Finite orbits of the braid group actions

We study the finite orbits of the braid group $B_n$ action on the space of $n\times n$ upper-triangular matrices with 1's along the diagonal. On one hand, we give a necessary condition for a matrix M to be in a finite orbit; on the other hand, we classify and provide lengths of finite orbits in low-dimensional matrices and some other important cases. As the finite orbits on $3\times 3$ matrix were crucial to finding the algebraic solutions of the sixth Painlevé equation, we hope the finite orbits on generic $n\times n$ matrices to be useful to finding solutions of higher order Painlevé type differential equations.