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This is a Quantitative Study study.

September 24, 2025Open Access

The characteristic quasi-polynomials of hyperplane arrangements with actions of finite groups

Key Points

Permutation characters are shown to be quasi-polynomials in mod q hyperplane arrangements.
The results detail the relationship between induced characters and Ehrhart quasi-polynomials in this context.
Coxeter arrangements, particularly for type A_ℓ, are extensively analyzed with specific computations.
Equivariant characteristic quasi-polynomials bridge finite group actions and combinatorial properties in geometric settings.

Abstract

In this paper, we introduce an equivariant version of the characteristic quasi-polynomials as the permutation characters on the complement of mod q hyperplane arrangements. We prove that the permutation character is a quasi-polynomial in q, and show that it can be expressed by the sum of the induced characters of an equivariant version of the Ehrhart quasi-polynomials. Furthermore, we consider the case of the Coxeter arrangements, and compute in detail for type A_.

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Ryo Uchiumi (Wed,) studied this question.

synapsesocial.com/papers/68d6e16f8b2b6861e4c4009d https://doi.org/https://doi.org/10.48550/arxiv.2508.10236

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