A generating function approach to new representation stability phenomena in orbit configuration spaces

TL;DR

This paper introduces a generating function approach using twisted commutative algebras to analyze the homology of orbit configuration spaces, revealing new stability phenomena and unifying existing results.

Contribution

It develops a novel factorization technique for generating functions that uncovers primary, secondary, and higher representation stability in orbit configuration spaces.

Findings

Established secondary and higher stability for configuration spaces on i-acyclic spaces.

Described a filtration revealing filtered representation stability in graph configuration spaces.

Unified and generalized known stability results using a geometric technique.

Abstract

As countless examples show, it can be fruitful to study a sequence of complicated objects all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of orbit configuration spaces: using the notion of twisted commutative algebras, which essentially categorify exponential generating functions. This idea allows for a factorization of the orbit configuration space "generating function" into an infinite product, whose terms are surprisingly easy to understand. Beyond the intrinsic aesthetic of this decomposition and its quantitative consequences, it reveals a sequence of primary, secondary, and higher representation stability phenomena. Based on this, we give a simple geometric technique for identifying new stabilization actions with finiteness properties, which we use to unify and generalize known stability results. As a first new application of our methods, we establish secondary and higher stability for configuration spaces on $i$-acyclic spaces. For another application, we describe a natural filtration by which one observes a filtered representation stability phenomenon in configuration spaces on graphs.