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

learn more… | top users | synonyms (1)

1
vote
0answers
32 views

How to tackle a research journal - level course in Lie Theory and Representation Theory?

I am taking a course in Lie Theory and Theory of Representations this year, where starting from the second lecture, Lie Theory is heavily bundled with Theory of Representations. It is pretty much a ...
5
votes
1answer
54 views

Base for symmetric group

Given symmetric group $S_n$, is it possible to find $k=\lceil\log_2S_n\rceil=\lceil\log_2n!\rceil$ members $\{\alpha_i\}_{i=1}^{k}$ in $S_n$ such that every member of $S_n$ can be written as ...
1
vote
0answers
34 views

Character of a Representation

Suppose that we have a representation $V$ of a group $G= SU(2)$ . Is it true that if $ \chi_V \cdot c \neq 0$ for some non-zero constant c, then the trivial representation must be one of the ...
5
votes
0answers
125 views

Is the left regular representation of an algebra, always faithful?

Let $\mathcal{A}$ be a unital associative algebra with a countable basis $\mathcal{b}$ over $\mathbb{C}$. Let $H=l^2(b)$ be the Hilbert space generated by $\mathcal{b}$. Let $H_0 = \{v \in H \ \vert \ ...
0
votes
0answers
30 views

Root Systems and Dynkin diagrams.

On page 142, the textbook An Introduction to Lie Groups and Lie Algebras (by Kirillov) determines the fundamental group of the root system $A_2$. Basically, the author says we have two simple roots ...
0
votes
0answers
15 views

A compact group with a finite dimensional faithful representation [duplicate]

Theorem: If $G$ a compact group has a finite dimensional faithful representation $W$, then any irreducible representation $V$ is contained in $W(k,l) = W^{\otimes k} \otimes (W^*)^{\otimes l}$ for ...
5
votes
1answer
84 views

Construction of a triple cover of $A_6$ in “Finite Simple Groups” by Wilson

I am reading The Finite Simple Groups by Robert Wilson: see page 29. I want to understand a construction of triple cover of $A_6$. On section 2.7.3., I don't understand the second paragraph, which is ...
1
vote
0answers
33 views

Schur-Weyl Duality - references

I'm trying to understand the Schur-Weyl duality. Unfortunately the lecture notes I have don't describe it very detailed. Any good references?
2
votes
1answer
49 views

Multiplicity of a dual simple module in the dual module?

Let $A$ be a finite dimensional $k$ algebra. Let $S$ be a simple left $A$-module and $M$ be any left $A$ module. Then my first question is that is it true $$[M:S]=[M^*:S^*]$$ where ...
1
vote
1answer
22 views

Is a virtual vector bundle the same as a vectorial bundle?

What is a virtual vector bundle? Is a virtual vector bundle the same as a vectorial bundle? The current entry in nLab states the following: "In one class of models for K-theory – generalized ...
5
votes
1answer
20 views

Are the weights of an irreducible representation of a simple Lie algebra in a single Weyl orbit?

When we consider the weights of an irrep of a simple Lie algebra, are they always in a single orbit under the Weyl group of the Lie algebra, or do they form a set of disjoint orbits? If they form ...
0
votes
0answers
86 views

How to construct $\operatorname{End}(V_{\pi})$ using a representation $\pi$

Let $(\pi, V)$ be a representation of the group $G$. To make the setting as general as possible, I will not put any restrictions on $\pi, V$, and $G$ from the beginning. By the very definition, for ...
0
votes
0answers
13 views

unique part of induced representation.

In the book about representation by Harris and Fulton . It proofs a proposition(3.17), where H is a subgroup of G: It only proofs existence at glance, why is it unique?
4
votes
2answers
60 views

A question in representations theory

My question is about irreducible representations of groups over the field $\mathbb{Q}$. Let $G$ be a cyclic or an abelian group. I want to check that under what conditions we have a ...
1
vote
1answer
33 views

Eigenspace of finite abelian group

Let $\rho: G\to {\rm GL}_n(\mathbb{C})$ be faithfull representation of finite abelian group $G$ and $V$ is the eigenspace of some $g\in G$. Is it true that $V$ is also eigenspace for all $G$ (that ...
5
votes
4answers
298 views

Polynomials as module over symmetric polynomials

Consider ring of polynomials $\mathbb{k} [x_1, x_2, \ldots , x_n]$ as a ring over symmetric polynomials $\Lambda_{\mathbb{k}}$ Is $\mathbb{k} [x_1, x_2, \ldots , x_n]$ free $\Lambda_{\mathbb{k}}$ ...
5
votes
1answer
42 views

A problem similar to Maschke's theorem

My question is about a problem that its assumptions are like Maschke's theorem in some ways. Let $\mathbb{F}$ be a field that $char \mathbb{F}$ doesn't divide $|G:H|<\infty$ and $M$ be an ...
1
vote
1answer
30 views

The Weyl group of $\widehat{\mathfrak{sl}}_2$.

