Adapted sequences and polyhedral realizations of crystal bases for highest weight modules

Abstract

The polyhedral realizations for crystal bases of the integrable highest weight modules of Uq(g) have been introduced in Nakashima (1999) [13], which describe the crystal bases as sets of lattice points in the infinite Z-lattice Z∞ given by some system of linear inequalities, where g is a symmetrizable Kac-Moody Lie algebra. To construct the polyhedral realization, we need to fix an infinite sequence ι from the indices of the simple roots. If the pair (ι,λ) (λ: a dominant integral weight) satisfies the ‘ample’ condition then there are some procedure to calculate the sets of linear inequalities.In this article, we show that if ι is an adapted sequence (defined in our paper [5]) then the pair (ι, λ) satisfies the ample condition for any dominant integral weight λ in the case g is a classical Lie algebra. Furthermore, we reveal the explicit forms of the polyhedral realizations of the crystal bases B(λ) associated with arbitrary adapted sequences ι in terms of column tableaux. As an application, we will give a combinatorial description of the function εi⁎ on the crystal base B(∞).