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In this paper we continue our study of complex representations of finite monoids. We begin by showing that the complex algebra of a finite regular monoid is a quasi-hereditary algebra and we identify the standard and costandard modules. We define the concept of a monoid quiver and compute it in terms of the group characters of the standard and costandard modules. We use our results to determine the blocks of the complex algebra of the full transformation semigroup. We show that there are only two blocks when the degree / 3. We also show that when the degree G 5, the complex algebra of the full transformation semigroup is not of finite representation type, answering negatively a conjecture of Ponizovskii.

The purpose of this paper is to develop a general theory of semilattice decompositions of semigroups from the point of view of obtaining theorems of the type: A semigroup S has property ~ if and only if S is a semilattice of semigroups having property ~.As such we are able to extend the theories of Clifford [3], Andersen [I], Croisot [5], Tamura and K~ura [14], Petrich [9], Chrislock [2], Tamura and Shafer [15], Iyengar [7] and Weissglass and the author [10]. The root of our whole theory is Tamura's semilattice decomposition theorem [12,13]. Of this, we give a new proof.

with ρ \ G -• GL(g) defined by p{g,t) = t Ad(g), then there is a one-to-one correspondence between {/ c Μ I / = GxG for some χ e M} and {U c G I U is the center of the unipotent radical of a parabolic subgroup of G}/ conjugacy defined by (p(J) = Z(R u ({g € G | ge = ege})) where e € E(J) is any idempotent. The lattice of inclusions among these subgroups can then be calculated with the algorithm of [7; Theorem 4.16].