Erdos-Ko-Rado theorems on the weak Bruhat lattice

Let L= (X, <=) be a lattice. For P subset of X we say that P is t-intersecting if rank(x boolean AND y) >= t for all x, y is an element of P. The seminal theorem of Erdos, Ko and Rado describes the maximum intersecting P, in the lattice of subsets of a finite set with the additional condition that P is contained within a level of the lattice. The Erdos-Ko-Rado theorem has been extensively studied and generalized to other objects and lattices. In this paper, we focus on intersecting families of permutations as defined with respect to the weak Bruhat lattice. In this setting, we prove analogs of certain extremal results on intersecting set systems. In particular we give a characterization of the maximum intersecting families of permutations in the Bruhat lattice. We also characterize the maximum intersecting families of permutations within the rth level of the Bruhat lattice of permutations of size n, provided that n is large relative to r. (C) 2018 Published by Elsevier B.V.