The combinatorics of affine crystals and the energy function
Abstract: Crystals are colored directed graphs encoding information about Lie algebra representations. Certain (not highest weight) crystals for affine Lie algebras known as Kirillov-Reshetikhin (KR) crystals are graded by the energy function. Since crystals have various combinatorial models, it is desirable to compute the energy as a related statistic. With A. Postnikov we defined the so-called alcove model for (highest weight) crystals. I will present a generalization which is conjectured to model tensor products of KR crystals of arbitrary Lie type. The conjecture implies that a related statistic computes the energy. There is reasonable evidence for this conjecture. For instance, it is proved for Lie types A and C. I rephrase the energy statistic in type A as a well-known word statistic (the charge), while in type C I define a similar one. The talk contains joint work with Anne Schilling and Arthur Lubovsky, and is largely self-contained.