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

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Universality of restricted representations

Suppose that $H$ is a subgroup of a finite group $G$. Given an irreducible representation $\rho$ of $G$, this creates a (possibly reducible) representation $\rho'$ of $H$ obtained by restricting ...
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Subring of $\mathbb{C}[S_4]$

I'm doing a question from an old exam paper and I'm stuck on the following: Does $\mathbb{C}[S_4]$ contain a subring isomorphic to $M_2(\mathbb{C})$? Here subring doesnt need to contain the unit 1. I ...
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intertwining map in schur's Lemma

I am learning schur's Lemma from page 4 here. It says Schur's Lemma 1. If $(\rho_1, V_1)$ and $(\rho_2, V_2)$ are irreducible representations of a group $G$, then any nonzero homomorphism $\phi : ...
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18 views

When does a sequence of finitely generated $k[G]$ modules split?

I am self studying some non-commutative algebra, and I want to make sure I don't confuse myself. Here is what I am thinking: Let A and B be finitely generated $k[G]$-algebras, for $G$ a finite group ...
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Quotient braid group as a representation of SU(n)

I am working with the quotient braid group $B_3 (3) = B_3 / \langle\sigma_1 ^3\rangle$, where I construct a vector space $V$ so that every element $a \in B_3 (3)$ has a corresponding basis vector ...
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34 views

Group algebra is a tensor product?

Am I correct in describing the group algebra $R[G]$ as $R \otimes_{Z} G$? (As a tensor product of $Z$-algebras.) There is clearly a map $R \times G$ to $R[G]$, just by sending $(r,g)$ to $rg$, and ...
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Can the equality $e^{-tY}Me^{tY} = e^{tX}M $ be shown by showing it only to 1st order? (Lie representations)

We have that A and B belong to different representations of the same Lie group. The representations have the same dimension. X and Y are elements of the respective Lie algebra representations. $$A = ...
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34 views

Representing Groups as matrices

How to represent any group as group of matrices ? Like how to represent dihedral (4) group (order $8$) as group of $2$ by $2$ matrices ? How to represent direct product of $Z_2$ and $Z_2$ as a group ...
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32 views

Injection from the permutation representation of $S_4$ to $\uparrow^{S_4}_{S_2 \times S_2}$?

Let $V$ denote the permutation representation of $S_4$. I want to know if there is an injection $\alpha: V \rightarrow \space \uparrow^{S_4}_{S_2 \times S_2} 1$. My Answer: I don't think we can find ...
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A question about character of symplectic group

Let $V$ be a vector space with symplectic two form $\Omega$. Then the character $\chi:Sp(V,\Omega)\to U(1)$ must be trivial?
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$\mathbb{C} [G] \longrightarrow \prod_{\rho} \text{End}(V_{\rho})$ an intertwining isomorphism

Consider the vector space $\mathbb C[G]$ of functions $f: G \longrightarrow \mathbb{C}$ where $G$ is a finite group, or equivalently a vector space of all formal linear combinations of elements of $G$ ...
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38 views

Conjugacy classes and centralizers of a SmallGroup

What is the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)? Would someone be willing to provide the complete lists of conjugacy classes and centralizers of ...
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1answer
60 views

Which non-Abelian finite groups contain the two specific centralizers? - part II

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other ...
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50 views

Which finite groups contain the two specific centralizers?

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers both of these two groups: i. the elementary group $Z_2^4$, ...
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11 views

About representations and transformations under an $SU(n)$ Lie Group

I think my problem is that I misunderstand what "transforms under" really means. Let's take $SU(3)$, for the $\mathbf{3}$ with Dynkin indices $(1,0)$, a state transforms like : $ψ→gψ$. For the ...
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1answer
24 views

Obtaining representations of $G$ from $\mathrm{Lie}(G)$.

Suppose $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$, and $\tilde{G}$ is the unique connected, simply connected Lie group whose Lie algebra is $\mathfrak{g}$. Let $C$ be any discrete ...
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A construction of $\mathfrak{e}_8$ in Fulton and Harris

In section $22.4$ of "Representation Theory: A First Course" by Fulton and Harris, the exceptional Lie algebra $\mathfrak{e}_8$ is constructed using a method of Freudenthal. For background, I will ...
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What are the finite groups with 8 or 16 conjugacy classes?

