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)

0
votes
0answers
27 views

Basic Representation Theorey: Bijective Correspondence Between Representations (Dummit and Foote 18.1 #3)

I am working on the following question from Dummit and Foote: Prove that the degree 1 representations of $G$ are in bijective correspondence with the degree 1 representations of $G/G'$ (where $G'$ is ...
1
vote
2answers
32 views

Basic Representation Theory: One Dimensional Representation (Dummit and Foote 18.1 #2)

I'm working on question 2 in 18.1 of Dummit and Foote. The question states: Let $\phi : G \to GL_n(F)$ be a matrix representation. Prove that the map $g \to det(\phi(g))$ is a degree 1 ...
5
votes
2answers
88 views

Prove that $Q_8 \not < \text{GL}_2(\mathbb{R})$

Problem 18.1.10 in Dummit and Foote's Abstract Algebra, third edition: Prove that $\text{GL}_2(\mathbb{R})$ has no subgroup isomorphic to $Q_8$. [EA: The quaternion group]. [This may be done by ...
0
votes
0answers
13 views

Checking the definition of absolutely irreducible representations

The definition of an irreducible representation $(\rho, V) $ is one with no subrepresentations. Am I correct in saying that a absolutely irreducible means "it is irreducible over the algebraic ...
1
vote
1answer
30 views

Show that $ (φ^G )_K = (φ_{H∩K})^K $ with Mackey's theorem

Suppose H,K ≤ G e θ $ ϵ $ Char(H). Show that Z(θ)≤H. Suppose H,K ≤ G and HK = G. Se $ φ $ ϵ Char(H) show that $ (φ^G )_K = (φ_{H∩K})^K $. For the proof I have to use the Mackey's theorem. How do I ...
2
votes
1answer
28 views

Differences between primitive central idempotents and primitive orthogonal idempotents

I asked this question in mathoverflow. But it was closed. So I ask it here. If we have a complete set of primitive orthogonal idempotents of an algebra $A$, then we can obtain simple modules, ...
-1
votes
0answers
34 views

Indecomposable Finite Abelian Groups are just Cyclic Groups of a Prime Power Order.

there is a structure theorem in group theory states that: indecomposable finite abelian groups are just cyclic groups of a prime power order. my question is about proving this theorem. how to prove? ...
1
vote
0answers
34 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
1answer
37 views

Functor from category of group representations to space of $G$ invariants

For a representation $(V,\rho)$ of a group $G$, define the subspace of $G$-invariants by $$ V^G=\{v\in V: \rho(g)v=v\quad \forall g\in G\} $$ and want to prove the following: $V\mapsto V^G$ ...
1
vote
1answer
29 views

Prove that $\mathrm{Ind}_{\mathbb{I}}^G \cong \mathbb{C}[G]$

Prove that $\mathrm{Ind}_{\mathbb{I}}^G \cong \mathbb{C}[G]$. Apparently: $$\langle \mathrm{Ind}_{\mathbb{I}}^G \mathbb{I}, \chi \rangle_G \overset{Frob.Rep.}= \langle \mathbb{I}, ...
1
vote
2answers
32 views

Action of universal R-matrix of U_q(sl_2)

My question is really simple but requires a few definitions. No special knowledge of quantum groups should be needed, it is more about tensor algebra. Let $q \in \mathbb{C}$ with $q \neq 0, \pm 1$. ...
2
votes
0answers
28 views

Schur and Weyl modules.

Let $m$ be a non-negative integer and $\lambda=(\lambda_1, \cdots, \lambda_s)$ a partition of $m$. If $V$ is a vector space of dimension $n$ (over a field $\mathbb{K}$), we can consider the Schur ...
2
votes
2answers
49 views

Replacing entries of dice by average of it neighbours

I am interested in Representation Theory. I came across the following answer while reading this question on Mathoverflow. An example from Kirillov's book on representation theory: write numbers ...
1
vote
0answers
8 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 ...
2
votes
0answers
27 views

Invariant subspace vs. irreducible subspace (terminology)

In a course in representation theory I was presented the following proposition: Let $(\pi,V)$ be a finite dimensional irreducible representation with a cyclic vector. $V$ has a unique max. proper ...
1
vote
1answer
25 views

explicit components of regular representation of $S_4$

Consider (left) regular complex representation of $S_4$. It has two 2-dimensional irreducible components. I need exact form of elements in those components (probably, having one element I may get ...
0
votes
1answer
11 views

Need definition of symmetric and antisymmetric tensor representations of a Lie algebra

I couldn't find a definitive answer online. Suppose we have a representation of a Lie algebra $(\pi,V)$. Consider the symmetric and antisymmetric vector subspaces of the $k$-th tensor product of ...
0
votes
0answers
9 views

