Ph.D. (2006) Cornell University
Abstract: The first part of this thesis studies the face semigroup algebra of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. Other algebraic structure is determined: a construction for a complete system of primitive orthogonal idempotents; projective indecomposable modules; Cartan invariants; projective resolutions of the simple modules; Hochschild (co)homology; and the Koszul dual algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A new cohomology construction on posets is introduced and it is shown that the face semigroup algebra is the cohomology algebra of the intersection lattice.
In the second part, attention is restricted to arrangements arising from finite reflection groups. The reflection group acts on the face semigroup algebra, and the subalgebra invariant under the group action is studied. The quiver is determined for the case of the symmetric group. Since the invariant subalgebra is anti-isomorphic to Solomon's descent algebra, the quiver of the descent algebra of the symmetric group is obtained.
The third part extends some of the results of the first part to a class of semigroups called left regular bands. In particular, a description of the quiver of the semigroup algebra is given, and it is used to compute the quiver of the face semigroup algebra of a hyperplane arrangement and of the free left regular band.