Spanning configurations and matroidal representation stability

Let V₁,V₂,… be a sequence of vector spaces where V_n carries an action of Ϭ_n for each n. Representation stability describes when the sequence V_n has a limit. An important source of stability arises when V_n is the d^th homology group (for fixed d) of the configuration space of n distinct points in some topological space X. We replace these configuration spaces with the variety X_n,k of spanning configurations of n-tuples (ℓ₁,…, ℓ_n) of lines in ℂ^k with ℓ₁ +… + ℓ_n = ℂ^k as vector spaces. That is, we replace the configuration space condition of distinctness with the matroidal condition of spanning. We study stability phenomena for the homology groups H_d(X_n,k) as the parameter (n, k) grows. We also study stability phenomena for a family of multigraded modules related to the Delta Conjecture.