Affine PBW Bases and Affine MV polytopes

TL;DR

This paper demonstrates how affine PBW bases can be used to construct affine MV polytopes, establishing their equivalence with recently developed models via algebraic and combinatorial methods.

Contribution

It generalizes the construction of affine PBW bases for arbitrary convex orders and clarifies their relation to affine MV polytopes, answering a key open question.

Findings

Affine PBW bases can be used to construct affine MV polytopes.

The constructed polytopes agree with those from preprojective and KLR algebra approaches.

Relations between affine PBW bases for different convex orders are explicitly described.

Abstract

We show how affine PBW bases can be used to construct affine MV polytopes, and that the resulting objects agree with the affine MV polytopes recently constructed using either preprojective algebras or KLR algebras. To do this we first generalize work of Beck-Chari-Pressley and Beck-Nakajima to define affine PBW bases for arbitrary convex orders on positive roots. Our results describe how affine PBW bases for different convex orders are related, answering a question posed by Beck and Nakajima.