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

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

2005-12-07

arxiv.org/abs/math/0512161v1

MSJ Memoirs 17 (2007) 75-104 (published by the Mathematical Society of Japan)

Mathematics

Quantum Algebra

31 pages, 1 figure, axodraw and youngtab style file necessary

Scientific paper

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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