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

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8
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3answers
826 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
1k views

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

Consider $G$ a cyclic group of order $n$ with prime $p\nmid n$. How do I construct all the irreducible representations over $\mathbb F_p$? How many irreducible representations are there and what are ...
89
votes
8answers
6k 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 ...
14
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. ...
8
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.
8
votes
3answers
547 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 ...
5
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 ...
8
votes
3answers
1k 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) = ...
7
votes
1answer
392 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 ...
2
votes
1answer
90 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 ...
27
votes
10answers
5k 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 ...
19
votes
0answers
173 views

Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc… & Representation Theory of Special Functions

Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question ...
15
votes
1answer
461 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
257 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, ...
5
votes
1answer
581 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 ...
3
votes
4answers
239 views

Alternate proof of Schur orthogonality relations

I am trying to find an alternate proof for Schur orthogonality relations along the following lines. Let $G$ be a finite group, with irreducible representations $V_1$, $V_2$, $\cdots$, $V_d$. Let $V$ ...
3
votes
1answer
444 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 $n$?
3
votes
2answers
462 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 ...
2
votes
1answer
78 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
287 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
1answer
233 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 ...
3
votes
2answers
223 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
84 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 $Z_2^4$, ...
24
votes
2answers
860 views

Surprising but simple group theory result on conjugacy classes

I have read that for any group $G$ of order $2m+1$ (odd) with $n$ conjugacy classes, it is always the case that $16$ divides the value $(2m+1)-n = |G|-n$. This seems to me like an astonishing ...
47
votes
2answers
973 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 ...
27
votes
2answers
666 views

What does the group ring $\mathbb{Z}[G]$ of a finite group know about $G$?

The group algebra $k[G]$ of a finite group $G$ over a field $k$ knows little about $G$ most of the time; if $k$ has characteristic prime to $|G|$ and contains every $|G|^{th}$ root of unity, then ...
7
votes
2answers
275 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
1k 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 ...
12
votes
4answers
901 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 ...
12
votes
3answers
421 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 ...
5
votes
0answers
101 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 ...
10
votes
2answers
283 views

History of the matrix representation of complex numbers

It is well-known to many that $\mathbb{C}$ can be represented by matrices of the form $\left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$. For example, see this question or this ...
10
votes
1answer
224 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
337 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 ...
8
votes
2answers
352 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 ...
8
votes
2answers
735 views

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

Why does the universal cover of $SL_{2}(\mathbb{R})$ have no finite dimensional representations?
6
votes
2answers
160 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 ...
7
votes
5answers
768 views

Shortest way of proving that the Galois conjugate of a character is still a character

Let $G$ be a finite group and $\chi$ a character of $G$. The values of $\chi$ generate an abelian Galois extension $K$ of $\mathbb{Q}$, and so one can consider the conjugate $\sigma(\chi)$ of $\chi$ ...
3
votes
0answers
73 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 ...
10
votes
2answers
748 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 ...
9
votes
3answers
554 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 ...
6
votes
4answers
403 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
1answer
505 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 ...
5
votes
2answers
308 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
381 views

Any irreducible representation of a $p$-group over a field of characteristic $p$ is trivial.

In general, we know that if $G$ is a finite group and $K$ is a field, then $K[G]$ (the group algebra) is semisimple whenever $\operatorname{char}(K)$ does not divide the order of $G$. However, this ...
4
votes
1answer
166 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
101 views

Centralizer of $\mathbb{C}[G]$ in $\mathbb{C}[H]$

I found this result, but can't understand how to prove. Let $H$ be a subgroup of $G$. Then prove $Z(\mathbb{C}[G],\mathbb{C}[H])$ is commutative iff every irreducible $G$ module when restricted to ...
3
votes
1answer
381 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
295 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 ...