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

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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 ...
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 ...
35
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4answers
6k 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 ...
31
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5answers
2k 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?
25
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2answers
592 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 ...
25
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0answers
462 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 ...
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 ...
22
votes
2answers
796 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 ...
22
votes
2answers
386 views

What is the least $n$ such that it is possible to embed $\operatorname{GL}_2(\mathbb{F}_5)$ into $S_n$?

Let $\operatorname{GL}_2(\mathbb{F}_5)$ be the group of invertible $2\times 2$ matrices over $\mathbb{F}_5$, and $S_n$ be the group of permutations of $n$ objects. What is the least ...
21
votes
0answers
373 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 ...
20
votes
4answers
963 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 ...
18
votes
4answers
539 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. ...
18
votes
5answers
286 views

$\sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 $

I came across this result while doing some representation theory of the permutation group $S_n$ $$ \sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 = 2 n!$$ This can be ...
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 ...
18
votes
1answer
731 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 $K_{\lambda\mu}$ ...
18
votes
2answers
594 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 ...
17
votes
1answer
213 views

Involutions, RSK and Young Tableaux

Let $S_n$ be the symmetric group on $n$ elements. The Robinson-Schensted-Knuth (RSK) correspondence sends a permutation $\pi\in S_n$ to a pair of Standard Young Tableaux $(P,Q)$ with equal shapes ...
17
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0answers
402 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 ...
16
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2answers
1k 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: ...
16
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2answers
413 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 ...
15
votes
2answers
298 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 ...
15
votes
1answer
123 views

Decomposing $V_1^{\otimes n}$, $\text{Sym}^2V_n$ into irreducibles, formula for all $n$?

$``$Let $G = \text{SU}(2)$, and let $V_n$ be the space of homogeneous degree $n$ polynomials in $\mathbb{C}[x, y]$. Decompose $V_1^{\otimes n}$, $\text{Sym}^2V_n$ into irreducibles.$"$ For ...
14
votes
2answers
274 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 : ...
14
votes
2answers
606 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, ...
14
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2answers
963 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 ...
14
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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 ...
14
votes
1answer
59 views

random walk on finite cyclic group

Suppose that I have a random walk on the finite cyclic group of order $d > 2$, where the initial probability distribution $Q$ assigns the values $p, q, r$ to $-1, 0, 1$, respectively, where $p + q ...
14
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0answers
34 views

Determinant of Character table as a matrix

I'm studying for finals and came across this problem in a book. Suppose $G$ is a finite group with conjugacy class representatives $g_1,...,g_k$ and character table $Z$. Consider $Z$ as a matrix. ...
13
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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. ...
13
votes
1answer
1k 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 ...
12
votes
3answers
254 views

Proving facts about groups with representation theory.

I was enrolled in a representation theory (of finite groups) course in the fall and throughout the class we focused on properties of representations and paradigms built around them. The whole time, I ...
12
votes
3answers
697 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 ...
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 ...
12
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1answer
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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 ...
12
votes
1answer
203 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 ...
12
votes
1answer
440 views

Plancherel formula for compact groups from Peter-Weyl Theorem

I'm trying to derive the following Plancherel formula: $$\|f\|^{2}=\sum_{\xi\in\widehat{G}}{\dim(V_{\xi})\|\widehat{f}(\xi)\|^{2}}$$ from the statement of the Peter-Weyl Theorem as given by Terence ...
11
votes
5answers
307 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 ...
11
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1answer
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What is the idea of a monodromy?

Is there a connexion between : 1) The monodromy group of a topological space. 2) The $\ell$-adic monodromy theorem of Grothendieck. 3) The $p$-adic monodromy conjecture of Fontaine (which is now ...
11
votes
1answer
221 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 $\mathbb{Q}$) ...
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 ...
11
votes
2answers
848 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
2answers
539 views

What is the standard definition of an ordinary (local) $p$-adic Galois representation?

Let $V$ be a $n$-dimensional $\mathbf{Q}_p$-vector space with a continuous action of $\operatorname{Gal}(\bar{L}/L)$, where $L$ is a complete discretely valued field of characteristic zero with ...
11
votes
1answer
161 views

Polynomials invariant under the action of $S_m \times S_n$

The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ has a maximal subring invariant under the action of $S_n$ on the variables. This is the ring of symmetric polynomials. Suppose we have ...
11
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1answer
255 views

Exercise 2.13, I. Martin Isaacs' Character Theory

I am trying to solve the exercise 2.13 in Isaacs' Character Theory Book. However I met some difficulties, let me sketch out what I am thinking so that you may tell me a hint. The problem 2.13 is ...
11
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1answer
704 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
228 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 ...
11
votes
2answers
375 views

Young diagram for exterior powers of standard representation of $S_{n}$

I'm trying to solve ex. 4.6 in Fulton and Harris' book "Representation Theory". It asks about the Young diagram associated to the standard representation of $S_{n}$ and of its exterior powers. The ...
11
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0answers
345 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 ...
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 ...
10
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2answers
327 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", ...