Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not know much about what is a repersentation (say knows the definition). The standard application that is usually mentioned is Burnside's theorem http://en.wikipedia.org/wiki/Burnside_theorem. The application may be of any kind, not necessarely in math. But math applications are of course very wellcome too!!! It will be very helpfull also if you desribe a bit this application.

7$\begingroup$ This should be biglist and comunitywikified, I guess... $\endgroup$– Mariano SuárezÁlvarezJan 14 '10 at 22:17

1$\begingroup$ Biglist is a classification of a "type of question", not the number of resopnses it receives. $\endgroup$– Harry GindiJan 14 '10 at 22:54

1$\begingroup$ The Erlangen program might be relevant. $\endgroup$– AnweshiJan 15 '10 at 0:17

1$\begingroup$ For a bit more see the cousin of this thread in Math.SE. $\endgroup$– Jyrki LahtonenJul 18 '18 at 20:06

$\begingroup$ In chemistry, you can use representation theory to decompose vibrations of a fairly symmetric molecule into modes. An example: consider a twoatom molecule in one dimension, whose atoms are vibrating. The motion can be decomposed into samedirection and oppositedirection vibrations. When you work it out, this corresponds to the projection of a representation $V$ of $\mathbf{Z}/2$ (the tensor product of $\mathbf{C}^2$ coming from the action on the atoms, and $\mathbf{C}$ coming from the action on the ambient space) into its irreducible components. $\endgroup$– MeowJun 12 '19 at 12:16
An example from Kirillov's book on representation theory: write numbers 1,2,3,4,5,6 on the faces of a cube, and keep replacing (simultaneously) each number by the average of its neighbours. Describe (approximately) the numbers on the faces after many iterations.
Another example I like to use in the beginning of a group reps course: write down the multiplication table in a finite group, and think of it as of a square matrix whose entries are formal variables corresponding to elements of the group. Then the determinant of this matrix is a polynomial in these variables. Describe its decomposition into irreducibles. This question, which Frobenius was asked by Dedekind, lead him to invention of group characters.
A function in two variables can be uniquely decomposed as a sum of a symmetric and antisymmetric (skewsymmetric) function. What happens for three and more variables  what types of symmetries do exist there?

1$\begingroup$ Can you provide reference information for Kirillov's book? $\endgroup$– alexJan 16 '10 at 1:32

4$\begingroup$ @alex: Elements of the theory of representations, Springer, 1976 $\endgroup$ Jan 18 '10 at 23:42
As Anweshi noted a moment ago, a classic answer is the use of character tables by chemists (as explained in this book for instance). The symmetry group of a molecule controls its vibrational spectrum, as observed by IR spectrosocopy. When Kroto et al. discovered $C_{60}$, they used this method to demonstrate its icosahedral symmetry.

2$\begingroup$ Also of interest is the use of representation theory for finite groups in understanding electronic structure: en.wikipedia.org/wiki/Term_symbol $\endgroup$ Jan 14 '10 at 22:58

$\begingroup$ Ah thanks a lot for the reference and explanations! You have a soft corner for Chemistry? $\endgroup$– AnweshiJan 14 '10 at 23:45

6$\begingroup$ Chemists really have courses on group representation theory, I had a friend who was a chemist who taught one. That said, if you have to know too much chemistry it might not be so easy to incorporate in a mathematics class! $\endgroup$ Jan 15 '10 at 1:16

$\begingroup$ The first part of Serre's book on representations of finite groups is written explicitly for chemists. When I saw a copy the other day, I was ashamed to see that a chemist who followed his course knew more about representations of finite groups than I do! $\endgroup$ Feb 2 '10 at 12:38
I'm surprised no one's mentioned this: the fact that a Frobenius kernel is a normal subgroup. It's not clear a priori that the set of elements not fixing any points should be a subgroup at all. I'm told there is no completely group theoretic proof (with no character theory) of this fact yet.
I love the proof of the theorem of Hurwitz that a normed division algebra has to have dimension 1, 2 or 4 using the representation theory of elementary 2groups.
Later: the original reference for the argument is [Eckmann, Beno. Gruppentheoretischer Beweis des Satzes von HurwitzRadon über die Komposition quadratischer Formen. (German) Comment. Math. Helv. 15, (1943). 358366. MR0009936 (5,225e)] I don't know of a more recent exposition, except some notes of a short course by Esther Galina a few years ago (which should be in her webpagein spanish, though)


