Questions tagged [representation-theory]

For questions about representations or any of the tools used to classify and analyze them. A representation linearizes a group, ring, or other object by mapping it to some set of linear transformations. A common goal of representation theory is classifying all representations of some type. Representation theory is a broad field, so questions not including the word "representation" may be appropriate.

Representation theory is the study of mathematical objects via their symmetries. The broad idea is to study all the ways to "linearize" an object by reinterpreting its elements as a collection of matrices. This reduces problems in abstract algebra to problems in linear algebra, which are better understood, and leads to new questions in turn.

Specifically, given some algebraic object $A$, a representation of $A$ is a vector space $V$ and a a structure-preserving function $\rho \colon A \to \mathrm{End}(V)$, the ring of endomorphisms, or linear transformations, of $V$. That is, we're viewing elements of $A$ as matrices in order to "linearize" $A$ and understand its symmetries.

While some representations can tell us a great deal about the structure of $A$ in themselves, representation theorists often aim to understand all of the representations of $A$. How many different possible ways can $A$ be linearized? Usually the collection of representations of $A$ forms an abelian category. One of the most common goals in representation theory is to classify representations, or to understand the structure of the category of representations.

The usual first example that students see of representation theory is in the case of finite groups, where $A$ is a finite group, $\rho$ is a group homomorphism and $V$ is a finite-dimensional vector space over the complex numbers. However, $A$ may be a Lie group, an associative algebra, a quiver, or virtually any object with an "algebraic" structure. On the other hand, $V$ may be a vector space with coefficients in any field, a Hilbert space, a module, or virtually any object with a "linear" structure.

Modern representation theorists frequently employ tools from disparate areas of math, including category theory, harmonic analysis, homological algebra, combinatorics, and algebraic geometry. Their results are frequently useful in physics, geometry, and number theory.

8400 questions
9 answers

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 can think of two mathematical reasons for studying…
Eric O. Korman
  • 18,051
  • 3
  • 52
  • 82
1 answer

Application of Hilbert's basis theorem in representation theory

In Smalø: 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 of $d$-dimensional modules over an algebra…
Julian Kuelshammer
  • 9,352
  • 4
  • 35
  • 78
1 answer

Solving Special Function Equations Using Lie Symmetries

The Lie group and representation theory approach to special functions, and how they solve the ODEs arising in physics is absolutely amazing. I've given an example of its power below on Bessel's equation. Kaufman's article describes algebraic methods…
1 answer

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 for the Fermat variety $X_m^r$, defined by the…
6 answers

The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things must a person know before they can understand the Langlands program and its geometric analogue? What are the good books…
1 answer

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 here. The gist of the theory is as follows: The…
13 answers

What's a good place to learn Lie groups?

Ok so I read the following article the other day: and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought "oh it must have something to do with groups" So I picked up a copy of Dummit…
1 answer

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 some cases, there are so many kinds of orbits of G on…
Jack Schmidt
  • 53,699
  • 6
  • 93
  • 183
3 answers

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 theory of $G$ over $K$, as for instance if…
5 answers

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?
  • 643
  • 6
  • 5
2 answers

Why are (representations of ) quivers such a big deal?

Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, Gabriel gives a complete classification of…
13 answers

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.
  • 13,820
  • 17
  • 72
  • 125
3 answers

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 $k[G]$ is a direct sum of matrix algebras, one for…
Qiaochu Yuan
  • 359,788
  • 42
  • 777
  • 1,145
2 answers

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 $n\in\mathbb{N}$ such that there is an embedding (injective…
Jack D'Aurizio
  • 338,356
  • 40
  • 353
  • 787
8 answers

Why do we care about two subgroups being conjugate?

In classifications of the subgroups of a given group, results are often stated up to conjugacy. I would like to know why this is. More generally, I don't understand why "conjugacy" is an equivalence relation we care about, beyond the fact that it…
2 3
99 100