Comparison of canonical bases for Schur and universal enveloping algebras

TL;DR

This paper demonstrates the compatibility of canonical bases in quantum groups and Schur algebras, extending to p-canonical bases, through categorification and connections between various algebraic and geometric frameworks.

Contribution

It establishes the compatibility of canonical and p-canonical bases in Schur and quantum algebras via a categorification approach, linking multiple categorification methods.

Findings

Canonical bases in $ ext{U}(rak{sl}_n)$ and Schur algebra are compatible.

Extension of compatibility results to p-canonical bases.

Connections between categorifications from parity sheaves, Soergel bimodules, and flag categories.

Abstract

We show that canonical bases in $\dot{U}(\mathfrak{sl}_n)$ and the Schur algebra are compatible; in fact we extend this result to $p$-canonical bases. This follows immediately from a fullness result from a functor categorifying this map. In order to prove this result, we also explain the connections between categorifications of the Schur algebra which arise from parity sheaves on partial flag varieties, singular Soergel bimodules and Khovanov and Lauda's "flag category," which are of some independent interest.