Bruhat Lattices, Plane Partition Generating Functions, and Minuscule Representations

Abstract

The Bruhat posets (arising from Weyl groups) which are lattices are classified. Seshadri's standard monomial result for miniscule representations is used to show that certain combinatorially defined generating functions associated to these lattices satisfy certain identities. The most interesting cases of these identities are known plane partition generating function identities. Independent combinatorial proofs of the other identities are given. Then the combinatorial proofs of these identities are used as a step in a simplified proof of Seshadri's standard monomial result. Partial results to the effect that the Bruhat lattices are the only distributive lattices with such generating function identities are quoted (‘Gaussian poset’ conjecture), a potential Dynkin diagram classification result. New proofs of the fact that Bruhat lattices are rank unimodal and strongly Sperner are given. Geometric interpretations (with respect to minuscule flag manifolds) of the combinatorial quantities studied are described.