The Coarse Structure of the Representation Algebra of a Finite Monoid

Mary Schaps

doi:10.1155/2014/529804

Abstract

Let M be a monoid, and let L be a commutative idempotent submonoid. We show that we can find a complete set of orthogonal idempotents L^0 of the monoid algebra A of M such that there is a basis of A adapted to this set of idempotents which is in one-to-one correspondence with elements of the monoid. The basis graph describing the Peirce decomposition with respect to L^0 gives a coarse structure of the algebra, of which any complete set of primitive idempotents gives a refinement, and we give some criterion for this coarse structure to actually be a fine structure, which means that the nonzero elements of the monoid are in one-to-one correspondence with the vertices and arrows of the basis graph with respect to a set of primitive idempotents, with this basis graph being a canonical object.

Highlights

When we speak of a coarse structure, we mean the decomposition of the monoid algebra into Peirce components corresponding to the elements of a commutative idempotent submonoid
The basic idea of this work is to try to understand the semigroup representation theory insofar as possible without delving into the group theory and to determine criteria for monoids for which there is a coarse structure which corresponds to the fine structure
The irreducible representations of the semigroup correspond to the disjoint union of the irreducible representations of the various maximal subgroups

Summary

Introduction

When we speak of a coarse structure, we mean the decomposition of the monoid algebra into Peirce components corresponding to the elements of a commutative idempotent submonoid. Deformation theory, the basis graph is preferable, and in this work we claim that for certain types of monoids, the basis graph gives a much clearer picture of the relationship between the monoid and the monoid algebra than the quiver does These are the monoids for which the coarse structure described above can be obtained for a complete set of primitive idempotents, and one of the main results of the paper is that this happens when the monoid is aperiodic and has commutative idempotents. Examples of this phenomenon given below include the matrix monoid and the set of order-preserving and extensive maps from a finite set into itself.

The Basis Graph of a Finite Dimensional Algebra

Finite Monoids with a Chosen Commutative Idempotent Submonoid

Examples

Findings

The Fine Structure