6$\begingroup$ Here is a more recent reference: math.uconn.edu/~kconrad/blurbs/linmultialg/hurwitzrepnthy.pdf $\endgroup$– KConradJan 23 '10 at 6:06

$\begingroup$ Very interesting! Was it the original proof? $\endgroup$ Jan 24 '10 at 17:50

1$\begingroup$ Not at all. Hurwitz's paper was in 1898 and Frobenius introduced characters of finite groups only in 1896. It took more than 2 years for representation theory to develop the tools necessary for Eckmann's argument to be found. Hurwitz's original proof used linear algebra. For a writeup of that, look at math.uconn.edu/~kconrad/blurbs/linmultialg/hurwitzlinear.pdf $\endgroup$– KConradJan 25 '10 at 4:39
Serre mentions in his book that the first part and examples are all very relevant to quantum chemists. If you can dig it up, it might turn out to be very exciting. Perhaps, it is for understanding crystal structure, etc..

$\begingroup$ Anweshi, thanks! I have the book of Serre "Linear Representations of Finite Groups" in my hands. Graduate Texts in Math. Serre mentions "quatnum Chemistry" in one phrase, but that is it... Is this the book? $\endgroup$ Jan 14 '10 at 23:05

1$\begingroup$ That is the book. The first part is written for Serre's wife, Josiane Serre, for teaching her students in quantum chemistry. Somehow it is all very relevant there, though I have no idea. Maybe someone else can explain it better, with references to Chemistry books. $\endgroup$– AnweshiJan 14 '10 at 23:42

$\begingroup$ I never understood Serre's phrase "quantum chemistry", let alone "quantum chemists". What other kinds are there  "deluded chemists"? $\endgroup$ Jan 14 '10 at 23:44

8$\begingroup$ @Tim Just because a mathematician does representation theory doesn't mean they think set theory is untrue, or that it isn't an underlying basis for their work. It's just that most of the set theory they see is boring, and other interesting problems dominate. I would guess the chemistry thing is analogous. $\endgroup$– Ben Webster ♦Jan 15 '10 at 16:37

4$\begingroup$ Quantum chemistry is the application of quantum mechanics to chemical problems, i.e. solving the Schrodinger equations for collections of atoms and molecules. It can be considered a subset of physical chemistry, which aims to reconcile what physics tells us about the universe with what chemists see molecules do. The boundary dividing it from chemical physics and molecular physics is not always meaningful. The examples of Chapter 5 are what us chemists call point groups. They are important to understand the spectra of molecules with symmetry, such as methane (Td), C60 (Ih), and benzene (D6h). $\endgroup$ Aug 19 '10 at 5:13
There's lots of stuff relating the representation theory of the symmetric group to sorting and shuffling. Persi Diaconis has worked on the latter to great effect.
Maybe this is not an "application", but I certainly found it fun when learning about representations of the symmetric group. Combinatorially, there is a clear correspondence between transposeinvariant partitions (ie partition diagrams that are symmetric along the main diagonal) and partitions involving distinct odd integers. (eg (3,2,1) corresponds to (5,1). )
Here is a sketch of the representation theory behind the result: the Specht modules from transposeinvariant partitions are precisely the irreducible representations of S_n that decompose into 2 irreducible representations when restricted to A_n. We view irreducible characters and conjugacyclassindicatorfunctions as two bases on the vector space of A_n class functions, and deduce that the number of these reducibleonrestriction representations is equal to the number of S_n conjugacy classes which split as two A_n conjugacy classes. An S_n conjugacy class splits into two A_n conjugacy classes precisely when it doesn't commute with any odd cycles, which is to say all factors of its cycle decomposition have distinct odd length.
I've seen other instances where representation theory "explains" a combinatorial coincidence (eg qdimension formula of various Lie algebras), so I think of this example as "typical" of the connection between representation theory and combinatorics.
EDIT: The background comes from pages 1825 of Anton Evseev's lecture notes here, and this particular statement is exercise 1 of sheet 3. I have what I believe is a complete solution to the exercise, but I'm having a little trouble pdfing it, hopefully it should be on my homepage (under "writings") by Tuesday (along with my own exposition of the background, which may expand on those official notes a little).

