Mohan S. Putcha

Semilattice decompositions of semigroups

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 propertyD if and only if S is a semilattice of semigroups having property β. As such we are able to extend the theories of Clifford [3], Andersen [1], Croisot [5], Tamura and Kimura [14], Petrich [9], Chrislock [2], Tamura and Shafer [15], Iyengar [7] and Weissglass and the author [10]. The root of our whole theory is Tamuras semilattice decomposition theorem [12, 13]. Of this, we give a new proof.

Monoids on groups with BN-Pairs

The theory of linear algebraic monoids has been developed in the last few years by L. Renner and the author [21-361 as a generalization of the theory of linear algebraic groups [16,40]. In applications of the theory of linear algebraic groups, one is naturally led to consider groups with BNpairs (or Tits systems) 12, 4, 9, 42,431. This class of groups includes in particular the groups of Lie type, so that the abstract properties of Tits systems are useful in studying the conjugacy classes and representations of groups of Lie type (see [4,9]). Our interest is in monoids. So let M be a monoid such that the group of units G has a BN-pair. In general G has very little to do with the rest of M. In fact, for a given G, M\G could be an arbitrary semigroup. So we introduce three axioms which intimately connect the idempotent structure of M with the structure of G. We then say that M is a monoid on G. The prime example is a connected regular linear algebraic monoid with zero. But there are other natural examples. For instance, Renner [37] is developing a theory of “monoids of Lie type” along the lines of Steinberg [41] for groups. Another example arises from the work of Smith [39] and Cabanes [3]. From their work, one can associate with certain irreducible modular representations, 4 : G + GL( V), a set of idempotents Es End(V). Then the monoid M generated by 4(G) and E is a monoid on 4(G) in the examples we have worked out, and very likely in general. In the situation of linear algebraic monoids, there is a natural type map from the finite lattice % of conjugacy classes of idempotents (g the lattice of f-classes) into the power set of the Dynkin diagram of G. The idea of the type map is due to Lex Renner and is exploited by Renner and the author [31] to show in particular that the system of idempotents can be constructed from the type map. In the present situation of monoids on G, there is again a type map I which characterizes the system of idempotents. 139 0021~8693/89

Complex representations of matrix semigroups

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Green's relations on a connected algebraic monoid†

Let M be a finite monoid of Lie type (these are the finite analogues of linear algebraic monoids) with group of units G. The multiplicative semigroup .4 (F) , where F is a finite field, is a particular example. Using HarishChandras theory of cuspidal representations of finite groups of Lie type, we show that every complex representation of M is completely reducible. Using this we characterize the representations of G extending to irreducible representations of M as being those induced from the irreducible representations of certain parabolic subgroups of G. We go on to show that if F is any field and S any multiplicative subsemigroup of .4 (F) , then the semigroup algebra of S over any field of characteristic zero has nilpotent Jacobson radical. If S = .4 (F) , then this algebra is Jacobson semisimple. Finally we show that the semigroup algebra of .4 (F) over a field of characteristic zero is regular if and only if ch(F) = p > 0 and F is algebraic over its prime field.

When is a matrix a sum of idempotents

Let S be a connected algebraic monoid with group of units G and idempotent set E(S). Let e f ϵ E(S),a,b ϵ S b regular. Then 1) if and only if x-1 ex = f for some x ϵ G.2) if and only if xay = b for some x y ϵ G. 3) if and only if ax = b for some x ϵ G. 4) if and only if ya = b for some y ϵ G. 5) if and only if ex = xe = a for some x ϵ G. 6) if and only if e′ ⩾ f for some e′ϵE(S) with . 7) bub = b for som u ϵ G. Several other related results are obtained.

The canonical compactification of a finite group of Lie type

It is shown that a matrix A, ovei a field of characteristic zero, is a sum of idempotents. exactly when the trace of A is an integer at least as large as rank(A).

When is a matrix a difference of two idempotents

Let G be a finite group of Lie type. We construct a finite monoid M having G as the group of units. M has properties analogous to the canonical compactification of a reductive group. The complex representation theory of M yields Harish-Chandras philosophy of cuspidal representations of G. The main purpose of this paper is to determine the irreducible modular representations of M. We then show that all the irreducible modular representations of G come (via the 1942 work of Clifford) from the one-dimensional representations of the maximal subgroups of M. This yields a semigroup approach to the modular representation theory of G, via the full rank factorizations of the «sandwich matrices» of M. We then determine the irreducible modular representations of any finite monoid of Lie type

Band of t-archimedean semigroups

Necessary and sufficient conditions are given for a matrix A to be a sum or a difference of two idempotents. These conditions are expressed in terms of (i) the block structure of A under similarity, as well as (ii) the elementary divisor structure of A.

Products of idempotents in algebraic monoids

A semigroup S is called t-archimedean if for all a,b∈S, there exists a positive integer i such that bi∈aS∩Sa. The purpose of this paper is to characterize semigroups which are bands of t-archimedean semigroups. We then apply this result to exponential semigroups.

Conjugacy classes in algebraic monoids

Let M be a reductive algebraic monoid with zero and unit group G. We obtain a description of the submonoid generated by the idempotents of M. In particular, we find necessary and sufficient conditions for M/G to be idempotent generated.