Some faithful algebraic braid twist group actions for 3-fold crepant resolutions

Let X(1,3,a) be a crepant resolution of the quotient singularity C^3/G, where G is a diagonal cyclic subgroup of SL(3,\C) acting on C^3 with weights (1,3,a). For each such X(1,3,a), we construct a (Q,W)-configuration of spherical objects in the bounded derived category of coherent sheaves. When a=9, the derived category D(X(1,3,9)) admits a faithful algebraic braid twist group action of type D induced by the associated (Q,W)-configuration. When a=13, the derived category D(X(1,3,13)) admits a faithful algebraic braid twist group action of type E. These two cases illustrate the emergence of type D and type E patterns from specific geometric data, supporting a broader conjectural framework.