The Quiver of the Semigroup Algebra of a Left Regular Band

Details The Quiver of the Semigroup Algebra of a Left Regular Band The Quiver of the Semigroup Algebra of a Left Regular Band

2006-08-28

arxiv.org/abs/math/0608698v1

Mathematics

Combinatorics

24 pages, 2 figures

Scientific paper

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.

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