Reordering indexed expressions (combinatorics)

To me, it appears always as a little 'magic' when people reorder expressions, indexed by highly complex combinations of permutations and I would like to know in deep and formally what really is going ...
1
vote
0answers
68 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 ...
3
votes
0answers
77 views

Regge symmetry and outer automorphisms of Dynkin diagrams

Quantum $6j$-symbols are the coefficients of the change of basis matrix in the central extension of Temperley-Lieb algebra(see the book by Kauffman and Lins). It is my understanding that Ocneanu has ...
1
vote
2answers
44 views

The set of non-conjugate elements

I have $H \leq G$ where $G$ is a group. Now for any $t \notin H$ we have $H \cap tHt^{-1} = e$ Now $N$ is a set of all elements of $G$ which are not conjugate to any element of $H$ I want to ...
2
votes
2answers
37 views

Is the trivial representation a subrepresentation of a tensor power of any irreducible complex representation of a finite group?

Let $G$ be a finite group, $V$ an irreducible complex representation and $\mathbb{1}$ the trivial representation. Question: $\exists n >0$ such that $\mathbb{1} \le V^{\otimes n}$?
0
votes
0answers
31 views

Characters of Linear Algebraic Groups

Reading about the semi-invariants of quivers, I see a fact which is frequently referred to in the literature, and is assumed to be trivial. However, I don't see that very easily. So, I was wondering ...
2
votes
1answer
35 views

Characters of (distinct) irreducible finite-dimensional representations of $A$

I need help to understand the proof of this theorem. The theorem can be found in the book Introduction to representation theory by Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex ...
3
votes
1answer
127 views

Computing character of a representation and irreduciblity

for a finite field $k$ I have $G = SL_2(k)$ a group. $H \leq G $ and $H = \lbrace $ $\begin{bmatrix} a & b \\ 0 & d\\ \end{bmatrix} \vert a,b,d \in k \rbrace $ Now $\omega : k^{*} ...
3
votes
1answer
41 views

To show that $A_4$ is solvable

I need to show that $A_4$ is solvable. From what i know the definition of solvable expects to give some chain of subgroups such that each subgroup in the chain is normal to the one in which it is ...
1
vote
1answer
29 views

Irreducible representation of $S_3$ on $\mathbb C^3$

Does there exists an irreducible representation of the group $S_3$ on $\mathbb C^3$? The representations that I can think of all have a $1$ dimensional subspaces that are fixed.
2
votes
1answer
49 views

Product of chracter

From Isaac's character theory book; $3.12$ Let $x\in Irr(G)$ and $g,h\in G$. Show that $$\chi(g)\chi(h)=\dfrac{\chi(1)}{|G|}\sum_{z\in G}\chi(gh^z)$$ I had thought that it was related to $3.9)$; ...
1
vote
0answers
46 views

Any continuous group homomorphism from $\mathbb{R}$ to $GL(n,\mathbb{C})$

Any continuous group homomorphism $\phi$ from $\mathbb{R}$ to $GL(n,\mathbb{C})$ is of the form $\phi (t)=exp(tX)$ for some $X\in M(n,\mathbb{C})$. Can anyone give hints for the proof of this fact? I ...
4
votes
1answer
61 views

Understanding the proof of the Jordan-Hölder Theorem.

I need some help to understand the proof of this theorem which can be found in the book of Introduction to representation theory by Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex ...
0
votes
0answers
27 views

Regular representation, representability of the fiber functor, and hom-distributivity for Hilbert spaces

I've culled together a slick proof of $\Bbb C[G]\cong\bigoplus_{V\in\widehat{G}}{\rm End}(V)$ (Peter-Weyl decomposition) for finite groups using the fact that the fiber functor (that is, the forgetful ...
1
vote
1answer
37 views

Dihedral group is supersolvable

I need to show that Dihedral group $D_n$ is supersolvable. My Approach : I think the existence of a normal chain $\{e\} = G_0 \leqslant G_1 \leqslant ... \leqslant G_n = G$ satisfying following ...
4
votes
2answers
55 views

Dimension of irreducible module divides the dimension of the algebra?

Fact: $\chi(1)$ divides order of $|G|$ where $\chi$ is an irreducible character of $G$. Above fact is equivalent to say that if $V$ is an irreducible $A=\mathbb C [G]$ module then $\dim(V)$ divides ...
2
votes
1answer
61 views

Is there a natural permutation representation of a wreath product of groups?

Is there a "natural" embedding of a $G \wr H$ into the group of permutation matrices? Like an element of $G\wr H$ looks like, $g=((g_1,g_2,..,g_{\vert H \vert}),h), \forall g_i \in G, h \in H$. Now ...
1
vote
1answer
20 views

When does a faithful representation remain faithful on a quotient representation?

