Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.
6
votes
3answers
465 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
326 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 ...
61
votes
8answers
3k 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 ...
5
votes
2answers
117 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 ...
4
votes
2answers
225 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 ...
2
votes
2answers
521 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.
9
votes
2answers
318 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 ...
5
votes
2answers
214 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 ...
4
votes
1answer
128 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 $C_3$, ...
5
votes
2answers
302 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
1answer
221 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
1answer
154 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
1answer
192 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 ...
4
votes
2answers
161 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 ...
2
votes
1answer
69 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
113 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 $\chi(\sigma) \in \mathbb Q$. I ...
1
vote
3answers
65 views
Show $U \otimes V$ is an irreducible G-module
Let $G$ is some group and $U$ is an irriducible $G$-module over the complex numbers. Now if $V$ is a $G$-module of dimension 1, I would like to prove $U \otimes V$ is an irriducible $G$-module.
My ...
1
vote
2answers
161 views
Exact Group Representation
Definition: Exact Representation of finite group $G$ in some field $V$ - is injective homomorphism
$$\rho : G\to GL(V).$$
(I don't know English terminology, so you may correct me; or probably exists ...
0
votes
2answers
278 views
Irreducibility of the standard representation of $S_n$.
The permutation representation of $S_n$ is $\mathbb C^n$ with elements of $S_n$ permuting the basis vectors $\{e_1, e_2, \ldots, e_n\}$. It has a trivial subrepresentation spanned by the vector $v = ...
27
votes
4answers
1k views
The Langlands program for beginners
Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its ...
17
votes
9answers
803 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 ...
34
votes
0answers
667 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 ...
14
votes
2answers
443 views
Categorical description of algebraic structures
There is a well-known description of a group as "a category with one object in which all morphisms are invertible." As I understand it, the Yoneda Lemma applied to such a category is simply a ...
10
votes
0answers
225 views
Subgroups as isotropy subgroups and regular orbits on tuples
Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω?
Motivation:
In ...
6
votes
1answer
166 views
What is the relationship between Mackey's theorem in character theory and Mackey's theorem in transfer theory?
Here are the statements of the two theorems. The first statement I took from a paper I have been reading, but I believe can also be found in Isaacs' Character Theory of Finite Groups as an exercise. ...
6
votes
2answers
168 views
Is $\sqrt 7$ the sum of roots of unity?
Let $a_n$ and $b_n$ be 2 sequences of $n$ rationals.
Is it possible that $\sqrt 7 = \sum_{m=1}^{n} a_m (-1)^{b_m}$ ? Is it possible that $\sqrt{17}$ = $\sum_{m=1}^{n} a_m (-1)^{b_m}$ ?
How to ...
16
votes
1answer
557 views
Why is a general formula for Kostka numbers “unlikely” to exist?
In reference to Stanley's Enumerative Combinatorics Vol. 2: right after he has defined Kostka numbers (section 7.10), he mentions that it is unlikely that a general formula for $K_{\lambda\mu}$ ...
8
votes
1answer
228 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 ...
6
votes
1answer
173 views
Linear Representations coming from Permutation Representations
Let $G$ be a finite group, and consider a permutation representation of $G$ on
some finite set $\Sigma$ with $|\Sigma| = n$. By considering the vector space $V$
over $\mathbf{C}$ of dimension $n$ ...
5
votes
1answer
174 views
Why is every $N$-invariant polynomial function on $n\times n$ matrices in the Plücker algebra?
Let $k$ be a field and $k[{\bf x}] = k[x_{ij}: 1 \leq i, j \leq n]$ be a polynomial algebra that I can think of as the algebra of functions on $n \times n$ matrices that are polynomial in each ...
4
votes
1answer
264 views
Definition of a “root” of a Lie Algebra
I am using the notation that $g$ is the Lie algebra of the Lie group $G$ and $T$ is the maximal torus of $G$ and $t$ is the Lie algebra of $T$ (and hence $t$ is the Cartan subalgebra of $g$). A ...
9
votes
5answers
280 views
$\varphi$ in $\operatorname{Hom}{(S^1, S^1)}$ are of the form $z^n$
I'd like to see a proof why $\varphi \in \operatorname{Hom}{(S^1, S^1)}$ looks like $z^n$ for an integer $n$.
