Hyperplane Arrangements and the Bernstein-Gelfand-Gelfand Correspondence

The Chen ranks conjecture has stimulated work that involves ideas from the theory of hyperplane arrangements and homological algebra, namely the Bernstein-Gelfand-Gelfand (BGG) correspondence. The conjecture is an attempt to give a combinatorial formula for the Chen ranks invariants of a hyperplane arrangement. In 2005, Schenck and Suciu proved half of the conjecture. First, we motivate the necessary definitions and explain the connections between the field of hyperplane arrangements and the field of homological algebra with the goal of explaining the Chen ranks conjecture to the reader. The reader is not assumed to have any background in hyperplane arrangements, but some familiarity with homological algebra. SINGULAR routines were used to drastically simplify the calculations for the Chen invariant of an arbitrary hyperplane arrangement. Apart from the Chen invariant, our routine can calculate two other invariants associated to an arrangement with a high degree of efficiency. Thus, it has proven itself to be a very useful tool in studying arrangements. The difficulty with proving the conjecture is that the formulae involved are extremely complicated and difficult to compute by hand. To overcome this, we tried to verify the conjecture through the examination of examples. So far in all the examples we have examined, we have not found any contradictions; rather, we are very optimistic about the validity of the conjecture.