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

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Let $\rho : G \rightarrow GL_n(\mathbb{C})$ be a representation show that $|\operatorname{tr} X| \leq \dim \rho$

Let $G$ be a finite group. Let $\rho : G \rightarrow GL_n(\mathbb{C})$ be a representation, pick $g \in G$ and write $X=\rho(g)$. Prove that all eigenvalues of $X$ are roots of unity, and deduce that ...
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27 views

How to rigorously show tensor identities using symmetry arguments?

I am wondering how to rigorously show tensor identities such as the following. Let $n$ denote the radial unit vector in $D$ dimensions. Then $\langle n_i n_j \rangle = \frac 1 D \delta_{ij}$ and ...
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30 views

What is the notion of “character” in the context of Cayley graphs?

I am looking at these notes, http://www.eecs.berkeley.edu/~luca/books/expanders.pdf On page 37, Lemma 5.16, the notion of "character" defined seems to be any map from the finite Abelian group to ...
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46 views

Example of $\det \rho(g)=\det \sigma(g)$ for all $g\in G$, but $\rho \not\simeq \sigma$

Give an example of a group $G$ and two representations $\rho$ and $\sigma$ of $G$ such that $\det \rho(g)=\det \sigma(g)$ for all $g\in G$, but $\rho \not\simeq \sigma$. At the moment but ...
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91 views

Ten dimensional representation of $S_6$

Let $S=\{1,2,3,4,5,6\}$. For every three-element subset $A\subset S$ and $B=S\setminus A$ consider the symbol $e_{(A|B)}$ for which we assume that $e_{(A|B)}=e_{(B|A)}$. Then the vector space $V$ ...
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27 views

Reducibility of Cyclic groups

Let $G$ be the cyclic group $C_{4}$ and consider the 2-dimensional representations of G. Why does extending scalars to the complex numbers let this representation become reducible? I understand how it ...
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30 views

Decomposing some representations as a direct sum of irreducibles

I'm taking $V$ to be the standard representation of $S^3$. I'm looking for the decomposition of the following representations as a direct sum of irreducibles. (a) $V\bigotimes V$ (b) $Sym^2$ $V$ (c) ...
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28 views

Descending representation on the quotient group

I have this homework question in introductory representation theory: Let $ψ : G → GL(V)$ be a representation of $G$ and let $N$ be a normal subgroup of $G$. Define $ρ : G/N → GL(V)$ by $ρ(gN) = ...
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33 views

number of irreducible representations

Prove that if $H$ is an abelian subgroup of $G$ then each irreducible representation of $G$ has degree at most $\frac{|G|}{|H|}$ One proof is given in serre, but I would like to see some different ...
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24 views

Is each multiplicative linear functional on $L1(SL(2,R):SO(2,R))$ triviall?

We know that if $G=SL(2,R)$ and $H=SO(2,R)$ as a compact subgroup of $G$, then $\{ξ∈\hat{G};ξ|_H=1\}$ is triviall ($\hat{G}$ is the characters group). Can we conclude that each multiplicative linear ...
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69 views

Inner product doesn't matter for Schur orthogonality?

I'm reading Knapp's Basic Algebra, specifically the section about Schur orthogonality relations. Given a representation $R: G \to \text{End}(V)$, he defines $V_R$ to be the vector space of matrix ...
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4answers
98 views

Quick question about formal definition of a representation

So I know that a linear representation is defined as $\rho : G \to GL(V)$ over some finite group $G$. So if we define the action of some group ring, say F[G] over some representation V, is this ...
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23 views

Iwasawa module: $O[[G]]$ is noetherian where $G$ is a compact $p$ adic Lie group

Let $G$ be a compact $p$ adic Lie group. Let $E$ be a finite extension of $\Bbb{Q}_p$ with ring of integers $O$. Then how to show that $O[[G]]$ is noetherian. I was reading the article here and on ...
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29 views

Canonical Decompostion

This is regarding the proof of proposition 24(page 61) of Serre's Linear Representations of finite groups. Line 3 of the proof says that for $s\in G$, $\rho(s)$ permutes $V_i$? Can someone be kind ...
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15 views

counting how many entries from a given Young tableau contribute to hook length made from two differtn YTs

Think of a Young diagram as a collection of rows with numbers of elements $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ and define for $s=(i,j)\in R$ (it makes also sense for $s$ outside ...
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79 views

