Séminaire Lotharingien de Combinatoire, 84B.57 (2020), 12 pp.
Brendan Pawlowski, Eric Ramos and Brendon Rhoades
Spanning Configurations and Matroidal Representation Stability
Let V1, V2, ... be a sequence of vector spaces where Vn carries an action of Sn
for each n. Representation stability describes when the
sequence Vn has a limit. An important source of stability arises when Vn is the
dth 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
Xn,k of spanning configurations
of n-tuples (ℓ1, ..., ℓn) of lines in Ck with ℓ1 + ... + ℓn = Ck
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
Hd(Xn,k) as the
parameter (n,k) grows. We also study stability phenomena for a family of multigraded
modules related to the Delta Conjecture.
Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.
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