Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.
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8answers
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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 ...
33
votes
0answers
662 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
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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 ...
22
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5answers
882 views
Can every group be represented by a group of matrices?
Can every group be represented by a group of matrices?
Or are there any counterexamples? Is it possible to prove this from the group axioms?
19
votes
2answers
483 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 ...
17
votes
4answers
472 views
How is $\operatorname{GL}(1,\mathbb{C})$ related to $\operatorname{GL}(2,\mathbb{R})$?
I am trying to get a grasp on what a representation is, and a professor gave me a simple example of representing the group $Z_{12}$ as the twelve roots of unity, or corresponding $2\times 2$ matrices. ...
17
votes
9answers
800 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 ...
17
votes
4answers
709 views
Irreducible representations of Poincaré group
I am looking for any reference on Wigner's classification of irreducible representations of the Poincaré group. I know the classification, but is there any reference where the representations ...
16
votes
1answer
554 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 $K_{\lambda\mu}$ ...
16
votes
0answers
166 views
Tate conjecture for Fermat varieties
I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...
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 ...
14
votes
1answer
336 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 ...
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 ...
13
votes
2answers
248 views
A question on partitions of n
Let $P$ be the set of partitions of n. Let $\lambda$ denote the shape of a particular partition. Let $f_\lambda(i)$ be the frequency of $i$ in $\lambda$ and let $a_\lambda(i) := \# \lbrace j : ...
13
votes
2answers
242 views
Proving finite dimensionality of modular forms using representation theory?
It is well known how to use algebraic geometry (differentials, divisors, and Riemann-Roch) in order to prove the finite dimensionality of the vector space of modular forms of some fixed weight and ...
13
votes
1answer
196 views
Writing a group element as $ghg^{-1} h^{-1}$ and as $g^2 h^2$
I recently read the elegant paper Generalized Frobenius Schur Numbers, by Bump and Ginzburg, which I learned about here. The results in this paper imply the following:
Let $G$ be a finite group ...
12
votes
2answers
447 views
Understanding the proof of Schur-Weyl Duality
I am teaching myself representation theory on $GL(V)$ and $S_n$ using my friend's lecture notes, and have reached a proof of the Schur-Weyl Duality theorem; on reading through I'm struggling to make ...
11
votes
1answer
558 views
Is the dual representation of an irreducible representation always irreducible?
Let $G$ be a group and let $V$ be a complex vector space which is a
representation of $G$. Let's write the (left) action of $g\in G$ on
$v\in V$ as $gv$.
The dual vector space of $V$ is the set of ...
11
votes
2answers
493 views
Dimensions of irreducible representations of finite groups over $\mathbb Q$
If $G$ is a finite group, then it is well known that there are finitely many inequivalent irreducible representations of $G$ over $\mathbb{C}$; moreover the sum of squares of dimensions of the ...
11
votes
2answers
457 views
Restriction to a normal subgroup
More exam preparation.
Let $A$ be a normal subgroup of a finite group $G$ and $V$ an irreducible representation of $G$. Show that either $\text{Res}_A^G V$ is isotypic (a sum of copies of one ...
11
votes
1answer
91 views
Introduction to the trace formula for people outside number theory
I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula.
I am aware that there are "standard references" on the ...
11
votes
1answer
401 views
Exercise 6.5 in Humphrey's Book on Lie Algebras
I am trying to solve Exercise 6.5 part 4 in James Humphreys' Introduction to Lie Algebras and Representation Theory. I added the (homework) tag because my question is about an exercise, but this is ...
11
votes
1answer
185 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 ...
10
votes
2answers
284 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 ...
10
votes
2answers
115 views
Path Algebra for Categories
For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", ...
10
votes
5answers
186 views
Applications of Character Theory
Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, , Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character ...
10
votes
1answer
178 views
Do all algebraic integers in some $\mathbb{Z}[\zeta_n]$ occur among the character tables of finite groups?
The values of irreducible characters of a finite groups are always sums of roots of unity; do all sums of roots of unity (i.e. algebraic integers in the maximal abelian extension of $\mathbb{Q}$) ...
10
votes
1answer
66 views
Do all representations of finite groups have one-dimensional subrepresentations?
