  In this paper, we relate combinatorial conditions for polarizations of powers
of the graded maximal ideal with rank conditions on submodules generated by
collections of Young tableaux. We apply discrete Morse theory to the
hypersimplex resolution introduced by Batzies--Welker to show that the
$L$-complex of Buchsbaum and Eisenbud for powers of the graded maximal ideal is
supported on a CW-complex. We then translate the "spanning tree condition" of
Almousa--Fl\o ystad--Lohne characterizing polarizations of powers of the graded
maximal ideal into a condition about which sets of hook tableaux span a certain
Schur module. As an application, we give a complete combinatorial
characterization of polarizations of so-called "restricted powers" of the
graded maximal ideal.

Commutative Algebra