A Note on Braid Group Actions on Semiorthonormal Bases of Mukai Lattices

We shed some light on the problem of determining the orbits of the braid group action on semiorthonormal bases of Mukai lattices as considered in \cite{GK04} and \cite{GO1}. We show that there is an algebraic (and in particular algorithmic) equivalence between this problem and the Hurwitz problem for integer matrix groups finitely generated by involutions. In particular we consider the case of $K_0(\mathbb P^n) \quad n \geq 4$ which was considered in \cite{GO1} and show that the only obstruction for showing the transitivity of the braid group action on its semiorthonormal bases is the determination of the relations of particular finitely generated integer matrix groups. Although we prove transitivity for an infinite set of Mukai lattices, our work, however, indicates quite strongly that the question of transitivity of semiorthonormal bases of Mukai lattices under the braid group action cannot be answered in general and can, at most, be resolved only in particular cases.