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

• Published in: arXiv.org 2017-02
• Main Authors: Donnelly, Robert G, Donovan, Elizabeth A, Schroeder, Timothy A
• Format: Article
• Language: English
• Subjects: Combinatorial analysis; Diamonds; Games; Graphical representations; Graphs; Lattices; Lie groups; Modular structures; Set theory
• ISSN: 2331-8422

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.