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

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Why does complex conjugation permute the rows (columns) of a character table

If $\chi$ is the character of $\rho$, then $\overline{\chi}$ is the character of $\rho^*$ (dual) and $\chi_{irreducible} \iff \overline{\chi_{irreducible}}$. This implies complex conjugation ...
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11 views

Decomposition of axial vector and vector representions of C$_{4v}$ group

Let $R$ be the orthogonal matrix corresponding to an operation in $O(3)$. If R is a proper rotation, then both vectors $\vec{V}$ and axial vectors $\vec{A}$ are transformed in the same way $$ \vec{V} ...
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Subrepresentations of $\mathbb{I} \oplus \xi$

$G=C_2=\{e,h \}$. $\mathbb{I}$ is the trivial representation and $\xi$ is the sign representation. Let us consider $\mathbb{I} \oplus \xi$ where $e \mapsto \begin{pmatrix} 1 & 0\\ 0 & 1 ...
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Find the character of $\mathbb{C}[S_4/D_8]$

Find the character of $\mathbb{C}[S_4/D_8]$. I am assuming with this question that the first step will be to compute the (left) cosets $\{ gD_8: g \in S_4 \}$. Then I'm assuming that then it will ...
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1answer
34 views

Definitions of $\mathrm{Hom}(V,W)$

I have the definition of a homomorphism as map such that $\varphi(g_1g_2)=\varphi(g_1)\varphi(g_2)$ I have the definition of $\mathrm{Hom}(V,W)$ as $$\begin{align}\mathrm{Hom}(V,W) &= ...
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Etingof problem 2.16.2: Irreps of Two-dimensional Lie algebra over a field of positive characteristic

This is problem 2.16.2 in Etingof's introduction to representation theory. Note that problem 2.16.1 is a proof of Lie's theorem. I'm having trouble with the second case, where the base field has ...
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15 views

Dimension of the restricted representation

My definition of restriction is: Let $H < G$, $\rho: G \rightarrow GL(V)$. The restriction of $\rho$ to $H$, $\rho: H \rightarrow GL(V)$. Its character is $$Res_H\chi(h)=\chi(h) \ \ \forall h \in ...
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23 views

Invertibility of character table

Corollary. The character table of a group is an invertible square matrix. The theorem that is a corollary to states that the character table is a square matrix and the explanation for invertibility ...
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30 views

How to get real irreducible matrix representations from the complex irreducible matrix representations?

I'm trying to get real symmetry adapted orbitals for molecules with icosahedric symmetry (point groups $I$ and $I_h$) using the complete projector operator (truly projector if i=j): \begin{equation} ...
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23 views

Show that a representation of a finite group is isomorphic to its dual if its character takes only real values

This appeared as a part of showing that a representation of a finite group is isomorphic to its dual if and only if its character takes only real values. The "only if" part was easy to show. For the ...
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1answer
30 views

Verify regular representation?

Let $G$ be a finite group and let $V$ be the vector space of functions from $G$ to $\mathbb{C}$. For $g \in G$ and $f \in V$, let $R(g)(f)$ be the function $$(R(g)f)(x) = f(xg^{-1}).$$ How can I show ...
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34 views

A Question About Group Representation

I’m studying the representations of finite groups.We all know that the group representations are very important tool in the study of finite groups by allowing many group theoretic problems to be ...
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7 views

How can Clebsch-Gordan Decompositions be combined?

In section 4 of this paper the authors use a given list of Clebsch-Gordan coefficents for the $27 \otimes 27$ of $E_6$ from an old paper and combine it with their own list of Clebsch-Gordan ...
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17 views

A Lie Algebra $L$ is reductive iff it is completely reducibile as an $\operatorname{ad}_L(L)$-module

Given a Lie Algebra $L$ we say it is reductive if $\operatorname{Rad}L=Z(L)$. How can we prove that $L$ is reductive iff it is an $\operatorname{ad}_L(L)$-module completely reducibile? Suppose $L$ ...
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1answer
19 views

Identification of the Lie algebra of an isotropy group with the tangent space - stuck with a statement

I think I am stuck with the following statement that I read on the Encyclopedia of Mathematics website regarding Isotropy representations: "If $G$ is a Lie group acting smoothly and transitively on ...
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1answer
33 views

Etingof problem 2.15.1 Representations of sl(2)

I'm studying from Etingof's Introduction to Representation Theory. This is problem 2.15.1, part a. I feel I'm close to the solution. Here's what I have. Problem: A representation of sl(2) is a vector ...
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1answer
40 views

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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1answer
51 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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44 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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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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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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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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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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39 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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1answer
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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64 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} $ [closed]

Is there any reference where I can find the $ \mathbb{C} $-character table of the dihedral group $ D_{14} $?
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50 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
53 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
72 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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17 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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1answer
22 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
26 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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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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1answer
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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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1answer
28 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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1answer
48 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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1answer
42 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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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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34 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
39 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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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
25 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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1answer
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
70 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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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
47 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 ...