Abstract

Fine Bruhat intersections for reductive groups have been studied by several au-thors in connection with Kazhdan-Lusztig theory, canonical bases and Lie The-ory. The purpose of this thesis is to study the analogous intersections for reductive monoids. We determine the conditions under which the following intersections BσB ∩ B − θB BσB ∩ B − θB − are nonempty. First we study these intersections for the monoid Mn(K). This work gives rise to two new orderings, ≤1 and ≤2, on the monoid of partial permutation matrices. More precisely, ≤1 and ≤2 are orderings that exist within a particular J-class for a reductive monoid M. The J-classes of Mn(K) consist of matrices of the same rank. Combinatorial descriptions of the orderings are given and their relation to the Bruhat-Chevalley order is discussed. These results are then generalized to an arbitrary reductive monoid. We show that ≤1 is a partial order on R, the Renner monoid, and that ≤2 is in general not a partial order on elements of R, but rather on equivalence classes of elements in R. We describe the equivalence classes for the matrices and conclude with theorems for the partial permutation matrices of rank r < n