Diamond-colored distributive lattices, move-minimizing games, and fundamental Weyl symmetric functions: The type $\mathsf{A}$ case

We present some elementary but foundational results concerning diamond-colored modular and distributive lattices and connect these structures to certain one-player combinatorial "move-minimizing games," in particular, a so-called "domino game." The objective of this game is to find, if possible, the least number of "domino moves" to get from one partition to another, where a domino move is, with one exception, the addition or removal of a domino-shaped pair of tiles. We solve this domino game by demonstrating the somewhat surprising fact that the associated "game graphs" coincide with a well-known family of diamond-colored distributive lattices which shall be referred to as the "type $\mathsf{A}$ fundamental lattices." These lattices arise as supporting graphs for the fundamental representations of the special linear Lie algebras and as splitting posets for type $\mathsf{A}$ fundamental symmetric functions, connections which are further explored in sequel papers for types $\mathsf{A}$, $\mathsf{C}$, and $\mathsf{B}$. In this paper, this connection affords a solution to the proposed domino game as well as new descriptions of the type $\mathsf{A}$ fundamental lattices.