Title: Affine and cyclotomic $q$-Schur categories via webs Show Chinese title

Title: 仿射和分圆 $q$-Schur 范畴通过网络

Abstract: We formulate two new $\mathbb Z[q,q^{-1}]$-linear diagrammatic monoidal categories, the affine $q$-web category and the affine $q$-Schur category, as well as their respective cyclotomic quotient categories. Diagrammatic integral bases for the Hom-spaces of all these categories are established. In addition, we establish the following isomorphisms, providing diagrammatic presentations of these $q$-Schur algebras for the first time: (i)~ the path algebras of the affine $q$-web category to R.~Green's affine $q$-Schur algebras, (ii)~ the path algebras of the affine $q$-Schur category to Maksimau-Stroppel's higher level affine $q$-Schur algebras, and most significantly, (iii)~ the path algebras of the cyclotomic $q$-Schur categories to Dipper-James-Mathas' cyclotomic $q$-Schur algebras. Show Chinese abstract

Abstract: 我们构造了两个新的$\mathbb Z[q,q^{-1}]$-线性图示单子范畴，即仿射$q$-网络范畴和仿射$q$-Schur 范畴，以及它们各自的循环商范畴。 建立了所有这些范畴的 Hom 空间的图示整数基。 此外，我们建立了以下同构，首次提供了这些$q$-Schur 代数的图示表示：(i) 仿射$q$-web 范畴的路径代数到 R. Green 的仿射$q$-Schur 代数，(ii) 仿射$q$-Schur 范畴的路径代数到 Maksimau-Stroppel 的高阶仿射$q$-Schur 代数，以及最重要的是，(iii) 交代$q$-Schur 范畴的路径代数到 Dipper-James-Mathas 的交代$q$-Schur 代数。