[論文レビュー] Counting points on braid varieties and the Deligne--Simpson problem

Masoud Kamgarpour, Bailey Whitbread|arXiv (Cornell University)|Mar 20, 2026

Advanced Algebra and Geometry被引用数 0

ひとこと要約

この論文は exceptional 群に対する isoclinic Deligne–Simpson 問題を、 braid variety の 非空性の有無と有限場上の点数計測に還元することによって解き、P^1 上の新しい rigid irregular G-コネクションを得る。

ABSTRACT

We solve the isoclinic Deligne--Simpson problem for exceptional groups, completing a program initiated by Sage et al. and Jakob--Yun. As a by-product, we obtain new examples of physically rigid irregular connections on the projective line. Our approach uses the Riemann--Hilbert correspondence to reduce the problem to determining the non-emptiness of certain braid varieties associated to periodic braids. We show that this can be achieved by counting points over finite fields. Our approach is inspired by Lusztig's construction of a map from conjugacy classes in the Weyl group to unipotent classes.

研究の動機と目的

• Solve the isoclinic Deligne–Simpson problem for exceptional groups.
• Describe the Riemann–Hilbert correspondence reduction to braid varieties.
• Count points on braid varieties over finite fields to determine non-emptiness.
• Produce new examples of physically rigid irregular G-connections on P^1.

提案手法

• Riemann–Hilbert 対応を用いて irregular G-コネクションを periodic braids に関連する braid variety へ翻訳する.
• braid を isoclinic slope ν=d/m に対応づく m-Springer 要素 w_m とそのリフト ṽ_w_m により関連付け、β_A=(ṽ_w_m)^d を循環シフト up to に示す.
• Lusztig の枠組みを適用: M(β, C) が非空であることは B(β)∩C ≠ ∅ に同値、そして有限体上の braid stacks の点数を数える。
• Hecke代数と有限群 G^F のトレースを計算して点数を表現する、文字理論の和として。
• 必要な文字計算を exceptional 型で CHEVIE を用いて実行。

実験結果

リサーチクエスチョン

• RQ1 When does an isoclinic G-connection of slope ν possess unipotent monodromy in a given unipotent class C?
• RQ2 Can one detect the non-emptiness of the corresponding braid stack by counting points over finite fields?
• RQ3 How does the m-Springer element and its lift govern the associated braid β_A for isoclinic irregular classes?
• RQ4 What new cohomologically rigid (physically rigid) isoclinic G-connections arise from exceptional types?
• RQ5 Can explicit Coxeter G-connections realize minimal monodromy within the isoclinic framework?

主な発見

• There exists a unique unipotent class C_ν such that an isoclinic G-connection of slope ν with unipotent monodromy C exists iff C ≥ C_ν (with C_ν = {1} when ν>1).
• For exceptional G and ν<1, the classes C_ν are listed in Appendix A.
• An isoclinic connection’s associated braid is (ṽ_w_m)^d where w_m is an m-Springer element and m is a regular number for W; ṽ_w_m is its lift to the positive braid monoid.
• Non-emptiness of braid stacks is reduced to whether (Bw_mB)^d ∩ C ≠ ∅, linking Deligne–Simpson existence to Bruhat double coset intersections.
• Counting points on braid stacks over finite fields yields the required non-emptiness criterion via character theory of G(𝔽_q) and the finite Hecke algebra H_q, computable with CHEVIE.
• As a byproduct, new cohomologically rigid isoclinic G-connections are produced, and some Coxeter G-connections realize minimal monodromy (partial progress on explicit realizations).