On page 5 of this paper, example 3.1, it is said that the Weyl group of $\widehat{\mathfrak{sl}}_2$ is $$ W= \langle s_1, s_2 \mid s_1^2 = s_2^2 = 1 \rangle. $$ Why the Weyl group of ...
1
vote
0answers
42 views

How to compute $Ext_A^{1}(S_1, S_2)$ and $Ext_A^{1}(S_2, S_1)$?

Let $A = kQ/\rho $, $Q$ is the quiver \begin{align} 1 \overset{a}{\underset{a^*}{\rightleftarrows}} 2 \end{align} $\rho$ is the relation $a a^* - a^* a = 0$. Question: compute $Ext_A^{1}(S_1, S_2)$. ...
1
vote
0answers
20 views

Unitary dual of $\mathrm{Sp}_4(\mathbb{R})$

We know the unitary dual of $GL_n(\mathbb{R})$, unitary dual of $SU(2,2)$, how about $\mathrm{Sp}_4(\mathbb{R})$? Is there any known result? If so, can anyone provide me any references? Thanks!
2
votes
1answer
31 views

Decomposing $\mathfrak{sl}_3(\mathbb{C})$

There is a pretty standard exercise on $\mathfrak{sl}_2 (\mathbb{C}$) representations that consists in decomposing the representation given by $\mathfrak{sl}_3(\mathbb{C})$ via $\operatorname{ad}$, ...
3
votes
1answer
96 views

Reference for a proof of a projective representation of $A_6$

I want to understand the proof of There is a projective representation of $A_6 \hookrightarrow PSU(3).$ I am looking for a reference, but could not find. Suggestions are welcome. EDIT: We ...
7
votes
0answers
156 views

Representation theory of the general linear group over a finite prime field

The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely classified and well-understood via Schur-Weyl duality, the algebraic Peter-Weyl theorem and the entire ...
2
votes
0answers
28 views

Normal basis theorem

Let $K$ be a finite Galois extension of, say, $\mathbb{Q}$. Then is known(and called normal basis theorem) that if i view $K$ as a representation of $Gal(K/\mathbb{Q})$ over $\mathbb{Q}$ it is ...
0
votes
0answers
12 views

Finding the 2D irreducible representation of quaternions $Q_8$ in the space of functions $f\colon Q_8 \rightarrow \mathbb{C}$

The space of functions $F=\{f\colon Q_8\rightarrow\mathbb{C} \}$ is 8 dimensional, since we can choose for each element of $Q_8$ an element in $\mathbb{C}$ to send it to. The action of $Q_8$ on this ...
1
vote
3answers
57 views

A question about the proof of a theorem in Representation theory of groups

My Question is about one part of the proof of theorem in the book "A Course in the Theory of Groups" by Derek J.S. Robinson. I highlight the part that my question is about. We know that if $G$ is a ...
2
votes
1answer
81 views

Tannaka reconstruction: reference request

What is a classical and perhaps even original reference for the following result, often called Tannaka reconstruction? Let $G$ be a group and $R$ be a commutative ring in which $0,1$ are the only ...
2
votes
1answer
23 views

Holonomy representation: is it actually a class of representations?

In D. Joyce's book "Riemannian Holonomy Groups and Calibrated Geometry" (2007) the author writes that if $\nabla$ is a connection on a vector bundle $E$ (over a connected base) with the fibre $\mathbb ...
3
votes
0answers
28 views

Cauchy Identity for a specialized product of Schur polynomials

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_d)$ be a partition, with $|\lambda|=n$. Let $\nu=\nu(\lambda):=(\lambda_1-1,\lambda_2,\cdots,\lambda_d).$ In other words, $\nu$ is obtained from ...
1
vote
0answers
19 views

Finite-dimensional, irreducible Representations of the Diffeomorphism Group $Diff(R^4)$

Is there any possible way to study the finite-dimensional, irreducible representations of $Diff(R^4)$ systematically? My interests stems from the fact, that the symmetry group of general relativity is ...
2
votes
0answers
23 views

Some questions on Langlands Classification of Irreducible Admissible Representation

I am trying to construct some representations using Langlands classification theorem. But I get confused and have some problems when constructing these representations..... i) In the classification ...
3
votes
0answers
16 views

Lusztig's $h$-function of a dihedral group

Following the notations in Hecke algebras with unequal parameters, let $(W,S,L)$ be a weighted Coxeter system, and $H$ be the corresponding Hecke algebra with $\{c_w |w \in W\}$ the Kazhdan-Lusztig ...
1
vote
0answers
40 views

Character as sum with regular representation

Suppose $G$ is a group and $\chi$ is a character of $G$ with $\chi(g_1)=\chi(g_2)$ for all non-identity $g_1,g_2 \in G$, and let $\chi_{reg}$ denote the regular representation character. I read that ...
2
votes
1answer
26 views

$Hom_G(\pi,\sigma)$ = $Hom_{\mathfrak{g}}(d\pi,d\sigma)$?

