From the weak Bruhat order to crystal po

We investigate the ways in which fundamental properties of the weak Bruhat order on a Weyl group can be lifted (or not) to a corresponding highest weight crystal graph, viewed as a partially ordered set; the latter projects to the weak order via the key map. First, a crystal theoretic analogue of the statement that any two reduced expressions for the same Coxeter group element are related by Coxeter moves is proven for all lower intervals in a simply or doubly laced crystal. However, it is shown that no finite set of moves exists for arbitrary crystal graph intervals, even in type A. Additionally, for crystals associated with Kac-Moody algebras, we demonstrate that for lower intervals the Möbius function is always 0 or ±1, and in finite type this is also proven for upper intervals, with precise formulas provided for each case. The order complex for these intervals is shown to be homotopy equivalent to a ball or sphere of some dimension, despite often not being shellable. We also derive new properties of the key map, determining it entirely by the edge-colored poset-theoretic structure of the crystal, and provide a recursive algorithm for calculating it.