The representation theory of the symmetric group is an old and rich subject. The modern perspective was developed by Gordon James in the 1970s. The Iwahori-Hecke algebra of type A is a deformation of the symmetric group, and one motivation for studying it is that it provides a bridge between the representation theory of the symmetric and general linear groups.
The Specht modules are a family of modules of fundamental importance for the Iwahori-Hecke algebra, and an open problem is to determine which Specht modules are decomposable. I will be discussing my attempts to answer this using a very new approach via Khovanov-Lauda-Rouquier algebras. These algebras at first glance seem completely different from the Iwahori-Hecke algebra, but an amazing result of Brundan and Kleshchev is that the Iwahori-Hecke algebra is isomorphic to a certain KLR algebra. I will explain how using the KLR algebra approach makes Specht modules easier to understand.