It is steep, but at the same time, representation theory is so damn beautiful; for anyone with the time and interest to pursue a pure mathematical topic as a ho

It is steep, but at the same time, representation theory is so damn beautiful; for anyone with the time and interest to pursue a pure mathematical topic as a hobby, I highly recommend combinatorial representation theory. Besides, representation theory seems to be of great importance to computer science. The first explicit examples of expander graph families, for instance, were given by Grigori Margulis using deep theorems from unitary representation theory (see A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures for instance). Regarding the sentence you posted, it is a bit less scary than it sounds. Let's break down the concepts. Symmetric group of order N: it is the collection of all rearrangements of N distinct symbols on a line. This collection satisfies the axioms of a permutation group, which very roughly means a self-contained collection of permutations. Representations of... (a group): Abstract groups are opaque objects in general, but some particular groups tend to express information vividly in their elements and structure. The most computationally rich example is groups of linear transformations on a vector space: rotations, reflections, shears etc. A linear representation of an abstract group is a realization of the group as a group of linear transformations: you pick a vector space, and let each group element correspond to some linear transformation on the vector space, so that the product of group elements corresponds to the composition of linear transformations. Now you can use the power of linear algebra to study your abstract group. Irreducible representations of... (a group): Now that you have realized your group as a group of transformations on a vector space, maybe you get the following behavior: the group takes a subspace of that vector space and transforms it into itself. And takes the complementary part of that vector space, and transforms that into itself. Well, then you can study the two subspaces separately because they are never "mixed" by the group. That is called a reduction, and such a representation is called reducible. If you cannot do that, you have an irreducible representation. These act like the prime numbers of representation theory, at least where finite groups are concerned: every representation splits as a sum of irreducibles, and the deepest linear information about the group is carried in them. Branching properties: so now you have an irreducible representation of a symmetric group. Every group of permutations of N elements is a subgroup of that group, so what happens if you focus your attention to the subgroup, and forget about all the transformations that are not in the subgroup? The resulting representation may not be irreducible anymore: some transformations that mixed vector spaces may not be in the smaller group. If it is not irreducible, how does it split into irreducible representations? In particular, how many irreducible components are there and how many times does each component appears? The answer to those questions are the branching rules for the restricted representation to the subgroup. In general, it is an extremely difficult problem, but it has been resolved for some families of groups, the symmetric groups being a primary example (where the rules are given in terms of, well, Young tableaux and the linked Young lattice!).