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

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Characters of transitive finite permutation group

I know that Frobenius reciprocity helps us to solve this problem, but I don't know why: Let $ G $ be a transitive finite permutation group with permutation character $ \pi $. If $\chi $ is an ...
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39 views

Frobenius Reciprocity and a character theory problem

How Frobenius Reciprocity can help us to solve these two problems: Let $ H $ be a subgroup with index $ m $ in the finite group $ G $. Let $ F $ be an algebraic closed field of characteristic $ 0 $. ...
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19 views

A question and a conjecture on $USp(N)$ group

$USp(N)$ with $N$ an even integer is defined as the group of unitary matrices $M$ that satisfy $M^TJM=J$, where $M^T$ is the transpose of $M$ and $J$ is the anti-symmetric $N$-by-$N$ matrix ...
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42 views

transitivity of induction

I want to prove the "transitivity of induction" property: Let $ H\leq K\leq G $ where $ G $ is finite. Let M be an $ FH $-module, where $ F $ is any field. Then $ (M^K)^G\simeq^{FG} M^G $. Would you ...
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51 views

Relation between the characters of subgroups of a finite group

Let $ H $ and $ K $ be subgroups of a finite group $ G $. Let $ \chi_1(H) $ and $ \chi_1(K) $ denote the trivial characters of $ H $ and $ K $ over an algebraically closed field of characteristic $ 0 ...
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17 views

Weight spaces of a irreducible representation of $\mathfrak{gl}(n, \mathbb{C})$.

Let $\mathfrak{gl}(n,\mathbb{C})$ be the general linear Lie algebra. Let $\{E_{s,t}\}_{1\leq s,t,\leq n}$ be the standard basis for it. And set its Cartan subalgebra $\mathfrak{h}$ to be ...
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10 views

Relation between induced and coinduced spaces

Let $G$ be a compact Lie group and $H$ a closed subgroup of it. Let $X$ be a $G-$space. The induced $G-$space is defined to be $$G\times_H X$$ with the equivalence $(gh, x)=(g, hx)$, for any $g\in G, ...
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32 views

Character of transitive finite permutation groups [on hold]

Let $ G $ be a transitive finite permutation group with permutation character $ \pi $ and let $ \chi $ be an irreducible $ \mathbb{C} $-character. I want to know why the degree of $ \chi $ is at least ...
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45 views

Irreducible $ \mathbb{C} $-character of a finite groups [on hold]

Let $ \chi $ be an irreducible $ \mathbb{C} $-character of a finite group $ G $ and let $ K $ denote the kernel of the associated representation. If $ \chi $ has degree $ n $, is it true that $ ...
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44 views

$ \mathbb{C} $-characters of $ A_5 $

How can we find all of five irreducible $ \mathbb{C} $-characters of $ A_5 $? Precisely how can we construct the character table with the aid of orthogonality relations?
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36 views

Facts on $ \mathbb{C} $-characters

My assumption: $ G $ is a finite group & $ \chi $ is a faithful $ \mathbb{C} $-character of $ G $ with degree $ n $ and $ r $ is the number of distinct values assumed by $ \chi $. Now is it true ...
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30 views

Characters of a compact group with uniform positivity over $G$

Let $G$ be a compact group and let $\widehat{G}$ denote the set of all equivalence classes of irreducible representations of $G$. For each $\pi \in \widehat{G}$, we use $\chi_\pi$ to denote the ...
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60 views

Symmetric and antisymmetric powers of SU(2) representations

Recently, I took a course in representation theory at Imperial College, and on the first homework the questions were about certain sneaky relationships when it came to representations of SU(2). ...
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1answer
70 views

$ \mathbb{C} $-character table of $ D_{14} $ [on hold]

Is there any reference where I can find the $ \mathbb{C} $-character table of the dihedral group $ D_{14} $?
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2answers
47 views

Finding the character table of this group

if $ G = <a,b| a^9 = b^3 = 1, bab^{-1} = a^4> $ of order 27 then know the following, that any element can be written as $b^ka^n$ with n $\in [0,8], k\in[0,2]$ and that the 11 conjugacy classes ...
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2answers
51 views

irreducible characters of a group

I am currently attempting a past exam paper and am stuck on the following question for part a) $\mu$ is an irreducible character iff it is equal to the character of an irreducible representation, ...
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1answer
70 views

character tables and solubility

I am currently going through a past exam paper for a group theory module and am unable to answer the following section of a question. The copy of my lecture notes doesn't seem to have a section on ...
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15 views

Duality between the highest weight vector and lowest weight vector.

