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

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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 ...
3
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117 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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22 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
33 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
103 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
72 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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0answers
55 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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1answer
96 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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1answer
77 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
58 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 ...
18
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312 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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74 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 ...
2
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1answer
293 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 ...
2
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0answers
29 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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1answer
62 views

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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6answers
909 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 / ...
0
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1answer
124 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 ...
3
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1answer
74 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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1answer
34 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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1answer
203 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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56 views

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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1answer
77 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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77 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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1answer
52 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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2answers
193 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 ...
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2answers
122 views

Characters of a faithful irreducible Module for an element in the centre

Basically here is my questions. We have a character $\chi$ which is faithful and irreducible of a group $G$. we have an element $g$ which i needs to show belongs to the centre $Z(G)$, i.e. $gh=hg$ for ...
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0answers
84 views

Matrix coefficients of representations of finite groups

In finite-dimensional complex representations of finite groups, I would like to understand what I can learn by looking at a single matrix coefficient. In particular, I would like to look at "diagonal" ...
6
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1answer
471 views

How to show a representation is irreducible?

I have a professor who says that I should be able to show a representation is irreducible simply by looking at its trace (with other possible conditions), but after researching this for a while, I ...
1
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1answer
54 views

Given certain set of symmetries of a tensor, how do you associate the corresponding young tableaux

I have a particular problem, the following. $T^{a_1 \dots a_p;b_1 \dots b_p}$ is a tensor with the following symmetries. 1) $a_i$'s and $b_i$'s are completely antisymmetric, ie restricted to ...
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32 views

When would a finite group be cosidered as Fp G -Module?

What conditions are necessary to think of a finite group as Fp G -Module ?
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49 views

Is $\widehat{\mathbb{R}/\mathbb{Z}} = \mathbb{Z}$? [duplicate]

Let $\widehat{\mathbb{R}/\mathbb{Z}}$ be the set of all homomorphisms from $\mathbb{R}/\mathbb{Z}$ to $\mathbb{C}$. Is $\widehat{\mathbb{R}/\mathbb{Z}} = \mathbb{Z}$? I think that ...
4
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1answer
108 views

Recognition of positive integral projections in a group algebra

Let $G$ be any finite Group and $e \in \mathbb{C}G$ be a central idempotent element which decomposes $\mathbb{C}G = R \times S$ into a direct product of rings $R = \mathbb{C}Ge$ and $S = ...
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2answers
41 views

A question about Lie group homomorphisms

Suppose I have a Lie group $G$ and a Lie homomorphism $ \phi : G \rightarrow GL_n(\mathbb{R})$. Can $ \phi $ be viewed as some sort of representation of $G$? Can anyone make this rigorous for me ...
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2answers
75 views

Why do we want *unitary* representations of locally compact groups into $B(H)$?

This is related to a previous question of mine but I have a more philosophical issue with the material. Everywhere I have looked for representations of locally compact groups into $B(H)$, everyone ...
3
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1answer
107 views

Generalized Schur-Weyl Duality

Schur-Weyl duality relates representations of the symmetric group to representations of $GL(n)$. Is there a generalization to arbitrary reductive groups?
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1answer
128 views

Exercise from Etingof's notes on Representation Theory

I am reading through these notes of Etingof on Representation theory and I am stuck with one exercise (it's problem 4.69 in the notes). The problem is the following. Consider the space ...
3
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1answer
61 views

$\mathrm{GL}_n$-representation theory question or a Tale of Two Determinants

The irreducible representations of $\mathrm{GL}_n(\mathbb C)$ are indexed by partitions $\lambda$. These representations are denoted by $\mathbb S_{\lambda}(V)$, where $V$ is the standard ...
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1answer
156 views

Schur's Lemma and the Center of a Group

If I have a group $G$ and a complex irreducible representation $g:G\rightarrow GL_n(\mathbb{C})$. I am trying to use schur's lemma to show that for $x\in Z(G)$ we have that $g(z)=\lambda_z I_n$. Now ...
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1answer
29 views

Dimensions of the conjugacy classes of $S_3$ in $\Bbb{C}S_3$.

Since the conjugacy classes of $S_3$ are $\{1\}$, $\{(1 2), (1 3), (2 3)\}$, and $\{(1 2 3), (1 3 2)\}$, I would think that they have dimensions 1, 3, and 2; respctively (because they are the basis ...
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1answer
93 views

Unitary representations of locally compact topological groups

Known: In the representation theory of finite groups (on finite dimensional vector spaces of course), given a finite group $G$ and a representation $\rho$, we can construct an inner product $\langle ...
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1answer
82 views

An example of finding the irreducible representations of G over C

I have an example in my lecture notes but I really don't understand what it is doing at each step so I was wondering if someone could help me work through it so that I can then do the problem sheet ...
6
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Closed orbits of complete flags in $\mathbb{C}^n$

Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the ...
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102 views

Definition of a splitting field of a finite group

This is a basic question from the journal 'Mathematische Zeitschrift' 208 (1991) page 243. Let $K/F$ be a finite Galois extension of number fields and $G={\rm Gal}(K/F)$. Also let $L$ be any number ...
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Reciprocity for branching rules of $\mathrm{GL}_n(\mathbb C)$

[Separated from another question] If I have information about the restriction of representations of the general linear group, can I make any statements about the induction (by Frobenius reciprocity)? ...
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How do non-semisimple modules over $\mathbb C\mathrm{GL}_n(\mathbb C)$ look like?

[Separated from another question] Can you give an example of a non-semisimple module over $\mathbb C\mathrm{GL}_n(\mathbb C)$? (Preferably one without direct summands, i.e. an indecomposable module) ...
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1answer
49 views

Orthogonal invariants of an irredubile GL-representation

Let $n\in 2\mathbb Z$ be an even number. Let $G=\operatorname{GL}_n(\mathbb{C})$ and $V_\lambda$ the irreducible complex $G$-module corresponding to the partition ...
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1answer
282 views

Irreducible representations (over $\mathbb{C}$) of dihedral groups

Find number of complex irreps of the group $D_n$. Find dimension of the irreps. I know that The number of complex irreps of a finite group is equal to the number of conjugacy classes of the ...
2
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1answer
68 views

Sum of representations and characters of the symmetric group

Hi I was wondering if I could have some help to go in the right direction. I want to show that $\displaystyle\sum\limits_{\sigma \in S_n} (sgn(\sigma)*\chi(\sigma)) =0$ where $sgn(\sigma)$ : $S_n ...
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1answer
141 views

A little bit of Intuition for Corepresentations from Representations

Hi folks I am trying to prove what I think should be a straightforward enough result but I am having to make a somewhat unnatural definition to do it. This unnatural definition is hinted at in a paper ...
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3answers
561 views

Do all Groups have a representation?

I know that many kind of groups can be represented by matrices; for example: rotation groups can be represented by matrices. Especially all elements of rotation groups can be represented by ...