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

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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
32 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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31 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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33 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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1answer
23 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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1answer
39 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
51 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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46 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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70 views

Multiplicity one theorem for GL(n) and SL(n) [closed]

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) ...
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38 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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46 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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6 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
31 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
81 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
76 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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22 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
48 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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15 views

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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43 views

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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1answer
45 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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29 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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62 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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30 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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25 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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35 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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1answer
72 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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29 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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41 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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45 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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37 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
37 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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25 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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26 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 $M=N\oplus N$, ...
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52 views

Number field attached to a finite group.

Let $G$ be a finite group. I know that the set of irreducible representations of $G$ over the complex numbers (up to isomorphism) is finite. Let us fix our attention on some irreducible ...
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1answer
17 views

Proof that $f:\mathbb CG \to S$ given by $f(a)=a\cdot s$ is surjective

I am going through a proof of the following result from my lecture notes: Suppose $\mathbb C G \cong \bigoplus_{i=1}^nU_i$ where the $U_i$ are simple $\mathbb CG $ modules. Then any simple $\mathbb ...
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44 views

Irreducible unitary representations of $ \Bbb{R}^{2} \rtimes_{\alpha} \Bbb{R} $.

Let $ \alpha $ be the action of $ \Bbb{R} $ on the group $ \Bbb{R}^{2} $ defined by $ \alpha_{t} \! \left( \begin{bmatrix} a \\ b \end{bmatrix} \right) = \exp \! \left( \begin{bmatrix} t & 0 \\ ...
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27 views

Irreducible unitary representation of a solvable lie group

Determine the equivalence classes of irreducible unitary representations of a solvable lie group. $$\begin{bmatrix}ae^t & 0\\ 0 & be^{-t}\end{bmatrix}$$ for $a,b\in \mathbb{R}, t\in ...
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24 views

What is the representation induced by $Z(G)$?

If $G$ is a finite group, and we have a representation of the center $\varphi\colon Z(G)\to GL(V)$, what is the induced representation on $G$? I've seen it mentioned in places, but not its definition. ...
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1answer
25 views

Character of action of permutations on a subset

I have a problem involving characters of a certain $\mathbb C$$G$-module, where $G = S_n$. The module is the vector-space $V$ with basis $\{v(I) | I\subset\{1,...,n\},|I| = k\}$, where $k\le n$. With ...
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16 views

scalar multiple of Young symmetriser

The following is a lemma on Fulton and Harris' book -Representation theory,a first course (page 53): Lemma: For all $x\in \mathbb{C}\mathfrak{S}_r$, $c_{\lambda}\cdot x\cdot c_{\lambda}= scalar ...
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42 views

What's the meaning of $d^{\times } a$?

In the lecture notes, the last line of on page 51, what is the meaning of $d^{\times } a$ in the integral? Thank you very much.
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42 views

How does GAP understand $SL_2(\mathbb{F}_3)$?

When one is asking for " IrreducibleRepresentations(SL(2,3))" it makes sense that the matrices it returns are over the complex field as thats the field on which the representations are. There GAP ...
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24 views

Relation between Jordan Normal Form and Irreducible Matrix Representations.

Ok so I have learned some very basic things about groups and matrix representations of groups. I have learned that it can be possible to find a "minimal basis" or "irreducible basis" for which a ...
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36 views

Representation of complex Clifford algebra on exterior algebras when quadratic form has odd index

Overview This problem entails the explicit construction of representation of Clifford algebra upon the exterior algebra, using orthogonal complex structure or polarization, namely, given a ...
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36 views

Projection matrix (C* algebra.. but linear algebra question) [closed]

The subject is $C^*$-algebra, but I think my question might be linear algebra related type. I have a question from the book Operator Algebras Theory of C*-Algebra by Blackadar. On page 351, in the ...
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2answers
58 views

Existence of projection in proof of Maschke's theorem

In a proof of Maschke's theorem, my lecturer writes "If $N$ is a $\mathbb C G$-module and $N\leq M $ is a submodule, let $\pi: M \to M$ be a projection (i.e. a map with $\pi^2=\pi$) with image $N$." I ...
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40 views

Proving $Ind_H^G1_H=\pi_X$

Lemma Let $\psi=1_H$ the principal character of $H$, then $Ind_H^G1_H=\pi_X$, the permutation character of $G$ on the set $X$ of left cosets of $H$ in $G$. Proof Let $\{t_i \}$ form a tranversal. ...
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1answer
36 views

Does there exist a non-quasi-split torus?

In a homework, I was asked to prove that any torus is isomorphic to a quotient of a finitely many product of Weil restrictions $Res_{L/k}\mathbb{G}_m$. While solving this, I got an impression that ...
2
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1answer
52 views

Prove that if G has a faithful complex irreducible representation

I am struggling with the proof that if G has a faithful complex irreducible representation then $Z(G)$ is cyclic: Let $\rho:G \rightarrow GL(V)$ be a faithful complex irreducible representation. ...
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21 views

Definition of a matrix by using coaction.

Let $V$ be a vector space with a basis $e_1, \ldots, e_n$. Let $C$ be a coalgebra and $V$ a $C$-comodule. Consider the coaction $\delta: V \to C \otimes V$ given by $\delta(e_i) = \sum_{j=1}^n c_{ij} ...