Let us consider a self conjugate unitary irreducible representation $D$ of a semisimple Lie group $G$ (though I'd be glad if there is a more general case). If $u$ is the highest weight vector of $D$, ...
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20 views

Structure constants for and the adjoint representation and meaning in $sl(2,F)$

First, what I know is that given the basis: $$e = \left(\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right),f = \left(\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}\right),h = ...
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1answer
25 views

Complete misunderstanding of Lie groups and representations

Consider a particular representation of $\operatorname{SO}(2,\mathbb{R})$: \begin{equation} \begin{pmatrix} \operatorname{cos}(\theta) & \operatorname{sin}(\theta) \\ ...
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30 views

The maps $\delta: V \to B \otimes V$ and $\delta': V \otimes V^* \to B$.

Let $B$ be a coalgebra and $V$ a vector space. Suppose that we have a coaction $\delta: V \to B \otimes V$. Is the map $\delta$ equivalent to a map $\delta': V \otimes V^* \to B$? Thank you very much. ...
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16 views

Thompson sporadic group presentation and conjugacy class representatives

So if you look here: http://web.mat.bham.ac.uk/atlas/v2.0/spor/Th/ they provide matrices, $a$ and $b$, which generate the Thompson sporadic group. They also give a representative for each conjugacy ...
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26 views

A reference for the Tannaka-Krein theorem

I am looking for a reference for the Tannaka-Krein theorem on compact groups. By the Tannaka-Krein theorem which is also called (classic) Tannaka duality (because of the quantum theory), I mean the ...
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46 views

An short exact sequence of $\mathfrak{g}$ of which head and tail are in category $\mathcal{O}$.

Let $\mathfrak{g}$ be a finite-dimensional, semisimple Lie algebra over $\mathbb{C}$. Let $$ 0\rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 $$ be a short exact sequence of ...
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35 views

Consequences of $G$ having exactly $3$ irreducible characters

Suppose a finite group $G$ has exactly irreducible characters $\chi_1 = \mathbb{I}, \chi_2,\chi_3$. i) Show that G is soluble and deduce it has a non-trivial one-dimensional character $\chi_2$. ii) ...
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46 views

Multiplicity one theorem for GL(n) and SL(n)

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) ...
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30 views

Can each of the following be a character of a finite group $G$

Can each of the following be a character of a finite group $G$? $(i) \ \ (2,0,\frac{1}{2},\frac{1}{2},0)$ $(ii) \ \ (3,-1,0,4,0)$ $(iii) \ \ (2,2,2,2)$ $(iv) \ \ (1,0,-1,0)$ I think that the ...
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1answer
36 views

Product of a character with an irreducible character a non-negative integer

Why is the inner product of a character with an irreducible character a non-negative integer? I can see that by properties of the inner product it will be non-negative but I cannot see why it would ...
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5 views

Spherical representation on locally compact group

What is the definition of a spherical representation of the the pair $(G \times G, G)$, where $G$ is a locally compact group?
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1answer
24 views

Jacobson's Density Theorem for Semisimple Algebras

I am trying to follow the proof of the following: Let $V=V_1\oplus \ldots \oplus V_r$, where $V_i$ are irreducible finitely dimensional representations of $A$ (where $A$ is an algebra over an ...
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72 views

An elementary property of tensor products.

I'm studying representations theory to start my Masters thesis. I'm using the book of J. P. Serre, Linear Representations of Finite Groups and in the pg. 55 He affirm: If $V$ is induced by $W$ and ...
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1answer
69 views

Why does applying this Galois automorphism $\sigma$ to each entry of a matrix representation $\varphi\colon G\to GL_n(F)$ preserve irreducibility?

Let $F\subset\mathbb{C}$ by the subfield of algebraic numbers, and let $\varphi\colon G\to GL_n(F)$ be a representation of a finite group $G$, with character $\psi$. Let $\mathbb{Q}(\varphi)$ denote ...
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15 views

Why do we need the Dynkin Basis to compute Branching Rules?

Given a representation $R$ of some Lie algbra $g$, we can compute the corresponding representation $R'$ (in general reducible) for some subgroup Lie algebra $ g \supset g'$ by utilizing the weights in ...
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1answer
45 views

If $\varphi$ is irreducible as representation over the algebraic numbers, then $\varphi$ is irreducible as a complex representation?

Let $F\subseteq\mathbb{C}$ be the subfield of algebraic numbers. Then a representation $\varphi\colon G\to GL_m(F)$ of a finite group $G$ may be also viewed as a complex representation. If ...
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Different Basis/Choices for $SU(3)$ generators?

