The Coarse Structure of the Representation Algebra of a Finite Monoid. (30th January 2014) - Schaps, Mary
Abstract : LetM be a monoid, and letL be a commutative idempotent submonoid. We show that we can find a complete set of orthogonal idempotentsL ^ 0 of the monoid algebraA ofM such that there is a basis ofA 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 toL ^ 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.