Rational parking functions and $(m, n)$-invariant sets

TL;DR

This work shows that rational parking functions arise as fixed points of a word action on the Weyl chamber $V^m$ and that this action extends coherently to $(m,n)$-invariant sets. By defining an action on $m$-invariant sets, the authors establish a tight correspondence between $(m,n)$-parking functions and invariant sets, with a unique associated monotone parking function emerging from the skeleton structure via the map $ G$. The fixed-point geometry is clarified through alcove centroids, the affine-braid/Sommers framework, and a unifying hyperplane-arrangement view, yielding a decomposable fixed-point set into unions of monotone parking-function fixed points; periodic-point behavior is characterized when $ ext{gcd}(m,n)=1$. Finally, the Pak–Stanley map inversion is made constructive by presenting two equivalent algorithms whose outputs coincide and terminate in finite steps, resolving a standing conjecture in the coprime case.

Abstract

An $(m, n)$-parking function can be characterized as function $f:[n] \to [m]$ such that the partition obtained by reordering the values of $f$ fits inside a right triangle with legs of length $m$ and $n$. Recent work by McCammond, Thomas, and Williams define an action of words in $[m]^n$ on $\mathbb{R}^n$. They show that rational parking functions are exactly the words that admit fixed points under that action. An $(m, n)$-invariant set is a set $Δ\subset \mathbb{Z}$ such that $Δ+ m \subset Δ$ and $Δ+ n \subset Δ$. In this work we define an action of words in $[m]^n $ on $(m, n)$-invariant sets by removing the $j$th $m$-generator from $Δ$. We show this action also characterizes $(m, n)$-parking functions. Further we show that each $(m, n)$-invariant set is fixed by a unique monotone parking function. By relating the actions on $\mathbb{R}^m$ and on $(m, n)$-invariant sets we prove that the set of all the points in $\mathbb{R}^m$ that can be fixed by a parking function is a union of points fixed by monotone parking functions. In the case when $\gcd(m, n) =1$ we characterize the set of periodic points of the action defined on $\mathbb{R}^m$ and show that the algorithm reversing the Pak-Stanley map proposed by Gorsky, Mazin, and Vazirani converges in a finite amount of steps.

Figures (7)

• Figure 1: Here is $\cS_3^4$ with alcoves labeled by affine permutations. The fundamental alcove is labeled by $[123]$.
• Figure 2: Periodic lattice path corresponding to the $(4,5)$-invariant set $\Delta=\{0,3,4,5,\ldots\}.$ The $5$-generators of $\Delta$ are $\{0,3,4,6,7\}$ in blue, and the $4$-cogenerators are $\{-4,-1,1, 2\}$ in red.
• Figure 3: This is $\cG(\Delta) = \cG(\Delta ')$ from Example \ref{['example: map G']}.
• Figure 4: Acting on the $(5, 3)$ invariant set $\Delta = \{ 0, 3, 5, 6, 7, 8, 9, 11, 12,\dots \}$ by the parking function $\p = 103$. The $5$-generators being replaced in each step is highlighted in red. The $3$-generators of $\Delta$ are $\{ 0, 5, 7\}$ which is exactly the set of elements that are removed by $\p$.
• Figure 5: In this example $\w = 2100$, we consider the four points, the fixed point $x_\w = (-5, 0, 5)$ along with $a = (-4, 1, 3), b= (-3, -1, 4),$ and $c = (-5, 0, 5)$. Arrows show where each point is mapped to by the parking function $\w = 2100$.

Theorems & Definitions (96)

• theorem 1
• theorem 2
• theorem 3
• definition 1
• definition 2
• definition 3
• definition 4
• theorem 4: mccammond_thomas_williams_2019, Theorem 1.1
• definition 5
• definition 6