Crystals and the RSK Correspondence

The theory of crystal bses allows algebraic questions about representations of semisimple Lie algebras to be answered by purely combinatorial means. For example, the decomposition of a tensor product of represenations into its irreducible components can be read off from the connected components of the corresponding crystal graph. I will give an introduction to crystal bases in Type $A$, focusing on the example of $\text{SL}_m$ and $\text{SL}_n$ crystal operators acting on $m\times n$ matrices. We will see that the Robinson--Schensted--Knuth (RSK) correspondence plays a crucial role in understanding this example.