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VOL. 17 | 2007 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.

## Information

Published: 1 January 2007
First available in Project Euclid: 24 November 2014

zbMATH: 1221.17018
MathSciNet: MR2269128

Digital Object Identifier: 10.2969/msjmemoirs/01701C040