Part 4. $X=M$ Theorem: Fermionic Formulas and Rigged Configurations under Review

Abstract

We give a review of the current status of the $X=M$ conjecture. Here $X$ stands for the one-dimensional configuration sum and $M$ for the corresponding fermionic formula. There are three main versions of this conjecture: the unrestricted, the classically restricted and the level-restricted version. We discuss all three versions and illustrate the methods of proof with many examples for type $A^{(1)}_{n-1}$ . In particular, the combinatorial approach via crystal bases and rigged configurations is discussed. Each section ends with a conglomeration of open problems.