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)

2
votes
0answers
47 views

notation question $Aut_{Modk}(A)$

What is $Aut_{Modk}(A)$ if A is a finite generated algebra, k is a field? Is it a group of automorphisms of A as a vector space over k or what?
1
vote
0answers
19 views

Etingof problem 2.15.1 Representations of sl(2)

I'm studying from Etingof's Introduction to Representation Theory. This is problem 2.15.1, part a. I feel I'm close to the solution. Here's what I have. Problem: A representation of sl(2) is a vector ...
1
vote
0answers
19 views

A question and a conjecture on $USp(N)$ group

$USp(N)$ with $N$ an even integer is defined as the group of unitary matrices $M$ that satisfy $M^TJM=J$, where $M^T$ is the transpose of $M$ and $J$ is the anti-symmetric $N$-by-$N$ matrix ...
1
vote
0answers
10 views

Relation between induced and coinduced spaces

Let $G$ be a compact Lie group and $H$ a closed subgroup of it. Let $X$ be a $G-$space. The induced $G-$space is defined to be $$G\times_H X$$ with the equivalence $(gh, x)=(g, hx)$, for any $g\in G, ...
1
vote
0answers
44 views

$ \mathbb{C} $-characters of $ A_5 $

How can we find all of five irreducible $ \mathbb{C} $-characters of $ A_5 $? Precisely how can we construct the character table with the aid of orthogonality relations?
1
vote
0answers
47 views

Multiplicity one theorem for GL(n) and SL(n)

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) ...
1
vote
0answers
15 views

Why do we need the Dynkin Basis to compute Branching Rules?

Given a representation $R$ of some Lie algbra $g$, we can compute the corresponding representation $R'$ (in general reducible) for some subgroup Lie algebra $ g \supset g'$ by utilizing the weights in ...
1
vote
0answers
23 views

Why $\int_{U} d(m n m^{-1}) = \int_{mUm^{-1}} dn = \omega^2_E(t_1) \int_U dn$?

I am reading the lecture notes. On page 5, formula (1.24) is $$ \int_{U} d(m n m^{-1}) = \int_{mUm^{-1}} dn = \omega^2_E(t_1) \int_U dn, $$ where $dn$ is the Haar measure on $N$, $U \subset N$ is ...
1
vote
0answers
31 views

What values can a character $\psi$ take on an element of order $2$?

If $\psi$ is the character of a degree $2$ complex representation $\varphi\colon G\to GL_2(\mathbb{C})$, and $x\in G$ has order $2$, then $\psi(x)=0,\pm 2$. I noticed this by seeing $\varphi(x)$ ...
1
vote
0answers
24 views

Infinitely many direct sum decompositions of $M$ into direct sum of irreducible $\mathbb{C}G$-modules?

I teaching myself character theory, but I don't understand a problem statement from Dummit and Foote, Exercise 18.3.4. Prove that if $N$ is any irreducible $\mathbb{C}G$-module, and $M=N\oplus N$, ...
1
vote
0answers
42 views

Irreducible unitary representations of $ \Bbb{R}^{2} \rtimes_{\alpha} \Bbb{R} $.

Let $ \alpha $ be the action of $ \Bbb{R} $ on the group $ \Bbb{R}^{2} $ defined by $ \alpha_{t} \! \left( \begin{bmatrix} a \\ b \end{bmatrix} \right) = \exp \! \left( \begin{bmatrix} t & 0 \\ ...
1
vote
0answers
23 views

Irreducible unitary representation of a solvable lie group

Determine the equivalence classes of irreducible unitary representations of a solvable lie group. $$\begin{bmatrix}ae^t & 0\\ 0 & be^{-t}\end{bmatrix}$$ for $a,b\in \mathbb{R}, t\in ...
1
vote
0answers
25 views

How does $\mathrm{Res}_{\{I\}}\chi = \mathbb{I} \oplus \dots \oplus \mathbb{I}$?

My definition of restriction is: Let $H < G$, $\rho: G \rightarrow GL(V)$. The restriction of $\rho$ to $H$, $\rho: H \rightarrow GL(V)$. Its character is $$Res_H\chi(h)=\chi(h) \ \ \forall h \in ...
1
vote
0answers
17 views

Does every Young diagram have a unique minimal major index?

