Spanning Configurations and Representation Stability

Let $V_1, V_2, V_3, \dots $ 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 $(W_1, \dots, W_n)$ 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.