Polyhedral realizations for crystal bases and Young walls of classical affine types

For affine Lie algebra g of type A⁽¹⁾ₙ₋₁, B⁽¹⁾ₙ₋₁, C⁽¹⁾ₙ₋₁, D⁽¹⁾ₙ₋₁, A⁽²⁾₂ₙ₋₂, A⁽²⁾₂ₙ₋₃ or D⁽²⁾ₙ, let B(λ) and B(∞) be the crystal bases of integrable highest weight representation V(λ) and negative part Uq⁻(g) of quantum group Uq(g). We consider the polyhedral realizations of crystal bases, which realize B(λ) and B(∞) as sets of integer points of some polytopes and cones in R∞. It is a natural problem to find explicit forms of the polytopes and cones. In this paper, we introduce pairs of truncated walls, which are defined as modifications of level $2$-Young walls and describe inequalities defining the polytopes and cones in terms of level $1$-proper Young walls and pairs of truncated walls. As an application, we also give combinatorial descriptions of εₖ^*-functions on B(∞) in terms of Young walls and truncated walls.