Demazure atoms are core components in the description of Demazure crystals.  It is natural to ask to which Demazure atom a given vertex of a crystal belongs.  The answer to this question is provided by the right key map (respectively left key for opposite Demazure atoms)....

Spectral graph theory is the study of the relationship between the graphical properties of a graph and the spectral properties (i.e., eigenvalues and eigenvectors) of various matrices associated with that graph, most commonly the adjacency matrix. The spectrum of the...