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arXiv · 2504.10270

Affine and cyclotomic $q$-Schur categories via webs

Yaolong Shen·Linliang Song·Weiqiang Wang
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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.

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Yaolong Shen, Linliang Song, Weiqiang Wang. 2025-04-14. Affine and cyclotomic $q$-Schur categories via webs. https://arxiv.org/abs/2504.10270

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