Year: 2026
Type: preprint
Abstract: We study integer-interval deformations of Coxeter arrangements of types $A$, $B$, and $D$, assigning an integer interval to each root direction through an integer-valued root function. We introduce a local compatibility condition on this root function. For compatible root functions, subject to additional local bounds in types $B$ and $D$, we derive characteristic-polynomial formulas as binomial sums over cyclic orders in type $A$ and signed permutations in types $B$ and $D$. The proofs use the finite-field method and reduce hyperplane avoidance to consecutive-gap inequalities. These formulas recover, in particular, the classical Catalan and Shi characteristic polynomials.
Source: arXiv (Cornell University)
Authors: Chen, Yanru, Li, Ang, Wang, Suijie
Institution: Hunan University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1711619002, G5678790074
OpenAlex ID: W7204893074