Conventionally, the generators of $SU(3)$ in the fundamental representation are written in terms of the Gell-Mann matrices. Wikipedia calls this a "particular choice of this representation". What do ...
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If $|\psi(g)|=\psi(1)$ for $\varphi$ a faithful complex representation, why is $g\in Z(G)$?

If you have a faithful complex representation $\varphi\colon G\to GL_n(\mathbb{C})$ with character $\psi$, why does $|\psi(g)|=\psi(1)$ imply that $g$ is central? I could show that if ...
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40 views

Spinors and Möbius strips

Consider a Möbius strip; draw on one side of it an arrow aligned vertically; now take it for a trip by around the strip; then when it comes back to the same position it has flipped direction; another ...
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23 views

Why $\int_{U} d(m n m^{-1}) = \int_{mUm^{-1}} dn = \omega^2_E(t_1) \int_U dn$?

I am reading the lecture notes. On page 5, formula (1.24) is $$ \int_{U} d(m n m^{-1}) = \int_{mUm^{-1}} dn = \omega^2_E(t_1) \int_U dn, $$ where $dn$ is the Haar measure on $N$, $U \subset N$ is ...
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50 views

Theorem about irreducible representation of $C^*$-algebra

I have been told, that there is a theorem about irreducible representation of $C^*$-algebras, but I have troubles finding it. It is also possible, that this theorem is consequence of some theorem I've ...
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1answer
27 views

Induced character and center of groups

If $Z$ is a subgroup of the center of G and $|G:Z|=m$, then $\chi^*(g)=m\chi(g)$ if $g\in Z$ where $\chi$ is a character of $Z$ and $\chi^*$ is the induced character of $G$. Let $\phi$ be the ...
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1answer
24 views

Distinction between algebra homomorphisms and $A$-module homomorphisms

I am getting quite confused about the distinction between algebra homomorphisms and $A$-module homomorphisms, where $A$ is an algebra. If $A=\mathbb CG$, the group algebra, then I have a result in my ...
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1answer
31 views

Decomposition of standard Borel subgroup.

I am reading the lecture notes. On page 3, formulas (1.7), (1.8), (1.9), let $P$ be the standard Borel subgroup, we have $P=MN$. Why $M, N$ must be (1.8), (1.9)? I know that $P$ should be a upper ...
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64 views

Group theory argument

I'm reading Group Theory in Physics by Wu-Ki Tung and on page 69 in the proof of Theorem 5.3 he makes a group theory statement that I don't get. Let me try give some notation and explanation ...
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28 views

Representation theory and point groups

Hello everyone :) I have a doubt. I have the point group $C_{3v}$, which is the group $$C_{3v}= \lbrace e, C_{3}, C_{3}^{2}, \sigma_{v_{1}}, \sigma_{v_{2}}, \sigma_{v_{3}} \rbrace$$ $C_{3}$ and ...
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31 views

What values can a character $\psi$ take on an element of order $2$?

If $\psi$ is the character of a degree $2$ complex representation $\varphi\colon G\to GL_2(\mathbb{C})$, and $x\in G$ has order $2$, then $\psi(x)=0,\pm 2$. I noticed this by seeing $\varphi(x)$ ...
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1answer
32 views

Product of characters in representation theory

Is it true that if $\phi_1$ and $\phi_2$ are characters then $\phi_1\phi_2$ is a character of a representation? I think this is true, say for instance if $R_1$ is a matrix representation with ...
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2answers
33 views

Does $\chi(g^{-1})=\overline{\chi (g)}$ hold for infinite groups

Let $\chi$ be the character of some representation $\rho:G \to GL(M)$ over $\mathbb C$. Suppose $G$ is a group, then $\forall g \in G$ of finite order $n$, $ \chi(g^{-1})=\overline{\chi (g)}$ ...
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1answer
34 views

Real regular representation of cyclic group

I am looking for help to answer the following questions: What are the irreducible real representions $ρ: C_n → GL(V ) $ of a cyclic group of order n? How does the real regular representation $RC_n$ ...
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24 views

Occurrences of trivial representation is equal to dimension of $\{v\in V:\varphi(g)v=v\}$.

Suppose $\varphi\colon G\to GL(V)$ is a complex representation with character $\psi$. If $W=\{v\in V:\varphi(g)v=v,\ \forall g\in G\}$, why is $\dim W=(\psi,\chi_1)$, where $\chi_1$ is the principal ...
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24 views

Infinitely many direct sum decompositions of $M$ into direct sum of irreducible $\mathbb{C}G$-modules?

I teaching myself character theory, but I don't understand a problem statement from Dummit and Foote, Exercise 18.3.4. Prove that if $N$ is any irreducible $\mathbb{C}G$-module, and $M=N\oplus N$, ...