Affine and Cyclotomic Schur Categories

Abstract

Abstract Using the affine web category introduced in a prequel as a building block, we formulate a diagrammatic $$\Bbbk $$ k -linear monoidal category, the affine Schur category , for any commutative ring $$\Bbbk $$ k . We then formulate the cyclotomic Schur categories . 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 higher level RSK correspondence. We show that the path algebras with the double SST bases are isomorphic to degenerate cyclotomic Schur algebras of Dipper-James-Mathas with their cellular bases, providing a first diagrammatic presentation of the latter. The presentations for the affine and cyclotomic Schur categories are simplified for $$\Bbbk =\mathbb {C}$$ k = C .