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

# Brendan Pawlowski, Eric Ramos and Brendon Rhoades

# Spanning Configurations and Matroidal Representation Stability

**Abstract.**
Let *V*_{1}, *V*_{2}, ... be a sequence of vector spaces where *V*_{n} carries an action of *S*_{n}
for each *n*. *Representation stability* describes when the
sequence *V*_{n} has a limit. An important source of stability arises when *V*_{n} is the
*d*^{th} 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
*X*_{n,k} of *spanning configurations*
of *n*-tuples (ℓ_{1}, ..., ℓ_{n}) of lines in **C**^{k} with ℓ_{1} + ... + ℓ_{n} = **C**^{k}
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
*H*_{d}(*X*_{n,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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