On semigroup algebras

Abstract

In the classical theory of representations of a finite group by matrices over a field , the concept of the group algebra (group ring) over is of fundamental importance. The chief property of such an algebra is that it is semi-simple, provided that the characteristic of is zero or a prime not dividing the order of the group. As a consequence of this, the representations of the algebra, and hence of the group, are completely reducible.

Representation theory of finite semigroups, semigroup radicals and formal language theory

“…By way of contrast, the theory of semigroup representations, which was intensively developed during the 50s and 60s in classic work such as Clifford [15], Munn [29,30] and Ponizovsky (see [16,Chapter 5] for an account of this work, as well as [26,56] for nicer treatments restricting to the case of finite semigroups), has found almost no applications in the theory of finite semigroups. It was pointed out by McAlister in his survey of 1971 [28] that the only paper applying representation theoretic results to finite semigroups was the paper [51] of Rhodes.…”

Regularity of Semilattice Sums of Rings

“…Using Theorem 5 it can be shown that if D has property Qj and a power of each element lies in a subgroup then RD is regular if and only if (i) D is a semilattice of groups Gx, a e £2, and (ii) RGX is regular for all a e £2. Since a union of groups has property Qj, this theorem generalizes a theorem of Munn [4,Theorem 9.5].…”

The category of representations of a completely 0-simple semigroup

“…The main theorem of this section shows that the category of basic representations' of S is equivalent to the category of proper representations of any of the maximal subgroups of S. From this theorem one can deduce, as corollaries, Theorems 5.48, 5.50 and 5.51 of [3]. In Section 4 we show how the theory given in [1], [2], and related results of Lallement and Petrich [4] and Munn [9], fits in with the theory given here.…”

“…Clifford [1 ] shows that every finite dimensional representation f of a completely 0-simple semigroup <J?°{G; I, A; P), over a field <P, is equivalent to one of the form where Q x = 0 = R v and Q x Ri = y(Pu-PuPu)'> ll ' s assumed t h a t p n = e the identity of G. Munn [9] has shown that S = ^°(G; m, n;P) has semisimple algebra over <P if and only if m = n and P is invertible over <P (G). If this is the case then he shows that the irreducible representations of S over <t> are the representations of the form X-W-* y{PX) where y is an irreducible representation of G, say of degree r. This follows easily from the theory given here.…”