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

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3
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88 views

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 ...
1
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1answer
167 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 $\widetilde{\lambda}=(\lambda_1,\lambda_2,\...
1
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1answer
36 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 ...
1
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0answers
76 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 v_{\pi^{-1}(...
2
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0answers
40 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

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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1answer
38 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
24 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) \...
4
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0answers
70 views

~ 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. ...
4
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0answers
96 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 $\mathbb C$). ...
2
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0answers
57 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 ...
2
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2answers
188 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 $T\...
5
votes
1answer
135 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 ...
2
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0answers
32 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 = \begin{...
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0answers
209 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 ...
2
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0answers
176 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 ...
4
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2answers
791 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 ...
5
votes
1answer
114 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 ...
5
votes
2answers
236 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 ...
0
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1answer
60 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 ...
3
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1answer
134 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?
0
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1answer
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 ...
2
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1answer
42 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 ...
5
votes
1answer
116 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
97 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 ...
3
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1answer
79 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, $\...
2
votes
1answer
154 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 ...
0
votes
1answer
101 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 don'...
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1answer
71 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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1answer
446 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 $\...
3
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0answers
82 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
471 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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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].
2
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1answer
69 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
2k 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
votes
1answer
228 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
votes
1answer
82 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, ...
1
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1answer
395 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 ...
2
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0answers
58 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 $G=...
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1answer
97 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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0answers
80 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 $a_i'=[\rho_1(\...
2
votes
1answer
63 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
283 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 ...
3
votes
2answers
152 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
113 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
1k 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 ...
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1answer
62 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 either$...
0
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1answer
37 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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52 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 $\mathbb{R}/\mathbb{...