Proof of Proposition 13.20 in Digne-Michel

I have some problems understanding a part of the proof of Proposition 13.20 in "Representations of Finite Groups of Lie Type" by Digne and Michel. It concerns a bijection between certain varieties ...
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31 views

cohomology of general linear group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let $\mathrm{GL}_n(\mathbb{Z}_2)$ be the group consisting of all $n\times n$ matrices with entries in $\mathbb{Z}_2$ with non-zero determinant. What is the ...
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45 views

Representation Theory over field of characteristic 2

How can we find a G-invariant form without getting something trivial by the usual averaging process?
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14 views

Classifying all irreducible admissible representations of $SL(2,\mathbb{C})$

I know that for $GL(2,\mathbb{R})$, any irreducible admissible representation is either finite dimensional, principal series, discrete series or limit of discrete series, up to equivalence as ...
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Given an irreducible representation, is there a *unique* unitary representation that it is equivalent to?

I might need help here in understanding my own question in places and please don't hesitate in asking for edits and clarifications. Background: A representation $\rho$ of a finite group $G$ is a ...
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32 views

$\overline\pi$ vs. $\check\pi$

Let $\pi$ be the automorphic representation of ${\rm GL}_2$ or $B^\times$ ($B$ an indefinite quaternion algebra over $\Bbb Q$) of central character $\varepsilon$ attached to some holomorphic newform ...
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63 views

Representation of Quaternion group in $GL(2,3)$

I am working with the representation of the quaternion group in $GL(2,3)$ generated by $A=\begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}, B=\begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}, ...
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$K$-basis of cyclic $KG$-module determined by group elements

Let $G$ be a finite group and $K$ a field with $\mathrm{char}(K) \nmid |G|$, and let $M$ be a cyclic $KG$-module. For a cyclic generator $v \in M$ we can pick group elements $g_1, \dots, g_d \in G$ ...
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40 views

Confusion in the tensor product homomorphism in representations

Let $k$ be a field and $G$ a group. Let $V$ and $W$ be two representations. And $V \otimes _k W$ be their tensor product which itself a representation with $G$ act on $V \otimes _k W$ by $g ( v ...
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67 views

If $N$ is semisimple and and $M/N$ semisimple then $M$ semisimple?

Let $R$ be finite dimensional $k$ algebra, $k$ is a field possibly $\mathbb{C}$. Let $_RN\leq{}_RM$ (submodule as a left module). Is the following statement true (if yes could you give me a hint) , ...
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134 views

Every representation of a finite groups decomposes into irreducible representations

I'm having trouble understanding this proposition from the first lecture in Fulton and Harris' representation theory book: Proposition 1.8. For any representation $V$ of a finite group $G$, there ...
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1answer
20 views

Relation between two sets of generators of SO(3)

I am working with the spin 1 representation of SU(2), which is just SO(3). The ordinary generators used in quantum mechanics are: $J_x = \left( \begin{array}{ccc} 0 & \frac{1}{\sqrt{2}} & 0 ...
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49 views

Group Action and “nice” Approximation!

Studying the action of a group $G$ on a set $X$ is naturally the same as looking at the group homomorphism $\alpha: G \rightarrow Perm(X)$. So, for a given group $G$, classifying all sets $X$, on ...
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1answer
76 views

Example of a linear algebraic group which is not a Lie group

I am trying to reconcile the notions of algebraic groups, linear algebraic groups, Lie groups, and Lie algebras, along with their notions of root systems, maximal tori, etc. To begin, I am trying to ...
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64 views

Proving complete reducibility of modular representations

Let $G$ = $S_{3}$ and consider the $3 \times 3 $ permutation representations. For example, we have $$ \psi (123) = \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ ...
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Is this extension of $Sp(4,2)$ a semidirect product?

Somebody I trust has been insisting to me that a certain extension of $Sp(4,2)$ is actually a semidirect product, and I'm inclined to believe him, but I haven't been able to convince myself he's ...
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1answer
32 views

The group action of $S_n$ given a partition of $n$

We know that irreducible representations of $S_n$ are given by partitions of $n$. I would like to know if there is a way to explicitly write down the action of some $g \in S_n$ on the representation ...
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38 views

