# Crystals and the RSK Correspondence

## LACIM

Abstract

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.