Séminaire Lotharingien de Combinatoire, 85B.43 (2021), 12 pp.
Matjaž Konvalinka and Vasu Tewari
Some Natural Extensions of the Parking Space
Abstract.
We construct a family of Sn modules indexed by c ∈ {1,...,n} with the property that upon restriction to Sn-1 they recover the classical parking function representation of Haiman.
The construction of these modules relies on an Sn-action on a set that is closely related to the set of parking functions.
We compute the characters of these modules and use the resulting description to classify them up to isomorphism.
In particular, we show that the number of isomorphism classes is equal to the number of divisors d of n satisfying
d =!= 2 (mod 4).
In the cases c=n and c=1, we compute the number of orbits.
Based on empirical evidence, we conjecture that when c=1, our representation is h-positive and is in fact the (ungraded) extension of the parking function representation constructed by Berget and Rhoades.
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
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