On Extending Type $B$ Parking Spaces

TL;DR

This work advances the theory of parking-space representations for Coxeter groups by proposing and validating a broad generalization of the type B parking space to $\mathbb{C}[ (\mathbb{Z}/m\mathbb{Z})^n ]$ and its extension to $B_{n+1}$. The authors derive explicit character-theoretic decompositions via Schur-Weyl duality, and construct concrete extensions in two regimes: fixed $m$ (notably $m=3$) and fixed $n$, supported by an invertible restriction-matrix framework. They provide constructive extension formulas, establish uniqueness properties in the fixed-$m$ case, and offer practical verification via ILP up to sizable $n$ and $m$, underscoring the approach's effectiveness and limitations. Collectively, the results deepen understanding of how generalized parking spaces behave under passage from $B_n$ to $B_{n+1}$ and lay groundwork for further extensions in type A and beyond. The findings have potential implications for representation-theoretic constructions tied to Weyl group actions and their combinatorial incarnations.

Abstract

Armstrong, Reiner, and Rhoades defined for all Weyl groups $W$ a natural representation of $W$ called the $W$-parking space. The type $B$ parking space is the representation $\mathbb{C}[(\mathbb{Z}/(2n+1)\mathbb{Z})^n]$ of the $n$th signed symmetric group. We consider more general representations of the form $\mathbb{C}[(\mathbb{Z}/m\mathbb{Z})^n]$; we conjecture that this representation extends to the $(n+1)$th signed symmetric group for all $n$ and $m$. We prove this conjecture when $m = 3$ or when $n \leq 2$.

Theorems & Definitions (53)

• Conjecture 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5