Perfect bases in representation theory: three mountains and their springs

TL;DR

This paper explores three different bases in representation theory that are compatible with Lie algebra actions, revealing their common combinatorial structures and introducing measures to distinguish them, with implications for cluster algebra interactions.

Contribution

It introduces measures supported on Mirkovic-Vilonen polytopes to distinguish three known bases and analyzes their interaction with cluster structures.

Findings

All three bases share the same crystal and polytope structures.

Measures help differentiate the bases despite their shared combinatorial shadow.

The bases interact with the cluster structure on the coordinate ring of the unipotent subgroup.

Abstract

In order to give a combinatorial descriptions of tensor product multiplicites for semisimple groups, it is useful to find bases for representations which are compatible with the actions of Chevalley generators of the Lie algebra. There are three known examples of such bases, each of which flows from geometric or algebraic mountain. Remarkably, each mountain gives the same combinatorial shadow: the crystal B(infty) and the Mirkovic-Vilonen polytopes. In order to distinguish between the three bases, we introduce measures supported on these polytopes. We also report on the interaction of these bases with the cluster structure on the coordinate ring of the maximal unipotent subgroup.