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

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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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63 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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28 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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36 views

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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32 views

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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1answer
74 views

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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42 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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34 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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1answer
22 views

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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36 views

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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36 views

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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2answers
85 views

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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110 views

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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62 views

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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30 views

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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121 views

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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38 views

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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109 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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176 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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1answer
94 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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2answers
133 views

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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56 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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97 views

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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20 views

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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1answer
26 views

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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1answer
95 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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2answers
55 views

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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48 views

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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65 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 ...
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49 views

GAP-Character table

I the following link I have found the character table of $S_8$ which is computed with the program GAP. http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_symmetric_groups But I ...
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1answer
46 views

Number of degree-$d$ representations of a perfect group?

It seems to be a standard result that the number of degree-1 representations of a group $G$ is equal to $[G : G']$ where $G'$ is the commutator subgroup (e.g. Lemma 6.2.7 in the 2012 textbook ...
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213 views

Involutions, RSK and Young Tableaux

Let $S_n$ be the symmetric group on $n$ elements. The Robinson-Schensted-Knuth (RSK) correspondence sends a permutation $\pi\in S_n$ to a pair of Standard Young Tableaux $(P,Q)$ with equal shapes ...
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55 views

Integration over uncountable set of characters

Let $G$ be a compact (assumed Hausdorff) group and $\hat{G}$ be the set of all characters of irreducible, finite-dimensional representations of $G$. It might occur that $\hat{G}$ is uncountable. It ...
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1answer
167 views

Definition/existence/uniqueness of a minimal projective resolution

I'm reading Dave Benson's book "Representations and Cohomology," Volume I, and I'm trying to understand the following discussion on page $32$ in which he introduces the notion of a minimal projective ...
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28 views

Rational representation of a 136 order group

Let G be the group of order 136 = 8 * 17 with presentation $$<x,y : y^{8}=x^{17}=1 \quad yxy^{-1}=x^{4} >$$ Find the simple summands of the group ring Q[G].
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Question concerning Morita equivalence and an algebra over a field which is not algebraically closed

I would like to know, whether there are a quiver $Q$ and an admissible ideal $I$ such that the quiver algebra $\mathbb{F}_3Q/I$ and the group algebra $\mathbb{F}_3 (C_3\times C_3)$ are Morita ...
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579 views

The use of conjugacy class and centralizer?

This is more or less for a conceptual and better-understanding question in group theory and in representation theory: (1) Why are conjugacy class and centralizer important concepts in the group / ...
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1answer
55 views

Exponents of a semisimple Lie algebra

I'd like to compute the exponents of a semisimple complex Lie algebra $\mathfrak{g}$. According to http://math.berkeley.edu/~theojf/LieQuantumGroups.pdf proposition 8.1.2.18, this amounts to ...
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1answer
67 views

Building invariants of non-fundamental $SU(2)$

Suppose you have two objects, $ \phi _i $ and $ \psi _j $ that form representations of $ SU(2) $. With both fields in the fundamental representation, I believe there is one invariant, \begin{equation} ...
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32 views

Suppose p : G → GL(n, C) is a representation. Suppose that g, h exist in G and that p(g)p(h) = p(h)p(g). Is it then true that gh = hg?

Suppose $p : G → GL(n, C)$ is a representation. Suppose that $g, h$ exist in $G$ and that $p(g)p(h) = p(h)p(g)$. Is it then true that $gh = hg$? I don't know if I am not understanding the question, ...
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136 views

Invariants for the $SU(2)$ representation

The quantities $\delta_{ij}a_ib_j$ and $\epsilon_{ijk}a_ib_jc_k$ are invariant under the transformation of the $j=1$ (fundamental) representation of $SO(3)$. What would be the analogous expressions ...
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What is the Lie algebra of $G=\mathbb{R}$

The question is updated as following. 1. Let $(\Phi,L^2(R))$ be left regular representation of $\mathbb R$ given by $$ \Phi(g)f(x)=f(x-g). $$ It is unitary representation of $\mathbb R$. 2. For ...
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16 views

Combining infinitesimal generators of diferent dimensions

I am reading a paper about ways in which you can get $SU(2)\times{}U(1)\times{}U(1)$ as a subgroup of $SU(3)\times{}SU(2)\times{}U(1)$. At a certain point, it starts considering ways of getting ...
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1answer
61 views

Showing that an $\mathfrak{sl}(2,\mathbb{C})$-module is determined by eigenvalues of $h$

This question is essentially exercise 8.4 from the book "Introduction to Lie Algebras" by Erdmann and Wildon: "Suppose that $V$ is a finite-dimensional $\mathfrak{sl}(2,\mathbb{C})$-module. Show that ...
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73 views

Invariants under a transformation

Consider a $j=1,\,SU(2)$ representation (or fundamental $SO(3)$ representation). Suppose that $a_1, b_i, c_i$ with $i=1,2,3$ are vectors transforming under this representation i.e ...
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39 views

$U(1)$ generators of $SU(2)$

I wanna get $U(1)$ out of $SU(2)$. I know for example that this can be done using the diagonal Pauli matrix, but I wonder if there are more $U(1)$-s in $SU(2)$. So, which are the all the ways in ...
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1answer
42 views

invariants of a representation over a local ring from the residual representation

Let $(R, \mathfrak m)$ be a local ring (not necessarily an integral domain) and $T$ be a free $R$-module of finite rank $n\geq 2$. Let $\rho: G \to \mathrm{Aut}_{R\text{-linear}}(T)$ be a ...
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12 views

How to transform the following direct product of the group representations?

Let's have 4-vector $A_{\mu}$ which transforms as $\left(\frac{1}{2}, \frac{1}{2}\right)$ representation of the Lorentz group. So the product $A_{\mu}B_{\nu}$ refers to the direct product $$ \tag 1 ...
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2answers
143 views

Dummit and Foote on Galois and Representation Theory?

At some point, I'd like to learn both Galois Theory and Representation Theory. I currently know a lot of Group Theory and Linear Algebra, as well as some Ring Theory. I was thinking of reading ...