Period-collapse conjecture for deleted Shi arrangements of root systems

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Let k∈Z>0k\in\mathbb{Z}_{>0}, let Φ\Phi be a root system, let Φ+\Phi^+ be its set of positive roots, and let δ∈Φ+\delta\in\Phi^+. Write δc\delta^c for the complement of δ\delta in the relevant root data, and let χShiδc[1−k,k]quasi(q)\chi_{\mathrm{Shi}_{\delta^c}^{[1-k,k]}}^{\mathrm{quasi}}(q) denote the characteristic quasi-polynomial associated with this deleted Shi arrangement. Period-collapse conjecture. Period collapse occurs in χShiδc[1−k,k]quasi(q)\chi_{\mathrm{Shi}_{\delta^c}^{[1-k,k]}}^{\mathrm{quasi}}(q) in all cases except when (i) Φ=Am\Phi=A_m, or (ii) Φ=B2\Phi=B_2 and δ\delta is a long root. The conjecture predicts, in the type-BB setting, that for m≥3m\geq 3 period collapse always occurs in the characteristic quasi-polynomial of Bm∖{H,H′}\mathcal{B}_m\setminus\{H,H'\} when HH and H′H' are parallel hyperplanes in Bm\mathcal{B}_m.

References

Primary source

Akihiro Higashitani and Norihiro Nakashima, “Characteristic quasi-polynomials of deletions of Shi arrangements of type B and their period collapse”, arXiv:2405.20102 (2026).

Additional references

2 papers in this index state this conjecture (2021–2024). The statement above is taken from the most recent of them; the others are arXiv:2105.02367.

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