Cell Modules for Type $A$ Webs

TL;DR

This paper studies the structure of cell modules in type A webs, introduces an orthogonal basis, and proves a conjecture about intersection forms in this context, advancing understanding of their algebraic properties.

Contribution

It constructs an orthogonal basis for cell modules in type A webs and proves Elias's conjecture on intersection forms for all type A cases.

Findings

Orthogonal basis for cell modules established

Determinants of Gram matrices computed explicitly

Conjecture on intersection forms proved for type A_n

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 $A_n$.