Zero actions and energy functions for perfect crystals

Abstract

We give a combinatorial description of the action of the crystal operators ${\tilde{e}}_0$, ${\tilde{f}}_0$ on certain perfect crystals of $U_q^{\prime }(C_n^{(1)})$, $U_q^{\prime }(D_n^{(1)})$ and $U_q^{\prime }(D_{n+1}^{(2)})$, by means of Dynkin diagram automorphisms and Schensted column insertion. Also, for certain level 1 perfect crystals of these algebras we give a definition of a combinatorial “charge” related to the energy function on homogeneous paths in such crystals.