Affine and cyclotomic Schur categories

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.