Characters and complexity of finite semigroups

AbstractLet S be a finite semigroup and let K be an algebraically closed field of characteristic zero. Herein, we derive a formula for the congruence ≡ induced on S by the direct sum of all the irreducible representations of S over K. This congruence ≡ is proved to be the same as the congruence induced by the minimal homomorphic image of S, which is one-to-one on the subgroups of S and such that two distinct principal ideals of S, each generated by an idempotent, have distinct images.Assume ψ is a faithful finite dimensional representation of S. Let R(ψ) be the associated completely reducible representation having the same character as ψ. That is R(ψ) is the direct sum of the Jordan-Hölder factors of ψ. Then by applying the Burnside-Steinberg theorem we prove that the congruence induced by R(ψ) on S equals ≡. That is, the operator R preserves “one-to-oneness as much as possible”.We next apply these results to compute the complexity, #G(S), of S when S is a union of groups. A linear transformation T=B(S) is defined on the character ring of S into itself. Then by a theorem of Krohn-Rhodes (which determines the complexity of S in terms of its homomorphic images), together with the previous character results, we prove that T is nilpotent and index(T)=#G(S).Finally the character results proved here imply that, if the Fundamental Lemma on Complexity is valid, then the complexity of S is the maximum of the images of all its irreducible representations. This is known to be the case for all regular semigroups