6$\begingroup$ I once told a friend of mine who was a logician about the tensor product in the representations of the symmetric group. She exclaimed "You mean I can multiply partitions! This is amazing!" $\endgroup$– Ben Webster ♦Jan 16 '10 at 15:57

This is maybe stretching it a little bit, but Tim Gowers' quasirandom groups describes and references some extremal combinatorial properties of graphs constructed from the groups $PSL_2(\mathbb{F}_q)$ which ultimately rely on the fact that they have no nontrivial lowdimensional irreducible representations.
Let $G$ be a finite group with $n$ elements and $k$ conjugacy classes. Denote by $m=G:[G,G]$ the index of the commutator. Then $n+3m\geq 4k$.
It is less impressive than many other answers, but I find this inequality particularly nice, especially having in mind that there are some nontrivial examples of equality, all are explicitly listed. I do not know the proof whithout using representations.

1

5$\begingroup$ At first, I may give a proof. There are $k$ irreducible representations of $G$, $m$ of them have dimension 1, the others have dimension at least 2. Sum of squares of dimensions equals $n\geq m+4(km)$ as desired. At second, classification of equality case, i.e. groups with at most 2dimensional irreducible representations, is given here: Amitsur, S.A. Groups with representations of bounded degree. II. (English) [J] Ill. J. Math. 5, 198205 (1961). [ISSN 00192082] emis.de/cgibin/Zarchive?an=0100.25704 $\endgroup$ Mar 25 '15 at 12:56
One very basic and fun application of representations of finite groups (or really, actions of finite groups) would be the study of various puzzles, like the Rubik Cube. David Singmaster has a nice little book titled "Handbook of Cubik Math" which could potentially be used for material in an undergraduate course.

$\begingroup$ Also relevant might be answers and comments on my last question: mathoverflow.net/questions/11758/… $\endgroup$ Jan 15 '10 at 1:44
Here's a blog post I wrote, based on Georgi's book. The example is solving for the normal modes of oscillation of a system of identical masses and identical springs. More generally, you can use the automorphism group of the graph they form to do it for more complicated configurations.
There are some fun problems in the beginning of these notes by Vera Serganova.
At the Joint Meetings, I heard a fun and very interesting talk by Michael Orrison on applications of representation theory in voting theory. It was really neat! You should be able to find out more here.
The decomposition of the curvature tensor of a (pseudo) riemmanian manifold into scalar+ traceless Ricci + Weyl (the latter into SD+ASD in dim=4) is an application of the representation theory of the orthogonal group. There are many more examples in differential geometry (eg the decomposition of the intrinsic torsion tensor of an almost hermitian manifold into 4 irreducibles etc).
Now you may object because the orthogonal group (say over R) is not a finite group, but Weyl showed that the theory of the tensor representations of the classical groups is intimately related to the representation theory of the symmetric group.

5$\begingroup$ Not to mention that any reasonable person knows that finite should just mean compact. $\endgroup$– Ben Webster ♦Jan 16 '10 at 15:56
Bosons and fermions. Quantum mechanics texts, such as Dirac's classic, explain that in a system of indistinguishable particles in space, exchange of particles is modelled by a change in phase of the state vector. These phases form a 1dimensional representation of the symmetric group. Since all transpositions are conjugate, there are just two possibilities: bosons (trival rep) and fermions (sign rep), and no other(on)s.

3$\begingroup$ I think this only holds in R^n for n>2, since you're representing the fundamental group of the configuration space of points in R^n. For n=2, you can get anyons. $\endgroup$– S. Carnahan ♦Jan 15 '10 at 8:16

$\begingroup$ Little gets past MO's collective eyes!:) Yes, my phrase "in space" was a little more pointed than it appeared... $\endgroup$ Jan 15 '10 at 14:41
I would like to add the McKay correspondence. The symmetry groups of regular solids are easy groups to introduce. Then you want the double covers. It amazes me you can construct the irreducibles and characters so simply from the twodimensional.
Another result I like is Molien's theorem. The action on the polynomial ring seems complicated at first sight. However this is a straightforward way to calculate the dimensions of the spaces of invariant polynomials.
Ising gauge theory on a finite lattice is basically determined by a coupling constant and a gaugeinduced unitary representation from $\mathbb{Z}^M_2$ to $U(\mathcal{H}_2^{\otimes L})$.
Here $\mathcal{H}_2$ is the Hilbert space of a single spin variable, $L$ is the number of links in the lattice, and $M$ is the number of them that comprise a maximal tree (for a periodic $d$dimensional lattice with period $N$ in each dimension, so that there are $N^d$ sites, $M = N^d – 1$ and $L = dN^d$, so in the infinitevolume limit, $L \sim d \cdot M$).