At first I thought I could argue that if I have a homomorphism that maps $e^{ix}$ to some ...
8
votes
2answers
147 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, ...
6
votes
2answers
173 views
Understanding induced representations
Let $G$ be a group and $H$ be a subgroup. Let $\phi:H\rightarrow GL(V)$ be a representation of $H$. There are three constructions in Wikipedia, but I am not really convinced by these.
My question is: ...
4
votes
4answers
583 views
Best books on Representation theory
What are some of the best books on Representation theory for a beginner? I would prefer a book which gives motivation behind definitions and theory.
3
votes
0answers
77 views
Conjugacy of projective representations
Given characters of the Schur covering group of $G$ of the same degree, how does one tell if the projective representations (as homomorphisms from $G$ into $\operatorname{PGL}$) are conjugate in ...
2
votes
1answer
425 views
Question on fundamental weights and representations
I am a bit confused about the notion of "fundamental weights".
In a complexified setting, I am thinking of my Lie algebra to be decomposed as, $\cal{g} = \cal{t} \oplus _\alpha \cal{g}_\alpha$ where ...
13
votes
3answers
686 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
285 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
2answers
254 views
Characters of Symmetric and Antisymmetric Powers
Let $V$ be a representation with character $\chi$. I would like to have a formula for the characters of the representations $\mathrm{Sym}^m[V]$ and $\wedge ^m[V]$ in terms of $\chi$. Fulton and ...
7
votes
1answer
44 views
Specific projective dimension of a module over bound quiver
Suppose $K$ is an algebraically closed field, and $A$ is the algebra presented by the quiver
$$\require{AMScd}
\begin{CD}
1 @>>> 2\\
@V{}VV @V{}VV \\
3 @>>> 4 @>>> 5
...
4
votes
1answer
144 views
The primitive spectrum of a unital ring
I'm trying to investigate the notion of primitive spectrum and its so-called Jacobson or hull-kernel topology, but I can only find references which define it for C*-algebras: see the Wikipedia page ...
3
votes
1answer
89 views
Converse of Schur's First Lemma
Suppose $G$ is a closed subgroup of $SU(d)$, $d>1$, and let $\rho$ be a $d$-dimensional special unitary representation of $G$. Suppose that if a matrix $A$ commutes with all of $\rho(G)$ for all ...
3
votes
3answers
130 views
Understanding the representations of group and Modules.
I am trying to understand and have a good grasp on Representation theory . I was asking to myself " what essentially is the difference between representation of some group $G$ and a $KG$ module, how ...
3
votes
2answers
182 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 ...
3
votes
1answer
62 views
Is relatively free the same thing as induced for finite group modules?
I was looking over Alperin's Local Representation Theory and I realized I remembered a definition that may not be there (or true).
Is a relatively H-free G-module exactly the same as a G-module ...
3
votes
2answers
479 views
Does regular representation of a finite group contain all irreducible representations?
I know that every irreducible representations of $S_n$ can be found in $\mathbb{C}S_n$. I wonder how can I prove that irreducible representations of a finite groups $G$ can be found in $\mathbb{C}G$. ...
2
votes
1answer
154 views
Complete reducibility of sl(3,F) as an sl(2,F)-module
I was reading the Weyl's theorem on the complete reducibility of a finite dimensional representation of semi-simple Lie algerba and wanted to apply the theorem to the following problem which was ...
2
votes
1answer
118 views
Projective Tetrahedral Representation
I can embed $A_4$ as a subgroup into $PSL_2(\mathbb{F}_{13})$ (in two different ways in fact). I also have a reduction mod 13 map $$PGL_2(\mathbb{Z}_{13}) \to PGL_2(\mathbb{F}_{13}).$$ My question is:
...
8
votes
1answer
143 views
Can $(\Bbb{R}^2,+)$ be given the structure of a matrix Lie group?
I have an assignment problem that is coming from Brian Hall's book Lie Groups, Lie Algebras and Representations: An Elementary Introduction.
Suppose $G \subseteq GL(n_1;\Bbb{C})$ and $H ...
