Fine Bruhat Intersections for Reductive Monoids

Taylor, Dewey Terese. Fine Bruhat Intersections for Reductive Monoids. (Under the direction of Mohan S. Putcha.) Fine Bruhat intersections for reductive groups have been studied by several authors in connection with Kazhdan-Lusztig theory, canonical bases and Lie Theory. The purpose of this thesis is to study the analogous intersections for reductive monoids. We determine the conditions under which the following intersections BσB ∩ BθB BσB ∩ BθB 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. July 20, 2004 FINE BRUHAT INTERSECTIONS FOR REDUCTIVE MONOIDS by Dewey Terese Taylor a dissertation submitted to the graduate faculty of north carolina state university in partial fulfillment of the requirements for the degree of doctor of philosophy department of mathematics raleigh, north carolina July 2004