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

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