  The purpose of this paper is to forge a direct link between the hit problem
for the action of the Steenrod algebra A on the polynomial algebra
P(n)=F_2[x_1,...,x_n], over the field F_2 of two elements, and semistandard
Young tableaux as they apply to the modular representation theory of the
general linear group GL(n,F_2). The cohits Q^d(n)=P^d(n)/P^d(n)\cap A^+(P(n))
form a modular representation of GL(n,F_2) and the hit problem is to analyze
this module. In certain generic degrees d we show how the semistandard Young
tableaux can be used to index a set of monomials which span Q^d(n). The hook
formula, which calculates the number of semistandard Young tableaux, then gives
an upper bound for the dimension of Q^d(n). In the particular degree d where
the Steinberg module appears for the first time in P(n) the upper bound is
exact and Q^d(n) can then be identified with the Steinberg module.
