Characterization of the braid group by preserved HN mass functions

Determine whether the braid group is precisely the group of piecewise-linear automorphisms of \(\Sigma\) that preserves the collection of functions formed by linear combinations of Harder–Narasimhan mass functions from finitely many stability conditions.

Background

The paper considers functions on $\Sigma$ obtained as linear combinations of HN mass functions associated with finitely many stability conditions. The braid group preserves this collection, suggesting a possible intrinsic characterization of the group as the full group of PL automorphisms preserving these functions.