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 $U_q(\mathfrak{g})$ have been introduced in ([T.Nakashima, J. Algebra, vol.219, no. 2, (1999)]), which describe the crystal bases as sets of lattice points in the infinite $\mathbb{Z}$-lattice $\mathbb{Z}^{\infty}$ given by some system of linear inequalities, where $\mathfrak{g}$ is a symmetrizable Kac-Moody Lie algebra. To construct the polyhedral realization, we need to fix an infinite sequence $\iota$ from the indices of the simple roots. If the pair ($\iota$,$\lambda$) ($\lambda$: 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 $\iota$ is an adapted sequence (defined in our paper [Y.Kanakubo, T.Nakashima, arXiv:1904.10919]) then the pair ($\iota$, $\lambda$) satisfies the ample condition for any dominant integral weight $\lambda$ in the case $\mathfrak{g}$ is a classical Lie algebra. Furthermore, we reveal the explicit forms of the polyhedral realizations of the crystal bases $B(\lambda)$ associated with arbitrary adapted sequences $\iota$ in terms of column tableaux. As an application, we will give a combinatorial description of the function $\varepsilon_i^*$ on the crystal base $B(\infty)$.