Cell Modules for Type $A$ Webs
Stuart Martin, Robert A. Spencer
TL;DR
The paper analyzes the cellular structure of the type $A_n$ web category and its cell modules, linking Gram determinants to Elias' intersection-form conjecture via traces of higher-order Jones-Wenzl clasps. It develops a clasped-light-ladder basis and employs Howe duality to identify preimages of the orthogonal basis, ultimately proving Elias' conjecture for all $A_n$. The authors provide a concrete determinant formula in terms of edge-wise intersection forms and give explicit computations, illustrating the interplay between monoidal, cellular, and diagrammatic structures. These results advance the understanding of modular representation theory in web categories and offer explicit tools for computing cell-module determinants in quantum group settings.
Abstract
We examine the cell modules for the category of type An webs and their natural cellular forms. We modify the bases of these modules, as described by Elias, to obtain an orthogonal basis of each cell module. Hence, we calculate the determinant of the Gram matrix with respect to such bases. These Gram determinants are given in terms of intersection forms, computed from certain traces of clasps - higher order Jones-Wenzl morphisms. Additionally, the modified basis is constructed using these clasps, and each clasp is constructed using traces of smaller clasps. Elias conjectures a value for these intersection forms and verifies it in types $A_1$, $A_2$ and $A_3$. This paper concludes with a proof of the conjecture in type $A_n$.