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The Weak Order and Flag h-Vector Inequalities May 2005

The following paper examines properties of the weak order on the groups Sr+1 and Br+1 as motivated by the search for a combinatorial proof of known h-vector inequalities. Several general results are developed with respect to descent set domination, and an explanation of the computer programs used to generate a number of these results is presented. While this paper primarily focuses on hand flag h-vector inequalities of the order complex of geometric lattices, such inequalities for the order complexes of distributive and supersolvable lattices are also explored. Finally, several courses of action for further study are considered.