The category of representations of a completely 0-simple semigroup

Abstract

Communicated by G. B. PrestonA. H. Clifford [1], [2] has shown that all finite dimensional representations of a completely 0-simple semigroup S over a field <P can be obtained as extensions of those of its maximal subgroups and has given a method for constructing all such representations. This representation theory depends strongly on the fact that the representations under consideration are finite dimensional and is not adequate to deal with the infinite dimensional case or with representations over arbitrary rings. In order to determine the structure of the (contracted) algebra <P(S) of S modulo its radical, one has to consider representations which are not finite dimensional or over fields; c.f. [6]. Hence Clifford's theory does not suffice for this purpose.In this paper we give a representation theory for a class of rings which we call Munn rings. These rings are infinite dimensional analogs of the Munn algebras considered in [9], Section 4; thus the algebra of any completely 0-simple semigroup is a Munn ring. The form of the representations which we obtain is simpler than that obtained by Clifford yet the theory does not require that the representations should be finite dimensional or over a field.Although the theory given here was developed to solve the problem of determining the structure of the algebra of a completely 0-simple semigroup modulo its radical, it has proved useful in several ways. Let S be an arbitrary semigroup and let F : S -> Horn (V, V) be an irreducible representation of S, where V is a finite dimensional vector space over a field <P. Then it is shown in [5] that F is the 'extension' of an irreducible representation F* : C r -* Horn (V, V) where C r is a completely 0-simple semigroup associated with S and F. To give an explicit construction for F from F* one has to make use of the fact that F*(<P(C r )) contains the identity \ v of H o m ( F , V). Proposition 2.9 of this paper gives a simple method for finding Ee $(C r ) such that F*(E) = \ v and hence for giving an explicit formula for F. The theory can also be used to give short proofs of some known results; Theorem 2.7 here gives a short proof of [5], Theorem 3.12 and [4], Theorem 2.1.As the title indicates, the approach adopted in this paper is that of studying 193 use, available at https://www.cambridge.org/core/terms. https://doi

Semigroups of rectangular matrices under a sandwich operation

“…(Here, the sandwich matrix A arises from the celebrated Rees structure theorem [81] for completely 0-simple semigroups.) Since their introduction in [68], Munn rings have been studied by numerous authors, and continue to heavily inflence the theory of semigroup representations: for classical studies, see [12][13][14]37,54,[62][63][64][68][69][70]76]; for modern accounts, see for example [1,29,47,75,78,79,85,86], and especially the monographs [73,74,77,82,87].…”

Cellularity of diagram algebras as twisted semigroup algebras

“…(iii) The form φ λ is nondegenerate (ie det Φ λ = 0) for each λ ∈ Λ.Theorem 5 allows us to obtain cell representations of R α [S] from the cell representations of the twisted group algebras R β [G D ]. In fact this is a special case of the following general result, the proof of which is a consequence of Theorem 2 and Green's Lemma, and is omitted (see also Theorem 2.3 of[13]).…”

Rings related to completely 0-simple semigroups

“…Let Q, R be epic and monic respectively such that QR = y(P). Then, by[9], Theorem 2.7, F = [y; Q, R] is a basic representation of J( with kernel B(M); hence B(M) is basic. Further, since the representation of 21 induced by F is equivalent to y, l(B(M)) = M. Thus Io Bis the identity on£?(%).…”

“…The results of [9] provide the tools with which to investigate the ideal struc-257 ture of a Munn ring and to find its primitive homomorphic images. Throughout this paper, we shall assume familiarity with the results and terminology of [9].…”

“…Throughout this paper, we shall assume familiarity with the results and terminology of [9]. In Section 1, the ideal structure of a Munn ring J( = ^#(91; I,A;P) is investigated and the primitive ideals of J( are determined in terms of the primitive ideals of 91.…”