Given a Young diagram, $Y_\rho$, corresponding to an irreducible complex representation $\rho$ of the symmetric group $S_n$, we can associate a set of major indices $\{ ...
1
vote
0answers
21 views

Complex irreducible representation of solvable lie algebra

How can one infer from the Lie's theorem (in terms of existence of a common eigenvector) that a complex irreducible representation of a solvable lie algebra has dimension 1? What I know is that one ...
1
vote
0answers
27 views

How can I find the Weights of a Subalgebra

I'm currently trying to understand how we can derive the weights of a subalgebra of a given representation of a Lie group. For example, if we start with the 16-dimensional representation of ...
1
vote
0answers
15 views

Completing the character table

Let $G=\{a,b|a^6=1,a^3=b^2,b^{-1}ab=a^{-1}\}$ be a group of order 12. $G$ has 6 conjugacy classes $$\{1\},\{a^3\},\{a,a^{-1}\},\{a^2,a^{-2}\},\{b,a^2b,a^4b\},\{ab,a^3b,a^5b\}.$$ Name them ...
1
vote
0answers
26 views

Natural representation of GL(V)

Let $V$ be a vector space over some field. Is the natural representation $V$ of the group $GL(V)$ irreducible? Is it absolutely irreducible? Is the span of $GL(V)$ inside $End(V)$ all of $End(V)$? I ...
1
vote
0answers
19 views

Complete the character table

Let $G$ be a group of order 10 having 4 conjugacy classes and the above character table. Complete the table. It's easy to get the degrees of the remaining 2 irreducible characters to ...
1
vote
0answers
28 views

Decomposition of the Regular Q8 Module

For a worksheet we were asked to find the decomposition of the regular $Q_8$ module into a direct sum of simple modules. This isn't me asking for help on homework though, the problem is that I already ...
1
vote
0answers
61 views

Show that the action is transitive

$G$ is a finite group with a subgroup $H$. Let $\rho_1:G \to GL(V)$ and $\rho_2:H \to GL(U)$ be irreducible representations. $Z=\mathbb{C}[G]^H$, i.e., $Z$ is the centralizer of $H$ in ...
1
vote
0answers
46 views

hard lemma from a paper

I was reading a paper, there its mentioned as lemma that : Let $G$ be a finite group and $H$ be a subgroup of $G$. Let $\phi_1 : G\to GL(V)$ is a irreducible representation of $G$ and $\phi_2 : H \to ...
1
vote
0answers
21 views

Convolution product in Borel-Moore homology

I have a question about Exampla 2.7.10 from the book "Representation theory and complex geometry" by N. Chriss and V. Ginzburg. It concerns the convolution product. In the example we have $M_1 = M_2 ...
1
vote
0answers
39 views

Understanding restriction in $S_3$

Let $G=S_3$. \begin{array}{|c|c|c|} \hline & e & (123) & (12) \\ \hline \chi_0 & 1 &1 & 1\\ \hline \chi_1 & 1 & 1 & -1 \\ \hline \chi_2 & 2 & -1 & 0 \\ ...
1
vote
0answers
37 views

Counterexample to exactness of functor from group representations to fixed points

I recently asked this question. Now, the answer there claimed that the functor $()^G:Rep_G\to Vect_{\mathbb{C}}$, where $Rep_G$ are complex representations of a group $G$, and $V^G=\{v\in V: ...
1
vote
0answers
10 views

Inverting the the decomposition of tensor product representation into irreps

Suppose I have two unitary representations $U_V, U_W$ of a group $G$ on finite-dimensional vector spaces $V$ and $W$. I know that the tensor product representation $U_V\otimes U_W$ need not be ...
1
vote
0answers
69 views

Why Jacobson, but not the left (right) maximals individually?

When we are working with Path Algebras, it does not need very sophisticated tools to prove that for a finite, connected, acyclic quiver $Q$, the Jacobson Radical of $KQ$ is nothing but the arrow ...
1
vote
0answers
45 views

Considering a permutation representation of a transitive $G$-set

Suppose $X$ is a transitive $G$-set, where the size is greater than $1$, and $\pi=\pi_X$ the associated permutation representation. What is its character $\chi$? I thought that the permutation ...
1
vote
0answers
17 views

Questions about the indivisible imaginary root in affine root system.

I am reading the paper. On page 5, $\delta$ is defined as the indivisible imaginary root in $\widehat{\Delta_+}$. $\Lambda_0 \in \widehat{\mathfrak{h}^*}$ is the unique element satisfying $\langle K, ...
1
vote
0answers
21 views

Why is 1/2+1/2 in the weight space for SO(5)

Let's consider $\mathfrak{so}(5)$ as the Lie algebra of $\mathrm{SO}(5)$, where the symmetric bilinear form is $x_1y_5+\cdots +y_1x_5$. Then the maximal torus is given by $$\left(\begin{array}{cccccc} ...
1
vote
0answers
20 views

What are the differences between the three editions of the book “The Structure of Compact Groups”?

meta pre-clarification: I looked into another question like this but the guy didn't mark any specific tags for this type of question. Here's a link to the amazon book: ...
1
vote
0answers
44 views

Artin Algebra Representation Chapter Resource Request

I am working through chapter 10 of Artin's Algebra 2ed which introduces Group Representations. However, I've found that the approach to introducing groups is unlike the usual method used by more ...
1
vote
0answers
29 views

How to rigorously show tensor identities using symmetry arguments?

