Séminaire Lotharingien de Combinatoire, 95B.76 (2026), 12 pp.
Jason Stack
Parking Spaces for Complex Reflection Groups
Abstract.
For W a Coxeter group, Q its root lattice, and h its Coxeter number, we call Q/(h+1)Q the standard W-parking space, which carries an action of W. The orbits of the W-action can be parametrized by the W-nonnesting partitions, defined as antichains in the root poset (Φ+,≤). In this setting, there is also the notion of a W-noncrossing partition. Replacing the W-nonnesting partitions with the W-noncrossing partitions, Armstrong, Reiner, Rhoades (2015) define a new combinatorial parking space: the W-noncrossing parking functions. They further define an algebraic W-parking space, with both parking spaces carrying a (W × C)-action, and give an explicit map between the two. Answering an open problem of theirs, we extend these definitions and results to irreducible
well–generated complex reflection groups W. We define a combinatorial W-noncrossing parking space
and an algebraic W-parking space for such W, and exhibit a
(W × C)-equivariant isomorphism between
the two. As a consequence of this isomorphism, we enumerate the W-noncrossing parking functions. We prove the results for all such complex reflection groups except G34, E7, and E8. A Fuss generalization of the real case is given by Rhoades (2014), which we also extend to well-generated complex reflection groups.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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