Parking Spaces

By Drew Armstrong, Victor Reiner, Brendon Rhoades. 2012

70 citations recorded by the source corpus.

Let W be a Weyl group with root lattice Q and Coxeter number h. The elements of the finite torus Q/(h+1)Q are called the W -parking functions, and we call the permutation representation of W on the set of W -parking functions the (standard) W -parking space. Parking spaces have interesting connections to enumerative combinatorics, diagonal harmonics, and rational Cherednik algebras. In this paper we define two new W -parking spaces, called the noncrossing parking space and the algebraic parking space, with the following features: • They are defined more generally for real reflection groups. • They carry not just W -actions, but W ×C-actions, where C is the cyclic subgroup of W generated by a Coxeter element. • In the crystallographic case, both are isomorphic to the standard W parking space. Our Main Conjecture is that the two new parking spaces are isomorphic to each other as permutation representations of W ×C. This conjecture ties together several threads in the Catalan combinatorics of finite reflection groups. We provide evidence for the conjecture, proofs of some special cases, and suggest further directions for the theory.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8df9e0acfb5b58473caefd8423ac3b3eb3228c10

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In Discrete Mathematics and Combinatorics

  1. James Haglund
  2. Mark Shimozono
  3. Brendon Rhoades
  4. E. Gorsky
  5. Thomas Lam
  6. Mike Zabrocki
  7. Sarah K. Mason

In Geometry and Topology

  1. James Haglund
  2. Mark Shimozono
  3. Brendon Rhoades
  4. Thomas Lam

In Statistics and Probability

  1. James Haglund
  2. Mark Shimozono
  3. Brendon Rhoades

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