Decomposition of Perverse Sheaves on Plane Line Arrangements

On the complement $X={\mathbb{C}}^2-{\bigcup }_{i=1}^n{L}_i$ to a central plane line arrangement ${\bigcup }_{i=1}^n{L}_i\subset {\mathbb{C}}^2$, a locally constant sheaf of complex vector spaces ${\mathcal{L}}_a$ is associated to any multi-index $a\in {\mathbb{C}}^n$. Using the description of MacPherson and Vilonen of the category of perverse sheaves (\cite{MV2} and \cite {MV3}) we obtain a criterion for the irreducibility and number of decomposition factors of the direct image $R{j}_{\ast }{\mathcal{L}}_a$ as a perverse sheaf, where $j: X \rightarrow {\mathbb C}^2$ is the canonical inclusion.