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.

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? 


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. 


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. 


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.. 


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. 


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) 


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 the lecture notes here: http://www.dpmms.cam.ac.uk/~ae284/characters.html , 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). 


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. 


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. 


Check out this book: Group theory and physics 


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. 


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. 


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. 


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. 


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, $L \sim d \cdot M$). 


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
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. 


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. 