Let V be a representation of a finite group G, and $v\in V$ - a nonzero vector. Put $$u = \sum_{g\in G} gv.$$
Then for any $g\in G$ we have $gu = u$ and therefore $<u>$ is a subrepresentation of ...
10
votes
2answers
132 views
Is $f(\operatorname{rad}A)\subseteq\operatorname{rad}B$ for $f\colon A\to B$ not necessarily surjective?
If I have two $K$-algebras $A$ and $B$ (associative, with identity) and an algebra homomorphism $f\colon A\to B$, is it true that $f(\operatorname{rad}A)\subseteq\operatorname{rad}B$, where ...
10
votes
0answers
172 views
Application of Hilbert's basis theorem in representation theory
In Smalo: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand:
Two orders are defined on the set ...
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 ...
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 ...
9
votes
2answers
374 views
finite subgroups of PGL(3,C)
The enumeration of finite subgroups of $\operatorname{PGL}_2(\mathbb{C})$ is one of the classic classification problems: mathematicians in many fields know well that the answer is cyclic groups, ...
9
votes
2answers
317 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 ...
9
votes
1answer
181 views
Class equation of subgroup of $SL(4,\mathbb{F}_2)$
Can you point me toward a computation-light derivation of the class equation of the subgroup of $SL(4,\mathbb{F}_2)$ consisting of upper-triangular matrices with 1's on the main diagonal?
The ...
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, ...
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 ...
8
votes
3answers
331 views
Representations of a cyclic group of order p over a field of characteristic p?
Let $p$ be a prime. My eventual goal is to prove that the only irreducible representation of a $p$-group over a field of characteristic $p$ is the trivial representation.
At the moment, I'm trying ...
8
votes
1answer
226 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 ...
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 ...
8
votes
1answer
69 views
Wedge pure product
Let $V$ be a vector space of $\dim n$ over $K$. Let $P$ be the set of all pure products of the form $v_1 \bigwedge v_2$. How to prove that there is a one-one correspondence between the one dimensional ...
8
votes
1answer
300 views
Osp, USp, SU(,) and PSU
I would be glad if someone can give me some (hopefully easy to understand!) references for learning about these groups Osp, USp and PSU and their representations.
I run into these mostly while ...
8
votes
1answer
278 views
Undergraduate roadmap for Langlands program and its geometric counterpart
What are the topics which an undergraduate with knowledge of algebra, galois theory and analysis learn to understand Langlands program and its goemetric counterpart? I would also like to know what are ...
8
votes
1answer
135 views
Schur -Weyl duality for $sl_2$ and $S_n$
$V$ is an $m$ dimensional vector space having a structure of $sl_2(\mathbb{C})$-module, where $sl_2(\mathbb{C})$ is the Lie algebra of the Lie group $SL_2(\mathbb{C})$. The symmetric group $S_n$ acts ...
8
votes
1answer
81 views
Flatness of residual representations associated to modular forms
Let $f\in S_k(\Gamma_1(N),\chi)$ be a Hecke eigenform of weight $k\geq 2$, $p$ an odd prime not dividing $N$, and $K_f$ the number field generated by the Hecke eigenvalues of $f$. Fixing a prime ...
8
votes
1answer
235 views
Field of definition of representations of symmetric groups
Can one show in an elementary way, without recourse to Young tableaux etc., that the complex representations of symmetric groups are realisable over $\mathbb{R}$? It is easy to show that they are all ...
8
votes
2answers
185 views
The Noether-Deuring Theorem
I have to solve the following exercise taken from the book "Introduction to Representation Theory" by P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina and S. ...
8
votes
1answer
187 views
Exercise on representations
I am stuck on an exercise in Serre, Abelian $\ell$-adic representations (first exercise of chapter 1).
Let $V$ be a vector space of dimension $2$, and $H$ a subgroup of $GL(V)$ such that ...
8
votes
0answers
58 views
Expression of basis vectors of permutation modules in different bases.
Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that corresponds to $\lambda$, i.e. the complex vector space spanned by all standard ...
8
votes
0answers
199 views
Character theory of $2$-Frobenius groups.
Edit Summary: I've posted this on MO and received a partial answer there. Can anybody help me expand on this?
Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and ...
