  Recently it has been noticed that many interesting combinatorial objects
belong to a class of semigroups called left regular bands, and that random
walks on these semigroups encode several well-known random walks. For example,
the set of faces of a hyperplane arrangement is endowed with a left regular
band structure. This paper studies the module structure of the semigroup
algebra of an arbitrary left regular band, extending results for the semigroup
algebra of the faces of a hyperplane arrangement. In particular, a description
of the quiver of the semigroup algebra is given and the Cartan invariants are
computed. These are used to compute the quiver of the face semigroup algebra of
a hyperplane arrangement and to show that the semigroup algebra of the free
left regular band is isomorphic to the path algebra of its quiver.