I am wondering how to rigorously show tensor identities such as the following. Let $n$ denote the radial unit vector in $D$ dimensions. Then $\langle n_i n_j \rangle = \frac 1 D \delta_{ij}$ and ...
1
vote
0answers
30 views

Decomposing some representations as a direct sum of irreducibles

I'm taking $V$ to be the standard representation of $S^3$. I'm looking for the decomposition of the following representations as a direct sum of irreducibles. (a) $V\bigotimes V$ (b) $Sym^2$ $V$ (c) ...
1
vote
0answers
70 views

Inner product doesn't matter for Schur orthogonality?

I'm reading Knapp's Basic Algebra, specifically the section about Schur orthogonality relations. Given a representation $R: G \to \text{End}(V)$, he defines $V_R$ to be the vector space of matrix ...
1
vote
0answers
32 views

$\overline\pi$ vs. $\check\pi$

Let $\pi$ be the automorphic representation of ${\rm GL}_2$ or $B^\times$ ($B$ an indefinite quaternion algebra over $\Bbb Q$) of central character $\varepsilon$ attached to some holomorphic newform ...
1
vote
0answers
29 views

problem book suggetion

I'm taking a course on basic representation theory, where upto midsem we're supposed to learn upto $5$ th chapter of serre. Can one please suggest some book which consists of nice problems ? I've ...
1
vote
0answers
50 views

Is there such a notion of “expansion” in groups?

Given a subset of elements of a finite group $G$, I would like it to be such that the set of all distinct words (as elements of $G$) that can be formed from this set is exponentially large in the size ...
1
vote
0answers
27 views

Filling in the last two rows of a character table?

I have a hopefully simple question about character tables. If I know all but two rows of a character table, and the character table is sufficiently large (say, at least 4 rows), do the orthonormality ...
1
vote
0answers
57 views

Simplifying a direct sum $\mathbf{3}\oplus\mathbf{3}\oplus\mathbf{2}$ etc

In particle physics, one often uses the dimensionality of the irrep to label the irrep (apparently this is not a very good idea since the dimension does not unambiguously determine the rep.). What are ...
1
vote
0answers
40 views

Relation between Poisson bracket and commutator.

In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b \in \mathbb{C}_q[T].$ It seems that we have $$ [a, b]=(q-1)\{a,b\}+o((q-1)^2). $$ ...
1
vote
0answers
30 views

What is the root structure of the Diffeomorphism Group?

Being a physicist, I think it'd be cool to have Coxeter plane projections of the root systems of the symmetry groups associated with the fundamental forces hanging on my walls (example for E8: ...
1
vote
0answers
25 views

Showing that a subrepresentation is isomorphic to the trivial representation

I'm considering $V$ to be the regular representation of a group $G$ and $W$ to be the 1-dimensional subspace of $V$ generated by the element $x=\sum_{s\in G} e_s$. I'm trying to show that $W$ is a ...
1
vote
0answers
22 views

When are all Gorenstein projective also pure-injective?

For an artin algebra of finite global dimension, each Gorenstein projective module is projective then is pure-injective. Are there any other examples having this property? That is, all Gorenstein ...
1
vote
0answers
51 views

Simple question on the unitary representation.

Let $\pi_1,\pi_2,\pi_3$ are all irreducible, unitary representation of some algebraic group G. Then is it ture that $$Hom_{G}(\pi_1,\pi_2 \otimes \pi_3) \simeq Hom_{G}(\pi_1 \otimes ...
1
vote
0answers
39 views

Irreducible Representations of Nilpotent Lie algebras

By Lie's theorem all irreducible representations of a solvable Lie algebra over $\mathbb C$ are one dimensional. What are all irreducible representations of a nilpotent Lie algebra ?
1
vote
0answers
101 views

Constructing irreducible representations of quaternion group over $\mathbb{Q}$

I am a beginner in studying the representation theory and I am doing some exercises in this field. So this is not a homework. My question is about constructing all irreducible representations of ...
1
vote
0answers
41 views

Matrix representations of free groups?

What is the general form of faithful matrix representations of free groups? How about for the simple case of $F_2$?
1
vote
0answers
29 views

Infinite dimensional representation such that every subrepresentation is reducible

Let $V$ be a nonzero finite dimensional representation of an algebra $A$. a) Show that it has an irreducible subrepresentation. b) Show by example that this does not always hold for ...
1
vote
0answers
45 views

Can we say that $A$ is a complement for a group $G$?

Let $A$ be a frobenius complement for a $G$ i.e. $A$ act on $G$ by automorphism s.t. $C_A(g)=e$ for all nonidentity $g$. Now, Action of $A$ can be linearly extended so that $A$ act on $F[G]$. As a ...