Let $G$ be a Lie group. Let $\mathfrak{g}$ be the corresponding Lie algebra. Let $(\pi,V)$ and $(\sigma, W)$ be representations of $G$, with corresponding differentials $d\pi$ and $d\sigma$, which are ...
0
votes
0answers
10 views

proof that Linear transformation is isometry

let ∑=set of all continous unitary representation and $ Ψ \in{ ∑}$ $π_Ψ: \frac{L^1(G)}{N_Ψ}→ B ( \oplus H_π) $ is definde by $$π_Ψ(f^0)=\oplus π(f) , π \in{ Ψ},f^0\in{\frac{L^1(G)}{N_Ψ} }$$ ...
1
vote
1answer
22 views

Existence of Irreducible Character s.t. $\chi(g) \neq 0, \chi(1) \neq 0 \text{ mod } |C(g)|$ for Elements in Conjugacy Class of Prime Order

Given a finite group $G$, and a non-identity representative $g$ in a conjugacy class of prime order $p$, I'm trying to show that some nontrivial irreducible character of $G$ must have $\chi(g) \neq 0$ ...
1
vote
1answer
32 views

Describing $GL(2,\mathbb{C})$ with generators and relations.

My question is : how can I describe $GL(2,\mathbb{C})$ with generators and relations ? I do not know how to start ? Thanks for your help in advance,
0
votes
1answer
18 views

Trivial representation from the row-shape Young diagram

For the Young diagram $\lambda$ which is the row with, say $d$ squares, i.e. $\lambda = (d)$, the corresponding Young symmetrizer is $c_\lambda = \sum\limits_{g\in\mathfrak S_d}g$ such that the ...
4
votes
1answer
126 views

How to prove that representations on $S^k(V), \bigwedge ^ k(V)$ are irreducible?

Given a $\mathbb{C}$ vector space $V$, let $GL(V)$ act on $\bigotimes^k(V)$ via: $GL(V) \times \bigotimes^kV \to \bigotimes^k(V), \ (A,v_1\otimes...\otimes v_k)\mapsto Av_1\otimes...\otimes ...
0
votes
0answers
11 views

Exercise 2.4.1 Automorphic Forms and Representations, Daniel Bump

I am working my way through D. Bump's Automorphic Forms and Representations. I was trying my hand at this problem. Problem: Let $K$ be a compact subgroup of $GL(n,\mathbb{C})$ and let $(\pi, H)$ be a ...
0
votes
0answers
20 views

Trivial representation in tensor square

Taken from another question in this website. I am not sure why the following statement is true. Suppose $G$ is a group and $V$ an irreducible representation of $G$. One has that ...
2
votes
1answer
44 views

Request a reference in group theory

Although the book "A Course in the Theory of Groups" by Derek J.S. Robinson is an excellent up-to-date introduction to the theory of groups and covers various branches of group theory, it is hard for ...
4
votes
1answer
74 views

Question about isotypical components

Consider $V=\bigotimes^3(\mathbb{C}^2)$ as a $\mathfrak{S}_3$ representation. One of its isotypical component is $S^3(\mathbb{C}^2)$, which is a linear subspace of symmetric tensors of ...
2
votes
0answers
92 views

Clifford algebra - Gamma matrices

Let's say we have $\gamma^{a}$ matrices $(a=1,2,...,D)$. They satisfy the following condition $$\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\delta^{ab}I^{N\times N}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ ...
2
votes
2answers
36 views

Semisimple modules and the radical

I don't need a proof, but can someone tell me whether it is true that for all $A$-modules $V$ we have that $V/\text{rad}V $ is semisimple, where we define $\text{rad} V$ as the intersection of all ...
5
votes
2answers
78 views

About integral binary quadratic forms fixed by $\operatorname{GL_2(\mathbb Z)}$ matrices of order $3$

I am reading this paper of Manjul Bhargava and Ariel Shnidman, and I want to prove this claim, which appear at the first paragraph of Theorem $14$: Up to $\operatorname{SL_2}(\mathbb Z)$ ...
3
votes
2answers
68 views

Quaternion–Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $ \mathbf{R}^3 $ by conjugation. I've recently encountered spinors and notice similarities in their definitions ...
1
vote
1answer
30 views

Induced representation

I'm doing the problem section of the induced representations chapter by Steinberg, and I'm having problems with the following one: Let $G$ be a group and $H$ subgroup. Given a representation ...
1
vote
1answer
37 views

Projective representaions of $(\mathbb{Z}/3\mathbb{Z})^2$

I have a very short question: is there a faithful projective representaion $\rho: \mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}\to {\rm PGL}(4,\mathbb R)$? Thanks!
1
vote
0answers
29 views

Adjoint representation of the isotropie group of a homogeneous space

I have difficulties seeing why is the following true: Let $G$ be a lie group and $H$ a closed subgroup, with $\tilde{g}$ and $\tilde{h}$ their lie algebras. The adjoint action of $g\in G$ is given by ...