  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.
