This dissertation is dedicated to the study of combinatorial characterizations of polarizations of powers of the graded maximal ideal in a polynomial ring, and applications of these characterizations to questions in algebra, geometry, and combinatorics. We first characterize polarizations of powers of the graded maximal ideal in terms of their graphs of linear syzygies, and apply this characterization to study their Alexander duals and the question of when the Stanley--Reisner ideals of polarizations are shellable. We then give a novel characterization of polarizations of the same class of ideals in terms of hook tableaux. Finally, we show that any triangulation of a product of simplices gives rise to a polarization of a power of a graded maximal ideal.