Data

• Title: Orthogonal webs and semisimplification
• Authors: Elijah Bodish and Daniel Tubbenhauer
• Status: preprint. Last update: Mon, 1 Jan 2024 09:13:45 UTC
• ArXiv link: https://arxiv.org/abs/2401.00704

Abstract

We define a diagrammatic category that is equivalent to tilting representations for the orthogonal group. Our construction works in characteristic not equal to two. We also describe the semisimplification of this category.

A few extra words

In this paper we define a diagrammatic category of orthogonal webs that is equivalent to tilting representations of the orthogonal group.
A closed orthogonal pre-web is a trivalent graph with edges labeled with integers \(\{1,\dots,N\}\) such that we have \(k\), \(l\) and \(k+l\) around every trivalent vertex. A closed orthogonal web is an immersion of a closed orthogonal pre-web such that each point of intersection is a crossing in the usual sense. As usual in diagrammatic algebra, cutting these graphs open and putting them into a strip with bottom and top boundary points gives a way to define morphisms, called orthogonal webs, in a monoidal category. Here are two examples, the left one being closed:

If in this or other illustrations an edge is not labeled, then its label is determined by other labels and we omitted it to avoid clutter.
Summarized, we:

1. Fix \(p=0\) or any prime \(p\) not equal to \(2\). Let \(\mathbb{F}\) be an infinite field of characteristic \(p\) containing \(\sqrt{-1}\).
2. We give a diagrammatic presentation of tilting \(O_{N}(\mathbb{F})\)-representations using orthogonal webs. This extends the result of Sartori to prime characteristic.
3. A main ingredient is Howe's orthogonal duality in prime characteristic.
4. As an application we describe the semisimplification of tilting \(O_{N}(\mathbb{F})\)-representations. Here \(p\neq 2\) is arbitrary and does not need to be bigger than \(N\).