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PreprintarXiv licencearXiv

Interval Deformations of Coxeter Arrangements

  • Yanru Chen
  • Ang Li
  • Suijie Wang
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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.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026interval,
  title = {{Interval Deformations of Coxeter Arrangements}},
  author = {Yanru Chen and Ang Li and Suijie Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.29203},
  eprint = {2608.29203},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29203},
}
RIS
TY  - UNPB
TI  - Interval Deformations of Coxeter Arrangements
AU  - Yanru Chen
AU  - Ang Li
AU  - Suijie Wang
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2608.29203
UR  - https://arxiv.org/abs/2608.29203
AB  - 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.
ER  -