Séminaire Lotharingien de Combinatoire, 84B.57 (2020), 12 pp.

Brendan Pawlowski, Eric Ramos and Brendon Rhoades

Spanning Configurations and Matroidal Representation Stability

Abstract. 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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