Irreducible Matrix Representations of Finite Semigroups
Gerard J. Lallement, Mario Petrich · Transactions of the American Mathematical Society · 1969
Munn [9] has shown that for a semigroup S satisfying the minimal condition on principal ideals, there is a natural one-to-one correspondence between irreducible representations of S and irreducible representations vanishing at zero of its 0-simple (or simple) principal factors; for the case of S finite, see Ponizovskii [11].On the other hand, Clifford, [3] and [4], has obtained all representations of a completely 0-simple semigroup as "extensions" of those of its maximal subgroups.Combining their results, one can, in principle, obtain all irreducible representations of a semigroup satisfying the minimal conditions on principal left and right ideals and thus of finite semigroups.However, in constructing the representations of a completely 0-simple semigroup S=J(°(G;I, A;F), one has to solve the problem in matrix theory of factoring the block matrixwhere y is an irreducible representation of G (see [5, §5.4]).The main object of this paper is to show that, when dealing with finite semigroups and irreducible representations, it is possible to avoid the factorization problem and give explicit expressions for these representations.Let S be a finite semigroup and J a regular ^-class of S. By M¡ denote the Schützenberger representation of S by row-monomial matrices over G°, where G is the Schützenberger group of J (isomorphic to the maximal subgroups of S contained in /) ([5, § §2.4,3.5], or [12]).For every x e S, let T(x) = y[Mj(x)], where y is a proper irreducible representation of G° by matrices over a field O, and y[Mj(x)] denotes the matrix over i> obtained by replacing each entry gKlt of M¡(x) by y(gAß).Then T is a representation of S by matrices over , and we prove (Theorem 1.7) that T has a unique nonnull irreducible constituent T* for which [T*(S)] = [T*(J)], where [r*(F)] denotes the linear closure of T*(F) (r* is given by ( 10)).The importance of this constituent Y* lies in the fact that every nonnull irreducible representation of S is equivalent to the constituent T* of some representation T relative to a suitable ./-class of S. This is an analogue to the well-known result in the theory of group representations : every irreducible representation of a group occurs as a constituent of the regular representation [1, 15.2]; this points to the fact that the direct sum of all Schützenberger representations of a semigroup is a suitable analogue of the right regular representation of a group.The proof depends essentially on an analogous property of finite