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

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.