Suppose I have a faithful complex representation of some finite group $(V,\pi)$. I can show that whenever this representation contains the trivial representation $(\mathbb{C}v,1)$, so that as a module ...
0
votes
1answer
23 views

Why is the $\mu_n$ representation rational?

In their paper "On the irregularity of cyclic coverings of algebraic surfaces" by F. Catanese and C. Ciliberto, the authors consider the following situation. Let $A = V/\Lambda$ be a $g$-dimensional ...
0
votes
0answers
37 views

Problem from Serre's book

Here is the Q. from serre's representation book I tried solving this the following way let $\Phi : W \rightarrow W_0$ be the given map which takes $w$ to $f_w$ now in $W_0$ we have $f_w(h) =0 ...
1
vote
1answer
33 views

Isomorphism between an group and its double dual

I wanted to prove that for an abelian group $G$ , $\phi : G \rightarrow \hat{\hat{G}}$ is an isomorphism where $\hat{G}$ is a set of all irreducible characters of $G$ for $x \in G$, ...
0
votes
0answers
29 views

Proof of Wigner-Mackey in Serre.

The question is regarding the proof of Wigner-Mackey given as Proposition 25 of Linear Representation of finite groups by Serre. It is on page 23. The fifth line of the proof of $(b)$ on page 63 ...
1
vote
1answer
15 views

Proof that the space of morphisms between equivalent irreps has dimension 1.

Schur's lemma says that for finite group representations, this space between non-equivalent irreps has dimension 0, and that the morphisms between identical irreps are homothety. Yet I forgot how to ...
2
votes
0answers
145 views

Matrix Elements of Real Represententations

Suppose that $G$ is a finite group and we have a unitary irreducible representation $\rho:G\rightarrow \hom V$. Suppose we fix a basis $\{e_i\}_{i\geq 1}$ of $V$ and with respect to this basis we have ...
0
votes
1answer
24 views

Algebraic Indepence of Functions over Infinite Field

Can someone point in the right direction to a reference or give me an idea of the proof of the following fact. My field theory is rusty: Let $K$ be an infinite field of arbitrary characteristic. ...
0
votes
0answers
20 views

Gauge transformation laws, proof in Kobayashi & Nomizu Foundations of Differential geometry

I have two questions about this proof found in K&N's Foundations of Differential Geometry. 1) Can someone please explain how they deduce ...
0
votes
0answers
34 views

Finite matrix power over $\Bbb Z$

$p=\text{prime}$. $p[\Bbb N_{T_1\leq T_2}]=\{0\}\cup \{p^t:t\in\Bbb Z, T_1\leq t\leq T_2\}$. Given $T\in\{0\}\cup\Bbb N$, what is largest $s\in\Bbb N$ such that there is a partition $$0=T_0\leq ...
1
vote
1answer
47 views

Finite matrix power over $\Bbb F_q$

What is largest $s\in\Bbb N$ such that a matrix $M\in\Bbb F_q^{n\times n}=\Bbb F_{p^r}^{n\times n}$ could satisfy $$M^i\neq I,\quad\forall i\in\Bbb Z_+:0<i<s$$ $$M^0=M^s=I?$$
1
vote
2answers
60 views

Conceptual description of the isotypical component

This is probably rather simple but I have not found it in the literature. Consider the category $C$ of representations of a finite group $G,$ over a field $k$ of characteristic not dividing the order ...
2
votes
0answers
15 views

Normal form over $\mathbb{Z}$ of matrices of order $2$

Suppose $M \in GL_k(\mathbb{Z})$ is of order $2$. That is, $M^2 = 1$ and $M \ne 1$. Then is it true that upto a change of $\mathbb{Z}$ basis, $M$ has the form $$\begin{pmatrix}J \\ & J \\ & ...
3
votes
3answers
73 views

Matrices over a finite field with given Jordan normal form over the algebraic closure

Can one describe the (conjugacy classes of) square matrices over a finite field such that over the algebraic closure of this finite field their Jordan normal form consists of one Jordan block? (Such ...
1
vote
2answers
48 views

Confusion about Lie groups in Fulton & Harris

Near the beginning of chapter 8 (titled Lie groups and Lie algebras) authors motivate the definition of Lie algebra. I'm confused by two things in just one sentence: ($G$ is a Lie group) The ...
3
votes
0answers
53 views

How much we know about the Group from its Complex character table?

Suppose $G$ is a finite group and suppose that complex character table of $G$ is given.It is well known that from character table we cannot determine the Group uniquely (For example $Q_8$ and $D_8$ ...