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

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7
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727 views

Classifying the irreducible representations of $\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}$

I need help with the following problem: Suppose $n$ is some positive integer, and $n|p-1$. Classify all irreducible representations of $$\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}.$$ ...
1
vote
1answer
887 views

Irreducible representations of a cyclic group over a field of prime order

Consider $G$ a cyclic group of order $n$ with prime $p\not|n$. How do I construct all the irreducible representations over $\mathbb F_p$? How many irreducible representations are there and what are ...
13
votes
2answers
1k views

Is $A \times B$ the same as $A \oplus B$?

When $A, B$ are $K$-modules, then $A \times B$ is the same as $A \oplus B$. Let $A, B$ be two $K$-algebras, where $K$ is a field. Is $A \times B$ the same as $A \oplus B$? Thank you very much. ...
80
votes
8answers
5k views

Importance of Representation Theory

Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I ...
8
votes
3answers
527 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 ...
4
votes
3answers
1k views

Permutation module of $S_n$

Let $G=S_n$ and let $V$ be the permutation module of $G$ with basis $\{x_1,\cdots,x_n\}.$ Let $\lambda, \mu \in \mathbb{C}$ to allow one to define a $\mathbb{C}G$-homomorphism $\rho:V \to V$ by ...
2
votes
1answer
74 views

Conjugacy classes and centralizers of a SmallGroup

What is the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)? Would someone be willing to provide the complete lists of conjugacy classes and centralizers of ...
6
votes
3answers
2k views

Symmetric and exterior power of representation

Is there exists some simple formula for characters $$\chi_{\Lambda^{k}V}~~~~\text{and}~~~\chi_{\text{Sym}^{k}V}$$ for some representation $V$ of finite group? Thanks.
14
votes
1answer
433 views

Low-dimensional Irreducible Representations of $S_n$

For $n\geq 7$, I would like to show that $S_n$ has no irredicuble representations of dimension $m$ for $2\leq m\leq n-2$. The catch is that I am not allowed to use any "machinery" (evidently, this ...
8
votes
2answers
885 views

Complex finite dimensional irreducible representation of abelian group

I'm supposed to show that each Complex finite dimensional irreducible representation of an abelian group is one dimensional. For any map $\phi: V \rightarrow V$ it holds that $\phi(\rho(g)v) = ...
8
votes
2answers
242 views

Some irreducible character separates elements in different conjugacy classes

Let $x$ and $y$ be elements that are not conjugate in $G$. Then there is some irreducible character $\chi$ such that $\chi(x) \not = \chi(y)$. Clearly the "irreducible" part isn't important, ...
3
votes
1answer
410 views

Faithful irreducible representations of cyclic and dihedral groups over finite fields

How to determine all the faithful irreducible representations of $\mathbb Z_n$ and $D_{2n}$ over $GF(p)$, where $p$ is a prime not dividing $n$?
2
votes
1answer
77 views

Which non-Abelian finite groups contain the two specific centralizers? - part II

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other ...
3
votes
2answers
282 views

Existence of a G-invariant matrix

Let $\phi: G \to GL(\mathbb{R}^n)$ be a homomorphism, $G$ finite. Prove that there is a positive-definite matrix $M$ such that $\phi(g)^tM \phi(g) =M$ $\forall g \in G $. This looks really ...
3
votes
2answers
217 views

Why is $ U \otimes \operatorname{Ind}(W) = \operatorname{Ind}(\operatorname{Res}(U) \otimes W)$?

If $U$ is a representation of $G$ and $W$ is a representation of $H$, then why is $$ U \otimes \operatorname{Ind}(W) = \operatorname{Ind}(\operatorname{Res}(U) \otimes W)$$ I've tried to simply use ...
2
votes
0answers
73 views

Which finite groups contain the two specific centralizers?

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers both of these two groups: i. the elementary group $Z_2^4$, ...
6
votes
1answer
317 views

Nonabelian group with all irreducible representations one-dimensional

All irreducible representations of an abelian group are one-dimensional. For a finite group, the coverse is also true - if all irreducible representations are one-dimensional then the group is ...
3
votes
1answer
217 views

A conjugacy class $C$ is rational iff $c^n\in C$ whenever $c\in C$ and $n$ is coprime to $|c|$.

