Spanning Configurations and Representation Stability

Let V₁, V₂, V₃, … be a sequence of Q-vector spaces where V_n carries an action of S_n. Representation stability and multiplicity stability are two related notions of when the sequence V_n has a limit. An important source of stability phenomena arises when V_n is the d^th homology group (for fixed d) of the configuration space of n distinct points in some fixed topological space X. We replace these configuration spaces with moduli spaces of tuples (W₁, …, W_n) of subspaces of a fixed complex vector space C^N such that W₁ + · · · + W_n = C^N. These include the varieties of spanning line configurations which are tied to the Delta Conjecture of symmetric function theory.