Counting points on braid varieties and the Deligne--Simpson problem

Masoud Kamgarpour; Bailey Whitbread

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arXiv · 2603.20499

Counting points on braid varieties and the Deligne--Simpson problem

Masoud Kamgarpour·Bailey Whitbread

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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.

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Masoud Kamgarpour, Bailey Whitbread. 2026-03-20. Counting points on braid varieties and the Deligne--Simpson problem. https://arxiv.org/abs/2603.20499

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