Light ladders and clasp conjectures

TL;DR

This paper introduces an explicit root-theoretic formula for local intersection forms in sl(n) quantum groups, proves it for n≤4, and explores the structure of sl(n)-webs and clasps.

Contribution

It provides a new explicit formula for intersection forms, proves it for small n, and develops recursive clasp expansion and cellular structure for sl(n)-webs.

Findings

Explicit root-theoretic formula for intersection forms

Proof of the formula for n≤4

Recursive triple-clasp expansion formula

Abstract

Morphisms between tensor products of fundamental representations of the quantum group of sl(n) are described by the sl(n)-webs of Cautis-Kamnitzer-Morrison. Using these webs, we provide an explicit, root-theoretic formula for the local intersection forms attached to each summand of the tensor product of an irreducible representation with a fundamental representation. We prove this formula for n at most 4, and conjecture that it holds for all n. Given two sequences of fundamental weights which sum to the same dominant weight, the clasp is the morphism between the corresponding tensor products which projects to the top indecomposable summand. Using our computation of intersection forms, we provide a recursive, ``triple-clasp expansion'' formula for clasps. In addition, we describe the cellular structure on sl(n)-webs, and prove that sl(n)-webs are an integral form for tilting modules of quantum groups.