Decorated Geometric Crystals and Polyhedral Realizations of type D_n

TL;DR

This paper demonstrates that for type D_n, the realization of crystal bases from decorated geometric crystals aligns with their polyhedral realizations, unifying two approaches in crystal basis theory.

Contribution

It establishes the equivalence between geometric crystal realizations and polyhedral models for type D_n, advancing understanding of crystal base structures.

Findings

Realization from decorated geometric crystals matches polyhedral realizations for type D_n.

Unification of two different crystal base realization methods.

Provides a foundation for further exploration of crystal bases in Lie algebra types.

Abstract

This is a continuation of [15, 16]. We shall show that for type D_n the realization of crystal bases obtained from the decorated geometric crystals in [2] coincides with the polyhedral realizations of crystal bases.