What are the list of finite groups with 8 or 16 conjugacy classes? I learned that dihedral groups $D_{10}$ and $D_{13}$ have 8 conjugacy classes. (Here the order of these groups are $|D_{10}|=20$, ...
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Lie Algebra of Reduced Heisenberg Group Identities

I am having problems trying to understand a statement by Howe in his paper "On the role of the Heisenberg group in harmonic analysis". Here is the setting: Howe defined the (reduced) Heisenber group ...
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24 views

Conjugacy classes of right cosets

Is it true that all elements of a right coset $Hx$, for a subgroup $H$ of $G$, contained in a unique $G$-conjugacy class? I mean if $Hx=\lbrace{x_1,...,x_s}\rbrace$, then is it true that ...
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43 views

what is the conjugate of irreducible character of $G\wr S_n $?

Assume $G$ is any finite group and field as a complex field. The index set of irreducible representations of $G\wr S_n$ is set of all $k$-tuble of partitions ...
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19 views

Showing that a subrepresentation generated by an element is actually a subrepresentation.

Let $G$ be a group and $V$ be a representation of $G$. For $v_0 \in V$, the subrepresentation of $V$ generated by $v_0$ is constructed as $\{g \cdot v_0 | g \in G\}$. However, I don't immediately ...
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Schur-Weyl duality from Double Commutant Theory

Let $V$ be a finite dim complex vector space. Then $V^{\otimes n}$ carries an action by $S_n$ by permuting factors $\sigma(\pi)(v_1\otimes...\otimes v_n)=v_{\pi^{-1}(1)}\otimes...\otimes ...
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Why the adjoint representation of an SU(2) is a real representation?

And also why the fundamental is pseudo-real ?
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On an invertible element for equivariant K-theory

Fix a positive integer $m$. Let $G = \lbrace h\in\mathbb C | h^m = 1\rbrace$ and $(X,\pi)$ the standard representation of $G$. Namely $X = \mathbb C$ and $\pi:G \to GL(X)$ is defined by $\pi(h)v=h v$ ...
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S-modules and Schur functors

I am reading the book "Algebraic Operads" by Loday and Vallette. (I will refer to their version 0.999 here : http://math.unice.fr/~brunov/Operads.pdf) In Chapter 5, they define an $\mathbb{S}$-module ...
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31 views

Finding the dimension of $Alt^2(V)$ and $Sym^2 (V)$, given that $V = \mathbb{C}^2$.

The question is quite clear, I think. I know that if I can count the basis elements, then I am done. Here is the information I was given about these two spaces: $Sym^2(V) = < a \otimes b + b ...
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30 views

Prove that $\chi_{V_1 \otimes V_2} (g) = \chi_{V_1} (g) \cdot \chi_{V_2} (g).$

Here, $\chi$ is the character of the sub-representation, i.e., Given $\rho : G \to GL(V)$ is a representation, then the function $\chi_{\rho}: G \to \mathbb{C}: \chi_{\rho}(g) \to Tr(\rho_g)$. I ...
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Questions about root operators.

I am reading the notes. On line 13 in the section Root operators, it is said that The operator $f_1$ maps from the space $V(\mu)$ to $V(\mu-(1,-1,0))$. I don't know why. We have $$ f_i V (\mu) ...
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~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
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Jucys-Murphy elements confusion

I am taking a class called "Harmonic Analysis on Finite Groups" and am studying for an exam. We have recently been talking about the representation theory of the symmetric group (over $\mathbb C$). ...
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Learning representation theory of real reductive lie groups

I am interested in any sources that can be helpful for learning the representation theory of real reductive groups. I am currently reading Wallach book, but I feel that I don't understand the subject ...
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Unitarily equivalent $C^*$-algebra representations

the situation i want to talk about is the following: $(H_1,\varphi_1),(H_2,\varphi_2)$ irreducible representation of a $C^*$-algebra $A$. A bounded operator $T:H_1\rightarrow H_2$ such that ...
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How to prove that $\zeta*\zeta=\zeta$?

Let $F$ be a non-archimedean local field and $\mathcal{O}_F$ the ring of integers in $F$. Let $G_F=GL_2(F)$. Let $\pi_i$, $i=1,\ldots,n$,be non-equivalent finite dimensional irreducible ...
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Isaacs exercise 10.1 (Character Theory of Finite groups)

I need help on this problem. (10.1) Let $H \le G$, $\theta \in \operatorname{Irr}(H)$ and $\chi \in \operatorname{Irr} (G)$. Suppose $F \subseteq \mathbb{C}$. (a) If $\chi_H = \theta$, show that ...
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What do diagonal matrices do in irreducible repns of SL$_2(\mathbb{Z}/N\mathbb{Z})$?