Solving Fulton and Harris exercise 2.4

Let $\rho$ be a representation of $G$ on $V$ of dimension $4$. let $g\in G$ be an element of order 4.Let $\chi$ the character of representation $V$ is known then find the eigenvalues of $\rho_g$. ...
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13 views

manipulations with SU(2) Nekrasov partition function

Think of a Young tableau $R$ as collection of rows $y_1 \geq ... \geq y_d > y_{d+1}=0$ and all others zero, with $\ell(Y):= \sum_j y_j$ and for a box $s=(i,j)\in R$ we have $a_Y(s):=y_i-j$ and ...
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60 views

Reducibility of Representations over Finite fields

So, there are several standard ways of proving irreducibility/reducibility for representations over fields where the characteristic doesn't divide $|G|$ such as Maschke's theorem, jordan normal form, ...
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33 views

Eigenvalues of operator on $S_n$'s group algebra

Take the group algebra of the symmetric group $S_n$ (or equivalently consider $S_n$'s regular representation) - I guess over $\mathbb{C}$. If $e_{i,j} \in S_n$ denotes the element which swaps only ...
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36 views

Inequivalent representations of a finite group

I'm looking for this result: A finite group has only finitely many inequivalent representations of given degree over a field of characteristic $0$. Do someone know where I can find a proof of ...
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20 views

How does the $10$ dimensional irrep (tensor) of $SU(3)$ look like?

We know that for $SU(3)$ the following tensors furnish the $\mathbf{d}$ dimensional irreducible representation: $$\phi^i\hspace{1cm} (\mathbf{3})\\ \phi^{ij}\hspace{1cm} (\text{asymmetric in ...
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15 views

Non-trivial characters of $SU(2)$

Are there non-trivial characters (or quasi-characters) of the special unitary group $SU(2)$? I couldn't find a straightforward answer by googling.
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25 views

Infinite dimensional FG-modules

So the way I understand FG-modules is that it is analogous to a vector space defined over a field F with G a basis. However, I encountered a problem given the hypothesis that V is a possibly infinite ...
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31 views

Eigenvalues of (restrictions of) the standard representation of $S_n$

Let the permutation group on $n$ elements $S_n$ act on a set $S$ of size $k < n$ via permutations. Fix some ordering on the elements of $S$ to make this sensible. Is there any way to understand ...
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problem book suggetion

I'm taking a course on basic representation theory, where upto midsem we're supposed to learn upto $5$ th chapter of serre. Can one please suggest some book which consists of nice problems ? I've ...
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35 views

Exercise on representations of the Dihedral group (Etingof 3.17)

I'm confronted once more with a problem on representation theory which I cannot fully solve (Problem 3.17 http://math.mit.edu/~etingof/replect.pdf): Let $G$ be the group of symmetries of a regular ...
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Exercise on formal deformations of representations (Etingof 2.24)

I'm trying to work out a few exercises in Etingof's book on representation theory of associative algebras (http://math.mit.edu/~etingof/replect.pdf) At the moment I'm looking at Problem 2.24. about ...
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46 views

Is there such a notion of “expansion” in groups?

Given a subset of elements of a finite group $G$, I would like it to be such that the set of all distinct words (as elements of $G$) that can be formed from this set is exponentially large in the size ...
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17 views

Rewriting a sum of Young Tableaux as Tensors

Is there a straightforward way (perhaps a software) that can write a direct sum of Young Tableaux in terms of tensors? For instance the direct product in $SU(3)$ (taken from this post) ...
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34 views

An integral with respect to the Haar measure over the unitary group

I am trying to find the answer of this integral: $$E:=\int dU \ (U^2 \otimes I) M (U^{ \dagger 2}\otimes I) $$ That is an integration with respect to the Haar measure and $U$'s are $d\times d$ ...
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64 views

A relation between the Jacobson radicals of a ring and those of a certain quotient ring

Let $R$ be a ring $J(R)$ the Jacobson radical of $R$ which we define for this problem to be all the maximal left ideals of $R.$ I'm trying to prove the following proposition with only the definition ...
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1answer
24 views

Conservation of bilinear forms and conjugation

Let $\omega,\omega'$ be non-degenerate skew-symmetric bilinear forms on $V$, a vector space over $\Bbb{C}$, preserved by $G,G' \subset GL(V)$ respectively. Must there be an element $\gamma \in GL(V)$ ...
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24 views

Functoriality of group homology

I understand that group homology $H_*(-)\colon \mathsf{Grps} \to \mathsf{Ab} $ is a functorial. In Weibel's homological algebra, there is an argument in 6.7 show this by using that $H_*(G;-)\colon ...