# Cell Modules for $A_n$ Webs

### Stuart Martin and Robert A. Spencer

**Abstract:**We examine the cell modules for the category of type $A_n$ webs and their natural cellular forms. We modify the bases of these modules, due to Elias, to form an orthogonal basis of each cell module. Hence, we calculate the determinant of the Gram matrix in these bases. These Gram determinants are given in terms of 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. There is a conjectured value for these traces, or “intersection forms” given by Elias and this paper concludes with a proof of the conjecture in type $A_n$. Thus the results on the Gram matrices are exact

A draft of this work is available on the ArXiv.

#### Cite with BibTex

```
@misc{martinspencer2022cellmodules,
doi = {10.48550/ARXIV.2210.09639},
url = {https://arxiv.org/abs/2210.09639},
author = {Martin, Stuart and Spencer, Robert A.},
keywords = {Representation Theory (math.RT), FOS: Mathematics, FOS: Mathematics},
title = {Cell Modules for $A_n$ Webs},
publisher = {arXiv},
year = {2022},
copyright = {arXiv.org perpetual, non-exclusive license}
}
```