Spanning configurations and representation stability (Q2111779)

Summary: Let \(V_1, V_2, V_3, \ldots\) be a sequence of \(\mathbb{Q} \)-vector spaces where \(V_n\) carries an action of \(\mathfrak{S}_n\). 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 \((W_1, \ldots, W_n)\) of subspaces of a fixed complex vector space \(\mathbb{C}^N\) such that \(W_1 + \cdots + W_n = \mathbb{C}^N\). These include the varieties of spanning line configurations which are tied to the Delta Conjecture of symmetric function theory.