Spanning subspace configurations and representation stability

Abstract

Let $V_1,V_2,V_3,\dots$ be a sequence of $\mathbb{Q}$-vector spaces where $V_n$ carries an action of ${\mathfrak{S}}_n$ for each $n$. {\em Representation stability} and {\em multiplicity stability} are two related notions of when the sequence $V_n$ has a limit. An important source of stability phenomena arises in the case where $V_n$ 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 the variety $X_{n,k}$ of {\em spanning configurations} of $n$-tuples $({\ell }_1,\dots ,{\ell }_n)$ of lines in ${\mathbb{C}}^k$ which satisfy ${\ell }_1+\cdots +{\ell }_n={\mathbb{C}}^k$ as vector spaces. We study stability phenomena for the homology groups $H_d(X_{n,k})$ as the parameter $(n,k)$ grows.

16 pages