Let $C$ be a conjugacy class of the finite group $G$. Say that $C$ is rational if for each character $\chi: G \rightarrow \mathbb C$ of $G$, for each $c\in C$, we have $\chi(c) \in \mathbb Q$. I am ...
23
votes
10answers
4k views

What's a good place to learn Lie groups?

Ok so I read the following article the other day: http://www.aimath.org/E8/ and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought "oh it must have something to ...
46
votes
3answers
948 views

How to think of the group ring as a Hopf algebra?

Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
7
votes
2answers
251 views

Estimates on conjugacy classes of a finite group.

In Character Theory Of Finite Groups by I Martin Issacs as exercise 2.18, on page 32. Theorem: Let $A$ be a normal subgroup of $G$ such that $A$ is the centralizer of every non-trivial element ...
18
votes
3answers
967 views

Representation theory of the additive group of the rationals?

What do the finite-dimensional continuous complex representations of the additive group $\mathbb{Q}$ with the usual topology look like? With the discrete topology? Which representations are ...
10
votes
2answers
1k views

The physical meaning of the tensor product

I have come across tensor products many times in physics, namely for matrices, vector-space elements, Hilbert-space elements (quantum states), and representations of groups and algebras. However, the ...
5
votes
0answers
98 views

— Cartan matrix for an exotic type of Lie algebra --

(1) Is there a notion of Cartan matrix for non-semisimple Lie algebra? For example, consider this Lie algebra: $$ [X_i, X_j] = f_{ij}{}^k X_k \qquad\qquad [X_i,Y^j] = - f_{ik}{}^j Y^k \qquad\qquad ...
12
votes
3answers
378 views

Complex Galois Representations are Finite

In A First Course in Modular Forms, Diamond and Shurman leave as an exercise ($9.3.3$) that every complex Galois representation is finite. While I think I have worked through this exercise here, this ...
10
votes
1answer
213 views

Galois representations and normal bases

I am not very familiar with the theory of Galois representations, but I do know a bit about both Galois theory and representation theory. Recently I learned about the notion of a normal basis for a ...
4
votes
2answers
315 views

How to generalise $(\wedge^2 \chi)(g) = \frac{1}{2}(\chi(g)^2-\chi(g^2))$?

One can decompose $\bigotimes^2 V = \bigvee^2 V \oplus \bigwedge^2 V$, getting a corresponding decomposition for representations, say when $V$ is a module for some finite group $G$. One then has the ...
11
votes
4answers
775 views

Proofs that the degree of an irrep divides the order of a group

It is a theorem in basic representation theory that the degree of an irreducible representation on $G$ over $\mathbb{C}$ divides the order of $G$. The usual proof of this fact involves algebraic ...
8
votes
2answers
337 views

What is an irrreducible character of a finite group?

Let $S_n$ be the group of permutations of $\{1, 2, \ldots, n\}$. A “character” for $S_n$ is a function $\chi\colon S_n \to \mathbb{C} \setminus \{0\}$ with $\chi(ab) = \chi(a)\chi(b)$ for all $a, b ...
6
votes
2answers
155 views

Correspondence of representation theory between $\mathrm{GL}_n(\mathbb C)$ and $\mathrm U_n(\mathbb C)$

If I know something about the representation theory of the general linear group $\mathrm{GL}_n(\mathbb C)$, what can I say about the representation theory of the unitary group $\mathrm U_n(\mathbb ...
6
votes
2answers
667 views

Universal Cover of $SL_{2}(\mathbb{R})$

Why does the universal cover of $SL_{2}(\mathbb{R})$ have no finite dimensional representations?
3
votes
0answers
55 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 ...
3
votes
2answers
418 views

Why is the group action on the vector space of polynomials naturally a left action?

When seeking irreducible representations of a group (for example $\text{SL}(2,\mathbb{C})$ or $\text{SU}(2)$), one meets the following construction. Let $V$ be the space of polynomials in two ...
10
votes
2answers
652 views

Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz

As the title suggests, could anyone either provide me with or direct me to a proof that affine n-space $\mathbb{A}^n$ is irreducible, without using the Nullstellensatz? This is an exercise in a ...
8
votes
3answers
502 views

Does every irreducible representation of a compact group occur in tensor products of a faithful representation (and its dual)?

