X=M for symmetric powers

The X=M conjecture of Hatayama et al. states that the generating function X of highest weight elements in tensor products of Kirillov-Reshetikhin crystals graded by energy can be expressed in terms of a fermionic formula M. In terms of physics, this identity expresses the equivalence between the corner-transfer method and the Bethe Ansatz. We will give a proof of this conjecture for tensor products of symmetric powers by proving a direct statistic preserving bijection for simply-laced algebras, and then use the method of virtual crystals and fermionic formulas to prove the conjecture for nonsimply-laced algebras. This is based on work in collaboration with Mark Shimozono (to appear soon).