THE QUIVER OF THE SEMIGROUP ALGEBRA OF A LEFT REGULAR BAND

FRANCO V. SALIOLA
2007 International journal of algebra and computation  
Communicated by S. Margolis 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
more » ... up 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.
doi:10.1142/s0218196707004219 fatcat:3qa7iqna5jflnndg3f4xha2pvu