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Finite diamond-colored modular and distributive lattices with applications to combinatorial Lie representation theory

A modular or distributive lattice is `diamond-colored' if its order diagram edges are colored in such a way that, within any diamond of edges, parallel edges have the same color. Such lattices arise naturally in combinatorial representation theory, particularly in the study of poset models for semisimple Lie algebra representations and their companion Weyl group symmetric functions. One of our goals is to gather in one place some elementary but foundational results concerning these lattice structures; this includes some new results as well as some new interpretations of classical results. We then develop many points of contact between diamond-colored modular/distributive lattices and combinatorial Lie representation theory, leading to some new Dynkin diagram classification results and some new results concerning minuscule and quasi-minuscule representations.