Decorated Geometric Crystals, Polyhedral and Monomial Realizations of Crystal Bases

TL;DR

This paper demonstrates that for type A_n, the geometric crystal-based realization of crystal bases aligns with polyhedral realizations, and explores relations with monomial realizations and decorations.

Contribution

It establishes the equivalence between geometric crystal realizations and polyhedral realizations for type A_n and investigates their relations with monomial realizations.

Findings

Geometric crystal realizations coincide with polyhedral realizations for type A_n.

Relations between decorations and monomial realizations are identified.

Provides a unified view of different crystal base realizations.

Abstract

We shall show that for type $A_n$ the realization of crystal bases obtained from the decorated geometric crystals intorduced by Berenstein and Kazhdan coincides with our polyhedral realizations of crystal bases. We also observe certain relations of decorations and monomial realizations of crystal bases.