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X=M Theorem: Fermionic formulas and rigged configurations under review

Anne Schilling
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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_{n-1}^{(1)}. In particular, the combinatorial approach via crystal bases and rigged configurations is discussed. Each section ends with a conglomeration of open problems.

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Anne Schilling. 2005-12-07. X=M Theorem: Fermionic formulas and rigged configurations under review. https://arxiv.org/abs/math/0512161

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