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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 felt a bit let down. Not in the sense that the material was not beautiful because it very much was but in the sense that I wanted to prove things about groups somehow using representations (not facts about the representations themselves like constraints on the dimension of the subrepresentations and such).

My question is this:

What are some classical examples of properties we specifically prove about a group from representation theory that we might not be able to do otherwise?

We had a couple of problems in which we were to prove facts about very specific instances of groups based on the representations but they could all be done with group theoretic notions anyway; I'm looking for maybe a more fundamental paradigm underlying representation theory.

Perhaps the subtext to my post is this:

Why study the representation theory of finite groups?

I vaguely understand why we do it for locally compact groups since they can be much harder to handle, but for finite groups, it seems like beautiful math but not necessary for the understanding of groups.

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Representation theory is a fundamental tool in studying the internal structure of finite groups. For instance, read Gorenstein's textbook on Finite Groups. Rep theory is chapter 3; it is used in the next 14 chapters. –  Jack Schmidt Jan 31 at 14:32
    
I'll have to check that text out. I guess my course was so focused on representations we lost sight of actual group theory. Thanks! –  Cameron Williams Jan 31 at 16:32

3 Answers 3

up vote 6 down vote accepted

To address the stated subtext of your post: many people, myself included, take the position that groups are important because they act on things. A representation is just a group action on a vector space (by linear operators). And whenever you have a group action, even if it isn't on a vector space, there is often a closely related representation lurking nearby.

For example, if $G$ is acting on a finite set $S$ then there is an induced representation on $\mathbb{F}^S$ for any field $\mathbb{F}$. More generally, if $G$ is acting on a space $X$ then there is an induced representation of $G$ on the algebra $\mathcal{O}(X)$ of functions on $X$, whatever we mean by "function" in this situation (e.g. perhaps $X$ is a topological space on which $G$ acts by homeomorphisms, and $\mathcal{O}(X)$ is the algebra of continuous real-valued functions on $X$).

To elaborate on this last point: suppose $X$ is a set on which $G$ acts. For any set $Y$, there is an induced action on the set $Y^X$ of maps from $X$ to $Y$: for $g\in G$ and $f:X\to Y$, the map $g\cdot f$ is defined by $$(g\cdot f)(x) = f(g^{-1}x).$$

If we equip $X$ with some additional structure, like a topology, we may insist that the action of $G$ preserve this structure. This guarantees that the action of $G$ on maps will take structure-preserving maps to structure-preserving maps. As an example, if $X$ and $Y$ are topological spaces and $G$ acts on $X$ by homeomorphisms, then the action of $G$ on $Y^X$ sends continuous maps to continuous maps. In particular, taking $Y=\mathbb{R}$, we have an action of $G$ on the algebra of real-valued continuous functions.

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I like this answer a lot. I just gave a talk on cyclic sieving and the idea of groups acting on the induced vector space by a finite set is pervasive. I hadn't thought of representations in the more general context you provided though. Can you elaborate? Particularly, why would $G$ act by homeomorphism? Is it because - since we don't have groups - we restrict ourselves to what we do have, which are open sets, and we want to have $G$ be structure preserving (mimicking group homomorphisms)? –  Cameron Williams Jan 31 at 5:45
    
It is indeed because we want the action of $G$ to preserve the topological structure. I'll edit my answer to elaborate on this. –  Brad Jan 31 at 17:15

Two examples of purely group-theoretic theorems proved using representation theory are:

  1. Burnside's theorem, which states that a finite group having no more than 2 distinct prime divisors must be solvable. (Much later, a proof without representation theory was found in the 1970's.)

  2. Theorem about Frobenius groups: Assume a finite group $G$ contains a subgroup $H$ such that for every $x \in G - H$, the intersection $H \cap x^{-1}Hx = \{1\}$. Then the set of elements of $G$ not in any conjugate of $H$, together with the identity, form a normal subgroup of $G$. No proof not using representation theory is known.

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I had a feeling Burnside would show up as an answer but I hadn't seen Frobenius groups before but the setting is very similar to something we did for Frobenius reciprocity. Thanks for the answer! –  Cameron Williams Jan 31 at 5:37
    
H. Bender found a very simple, purely group-theoretic proof for 2. in case that $H$ as even order. A Fourier-analytic proof can be found here. –  j.p. Jan 31 at 17:33
    
A short, character-free proof for H of even size was given by Burnside in 1898 in ¶105 page 143 (or ¶134 page 172 in the 2nd ed). I don't believe Bender ever did this, but if he did it would be at least 60 years later. The proof given Terry Tao is Burnside's proof. –  Jack Schmidt Jan 31 at 19:40
    
I think the confusion might be that Bender was the one that gave a character-free proof of number 1, rather than number 2 (note that the character-free proof is extremely long compared to the original one). –  Tobias Kildetoft Feb 1 at 10:55
    
@TobiasKildetoft: Purely group-theoretic proofs of Burnside's theorem were given by Goldschmidt (1970, odd), Bender (1972, general) and Matsuyama (1973, even). But Bender provided also a purely group-theoretic proof for Frobenius groups with $H$ of even order. –  j.p. May 15 at 11:17

Just a few unsorted comments to add to what's been said above...

  1. The classification of finite simple groups relied heavily on modular representation theory, so there is that. Many of the sporadics are only described reasonably through their representations.

  2. Symmetric groups have nicely describable representations that correspond to a combinatorics construct called Young Tableaux. You can also determine certain things about conjugacy classes in symmetric groups using pretty basic representation theory, for example rational conjugacy classes have a basic, group theoretic description.

  3. More generally, there are a lot of concepts in finite group theory which generalize to representation theory and are often easier to understand there. Transfer, for example, can be realized as the determinant of an induced representation over $\mathbb{F}_2[G]$.

  4. This is related to Brad's answer: in particular, we are often concerned about groups acting on other groups, particularly subgroups of some supergroup. If the group being acted upon is abelian, we can use representation theory over finite fields to describe what's going on. This is particularly important in the study of solvable groups, for which the minimal normal subgroups are elementary abelian.

  5. A lot of computational group theory is done using representations, so this is one way representation theory is useful for its own sake. You don't want to be doing computations Cayley's theorem style in a group with a large number of elements.

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Adding to the part about symmetric groups: Combining that with Schur-Weyl duality relates those to representations of $GL_n(\mathbb{C})$, and this results in a very nice proof that certain numbers (Littlewood-Richardson coefficients) which show up when studying symmetric functions are in fact non-negative (they turn out to be composition multiplicities of certain simple modules in some tensor products). –  Tobias Kildetoft Jan 31 at 8:51

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