \left( \eta_{\nu\rho} M_{\mu\sigma} + \eta_{\mu\sigma} M_{\...
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$\text{Hom}$ to a projective $D[G]$-module for a complete DVR $D$

Suppose you have a complete DVR $D$ and a finite group $G$ with $D[G]$-modules $A$ and $B$. Does $B$ being projective imply that $\text{Hom}_{D[G]}(A,B)$ is $D$-free? Or should it be $A$ that's ...
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34 views

Representation of a group on a vector space induces a representation on another representation space?

Caveat: this is a very basic question. Suppose you have a representation of a group $G$ on a vector space $V$, let's say to be concrete $\mathbb{R}^n$. How is this representation related to the one ...
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Primitive of the matrix elements of irreducible representations of Lie groups

I am interested in the matrix coefficients $U_{ij}(g)$ of unitary irreducible representations of a Lie group $G$. In my case, these coefficients arise from the Peter-Weyl theorem. I would like to ...
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28 views

Tensor product of $Spin(2k)$ representations

I am trying to find the tensor product of spinor representations of $SO(2k)$. Labels are given as $$(n+I/2,I/2,\ldots,I/2,s)\otimes(I/2,\ldots,I/2).$$ Where $I$ and $n$ positive integers. How can ...
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$G$ and $G/H$ representations

It is known that if a group $G$ has an invariant subgroup $H$ and the factor group $G/H$ has a known representation then this representation is also a representation of group $G$. But, how can we ...
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1answer
34 views

Bases for irreducible representations $V$ and $W$ are linearly independent implies basis for $V \oplus W$ is linearly independent.

Let $\rho: G \to GL(U)$ be a reducible representation with dimension $n$ of a finite group $G$ such that $U= V \oplus W$, with $V$ and $W$ irreducible. If $\{v_1, v_2, ..., v_k\}$ and $\{w_{k+1}, w_{k+...
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24 views

Integration on compact group

Let $K$ be a compact topological group, and let $(V,\pi)$ be a continuous representation of $K$ over the complex field $\mathbb{C}$. Denote by $\mathrm{d}$ the Haar measure on $K$. If $v\in V$ ...
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Young tableaux to Specht polynomial to Irreducible representation for $(1,3,5) \in S_5$

What I am trying to do? Work out the irreducible representation of the group element $(1,3,5) \in S_5$ for the partition $2+2+1$ . Motivation: Learn how to calculate irreducible representation from ...