  Over a field of characteristic two, we develop a theory of standard monomials
for polynomial rings modulo a Frobenius power of the maximal ideal generated by
all variables. As a result, we obtain a filtration by modular
GL_n-representations whose characters are given by particular truncated Schur
polynomials, thus proving a conjecture by Gao--Raicu--VandeBogert in the
characteristic two case.
