Computations of Stable Multiplicities in the Cohomology of Configuration Space

TL;DR

This paper presents an algorithm for computing stable multiplicities of irreducible representations in the cohomology of configuration spaces, providing explicit calculations up to certain degrees and formulating related conjectures.

Contribution

It introduces a novel algorithm for calculating stable multiplicities in configuration space cohomology and performs extensive computations for Young diagrams with up to 23 boxes.

Findings

Computed stable multiplicities for all irreducible families with up to 23 boxes.

Determined stable cohomology in degrees 0 to 11.

Proved related qualitative results and proposed conjectures.

Abstract

We describe an algorithm to compute the stable multiplicity of a family of irreducible representations in the cohomology of ordered configuration space of the plane. Using this algorithm, we compute the stable multiplicities of all families of irreducibles given by Young diagrams with $23$ boxes or less up to cohomological degree $50$. In particular, this determines the stable cohomology in cohomological degrees $0 \leq i \leq 11$. We prove related qualitative results and formulate some conjectures.