$\begingroup$ Steve, thanks a lot! Though it is a bit dense for me to understand :)... Could you please expand it just a bit, or give a refference? For example, are you considering a periodic lattice in R^2? $\endgroup$ Jan 14 '10 at 22:43

$\begingroup$ I have some notes on this from four or five years ago that elaborate on work from the seventies (e.g. Fradkin, E. and Susskind, L. “Order and disorder in gauge systems and magnets”. Phys. Rev. D 17, 2637 (1978)). I will email you a PDF of these notes if you want. $\endgroup$ Jan 14 '10 at 22:48

$\begingroup$ Steve, please email me, this will be nice! dpanov@imperial.ac.uk $\endgroup$ Jan 14 '10 at 22:57

$\begingroup$ Done. Hope you find them worthwhile. Best/SH $\endgroup$ Jan 14 '10 at 23:03
Wallpaper groups and the crystallographic restriction theorem for the plane are a wonderful application/example of finite group theory and group actions.
This is a really good relevant clip: http://www.youtube.com/watch?v=7zLi47yYlcc#t=7m43s (queued up at the relevant point, whence came the #t=7m34s).
which continues: http://www.youtube.com/watch?v=xP52g6eQRmY&feature=related
Also, Bronowski was a mathematician.
You might want to look at Section 3.1 of "Group Theory and Physics" by Shlomo Sternberg, Cambridge Univ Press, 1994. This explains, through a simple example, how (in Sternberg's words) "molecular spectroscopy is an application of Schur's lemma". The argument is elementary in nature. The last chapter of the book by James & Liebeck (Representations and Characters of Groups 2e, Cambridge Univ Press, 2001) is a longer exposition of the same idea. I notice another post here about work by Diaconis  Diaconis has a book called "Group Representations in Probability and Statistics" which is available for free download. See the link at
http://www.math.columbia.edu/~khovanov/resources/
This page has links to dozens of useful articles. Also, there is the book "Unitary Group Representations in Physics, Probability and Number theory" by George W Mackey (Benjamin/Cummings Publ Co, 1978). This is more advanced than the others though. For applications to quantum chemistry there is (amongst many) "Chemical Applications of Group Theory" by F Albert Cotton, published by John Wiley. If you want to see how Section 2.7 of Serre's book is actually used in practice by chemists, see Chapter 6 in Cotton's book.

$\begingroup$ Apologies for the intrusive question. Just to satisfy curiosity  Are you the author of "“An arithmeticgeometric method in the study of the subgroups of the modular group”? $\endgroup$– AnweshiJan 24 '10 at 11:50

$\begingroup$ No, I am not. It is a fairly common name... :) $\endgroup$ Jan 24 '10 at 14:56

$\begingroup$ @Anweshi Ravi Kulkarni taught at Queens College at the City of New York as well as the Graduate Center of CUNY for nearly 40 years after getting his PHD at Harvard under Shlomo Sternberg.He's an expert in differential and classical geometry,as well as complex analysis.I had the priviledge of being his student on more then one occasion. Sadly,my role as his student did not match his as a teacher the final time.I'm happy to report for future mathematics majors that he's back there after several years in his native India.I'm hoping he'll remain there until his official retirement. $\endgroup$ Oct 5 '10 at 22:08

$\begingroup$ @Ravi First,welcome back,I didn't get a chance to get caught up with you at length yet.Secondto your excellent answer and recommendations,I'll add only the following: "Elements of Molecular Symmetry" by Yngve Ohm. This is a graduate level presentation of group representation theory for chemists that's not only much more readable then Cotton,but much more mathematicalit develops a great deal of formal group theory along the way.Both Sternberg and Serre should be in every mathematican's,physicist's,and chemist's library in my opinion. $\endgroup$ Oct 5 '10 at 22:14