Let $G$ be a compact (Hausdorff) group and $V$ a faithful (complex, continuous, finite-dimensional) representation of it. (Hence $G$ is a Lie group.) Is it true that every irreducible representation ...
5
votes
4answers
333 views

Ring of polynomials as a module over symmetric polynomials

Consider the ring of polynomials $\mathbb{k} [x_1, x_2, \ldots , x_n]$ as a module over the ring of symmetric polynomials $\Lambda_{\mathbb{k}}$. Is $\mathbb{k} [x_1, x_2, \ldots , x_n]$ free ...
5
votes
2answers
277 views

Faithful representation implies group is cyclic

Suppose there is a faithful representation $\rho:G\to SL_2(\mathbb{R})$. Prove that $G$ is cyclic. I know there has to be something special about its representation being special (no pun ...
4
votes
1answer
477 views

Determinant of the character table of a finite group $G$

This is an exercise from the book "Groups and Representations" by Alperin & Bell. This quantity is well defined upto a sign. By column orthogonality relations, its squared norm is ...
4
votes
1answer
163 views

Valuations on number fields

I'm trying to explicitly compute modular representations of some finite groups -- the easiest example to discuss is the cyclic group $C_3$ when $p=3$. The three ordinary irreducible modules for $C_3$, ...
3
votes
1answer
290 views

the converse of Schur lemma

I am interested in the converse of the following form of Schur's lemma: Lemma. (Schur) A group G, a $\mathbb{C}$-vector space V and a homomorphism D : G $\rightarrow$ GL(V) is given. Suppose that D ...
3
votes
1answer
287 views

Normal subgroups of $S_N$

Is there a list of all normal subgroups for $S_N$? What is a criteria for a finite group to be a normal subgroup of $S_N$? Which of them are kernels of irreducible representation? From a partition ...
2
votes
1answer
172 views

Question of Clifford theory

I have some questions about thist post: faithful irreducible representations of cyclic and dihedral groups over finite fields I would appreciate it really if someone could help me. 1) Do I get with ...
4
votes
2answers
268 views

Group of order 24 with no element of order 6 is isomorphic to $S_4$

Proposition: Given a group $G$ with $|G|=24$ such that $\nexists g\in G$ with $|g|=6$, then $G\cong S_4$. I understand methods you can employ to deduce the number of Sylow $p$-groups in $G$ by ...
4
votes
3answers
401 views

Reference request for algebraic Peter-Weyl theorem?

It seems that, for $GL_n$, and possibly for something like complex reductive groups $G$ in general, there's an algebraic version of the Peter-Weyl theorem, which might say that the coordinate ring of ...
3
votes
0answers
36 views

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)? ...
3
votes
0answers
155 views

Link between representation theory and Galois theory: Trivial representation in field towers.

Let $K|F$ be a finite cyclic Galois extension of number fields of degree prime to $p$ with Galois group $H$, where $p$ denotes a rational prime. Let $L|K$ denote a pro-$p$-extension (possibly ...
3
votes
2answers
185 views

Example of a simple module which does not occur in the regular module?

Let $K$ be a field and $A$ be a $K$-algebra. I know, if $A$ is artinain algebra, then by Krull-Schmidt Theorem $A$ , as a left regular module, can be written as a direct sum of indecomposable ...
2
votes
1answer
374 views

Sum of squares of dimensions of irreducible characters.

For anyone familiar with Artin's Algebra book, I just worked through the proof of the following theorem, which can be seen here: (5.9) Theorem Let $G$ be a group of order $N$, let ...
2
votes
1answer
142 views

Finite dimensional algebra with a nil basis is nilpotent

Prove that a finite-dimensional algebra $A$ over a field $K$ of characteristic zero having a basis consisting of nilpotent elements $\{e_1,...,e_n\}$ is nilpotent. My approach: Let $m_i$ be the ...
2
votes
2answers
422 views

The Ext functor in the quiver representation

First take a question as an example: Let $f:L\to M$ be an irreducible morphism in $\mathrm{mod}-A$ and $X$ be a right $A$-module. Show that ...