Affine and cyclotomic Schur categories

TL;DR

This paper introduces diagrammatic affine and cyclotomic Schur categories over any commutative ring, providing bases, isomorphisms to cyclotomic Schur algebras, and conjecturing a higher level RSK correspondence.

Contribution

It formulates new diagrammatic categories for affine and cyclotomic Schur algebras, establishes bases, and connects these to cellular bases and conjectural combinatorial correspondences.

Findings

Constructed integral bases for affine and cyclotomic Schur categories.

Established a double SST basis leading to a conjectural higher level RSK correspondence.

Proved isomorphisms between endomorphism algebras with these bases and degenerate cyclotomic Schur algebras.

Abstract

Using the affine web category introduced in a prequel as a building block, we formulate a diagrammatic $\Bbbk$-linear monoidal category, the affine Schur category, for any commutative ring $\Bbbk$. We then formulate diagrammatic categories, the cyclotomic Schur categories, with arbitrary parameters at positive integral levels. Integral bases consisting of elementary diagrams are obtained for affine and cyclotomic Schur categories. A second diagrammatic basis, called a double SST basis, for any such cyclotomic Schur category is also established, leading to a conjectural higher level RSK correspondence. We show that the endomorphism algebras with the double SST bases are isomorphic to degenerate cyclotomic Schur algebras with their cellular bases, providing a first diagrammatic presentation of the latter. The presentations for the affine and cyclotomic Schur categories are much simplified when $\Bbbk$ is a field of characteristic zero.