ROOT SEMIGROUPS IN REDUCTIVE MONOIDS

It is well known that in a reductive group, the Borel subgroup is a product of the maximal torus and the one-dimensional positive root subgroups. The purpose of this paper is to find an analog of this result for reductive monoids. Via a study of reductive monoids of semisimple rank 1, we introduce the concept of root semigroups. By analyzing the associated root elements in the Renner monoid, we show that the closure of the Borel subgroup is generated by the maximal torus and positive root semigroups. Along the way we generalize the Jordan decomposition of algebraic groups to reductive monoids.

REPRESENTATIONS OF THE RENNER MONOID

We describe irreducible representations and character formulas of the Renner monoids for reductive monoids, which generalize the Munn–Solomon representation theory of rook monoids to any Renner monoids. The type map and polytope associated with reductive monoids play a crucial role in our work. It turns out that the irreducible representations of certain parabolic subgroups of the Weyl groups determine the complete set of irreducible representations of the Renner monoids. An analogue of the Munn–Solomon formula for calculating the character of the Renner monoids, in terms of the characters of the parabolic subgroups, is shown.

H-POLYNOMIALS AND ROOK POLYNOMIALS

The purpose of this paper is twofold. First we describe a useful procedure for computing the H-polynomials of reductive monoids. Second we use this procedure to compute the H-polynomial of the monoid of n × n matrices in terms of the q-analogues of the rook polynomials of Garsia and Remmel.

FINE BRUHAT INTERSECTIONS FOR REDUCTIVE MONOIDS

This paper concerns Bruhat intersections for a reductive monoid M. A comparison is made to the known results for Bruhat intersections for a reductive group G.