Let $N \in \mathbb{N}, \mathbb{Z}_N := \mathbb{Z}/N\mathbb{Z}$. For every $a \in \mathbb{Z}_N^\times$ put $R_a = \begin{pmatrix} a^{-1} & 0 \\ 0 & a\end{pmatrix}$ and also set $T = ...
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The natural representation of $SO(n)$ is irreducible for $n\ge 3$

The natural representation $(\pi,\mathbb C^n)$ of $SO(n)$ is the one for which $$\pi (g)z = g^{-1}z$$ for $g\in SO(n)$ and $z \in \mathbb C^n$ (the product $g^{-1}z$ is just the usual matrix ...
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Powers of traces, integrals over spheres and class functions

Let $V$ be a complex vector space of dimension $\operatorname{dim}_{\mathbb C} V = n$, equipped with a Hermitian inner product $\langle \,\cdot\,,\,\cdot\, \rangle$. Let also $A$ be an endomorphism of ...
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70 views

What are the consequences of presentation of an algebra by generators and relations?

Let $A$ be a finite dimensional associative $K$-algebra, where $K$ is a field. I wonder how the presentation of $ A $ by generators and relations helps in the study of structure of the algebra ...
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67 views

Why does the tensor product of an irreducible representation with the sign representation yield another irreducible representation?

I was writing this question, and I came up with an answer, so I thought I would answer it myself: In considering representations of $S_n$, among others, we have the "sign representation", that is the ...
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81 views

Character Tables of $D_{4}$ and $Q_{8}$

Is there an intuitive reason that the Quaternion group and the Dihedral group on four vertices have the same character table? Does this indicate something special about the two groups? Or is it more ...
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Characters and conjugacy classes [duplicate]

This comes up in reading David Speyer's answer to this question. Given a finite group $G$ and two non-conjugate elements $x, y,$ how does one construct a unitary representation $\rho$ of $G$ such that ...
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23 views

Summing the traces of matrix powers

Let $G=\langle h\rangle_n\subset{\rm GL}(m,\mathbb{C})$ be a cyclic group of order $n$. I wonder if there is a good formula for calculating the sum $\sum_{g\in G}{\rm Tr}(g)$ via ${\rm Tr}(h)$, for ...
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Representation theory and particle physics

Are there good books which explain clearly explain the connections between modern particle physics and representation theory of groups and lie algebras?
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Linear Representations: Show that no $W^0$ exists.

Given the following linear representation and subrepresentation $W$, show that there exists no $W^0$ such that $\mathbb{R}^2 = W \oplus W^0$. Let $\rho: (\mathbb{Z}, +) \to GL(\mathbb{R}^2)$ be ...
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Uniqueness of decomposition of $\mathfrak{sl}(2,\mathbb{C})$-modules

By Weyl's Theorem, I know that every $\mathfrak{sl}(2,\mathbb{C})$-module is completely reducible. I'm under the impression that, up to isomorphism, this decomposition is unique, and I would go about ...
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80 views

Sums of products of average character values on cosets

Consider a finite group $G$, a subgroup $H\leq G$, and a transversal $G/H = \{t_1H, t_2H,\ldots,t_rH\}$. Given three characters $\chi_1,\chi_2$ and $\chi_3$ of $G$, I'd like to compute: $$ \sum_{i}^r ...
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Composition of Irreducible Representation and Surjective Homomorphism

Let $\varphi:G\to H$ be a epimorphism and let $\psi:H\to GL(V)$ be an irreducible representation. We wish to show that $\psi\circ\varphi$ is an irreducible representation of $G$. I have started this ...
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Questions about the proof of generalized Poisson summation formula.

The generalized Poisson summation formula is $$ \sum_{\gamma \in \Gamma} f(\gamma) = \sum_{ \pi \in \widehat{\Gamma \backslash G}} \hat{f}(\pi), $$ where $G$ is a locally compact Abelian group, ...
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40 views

Why a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$?

It is said that a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$. Why a modular form is a highest weight vector of a ...