  We show that combinatorial objects called row-strict composition tableaux,
introduced by Mason and Remmel in 2014 and closely related to the
quasi-symmetric Schur functions of Haglund-Luoto-Mason-van Willigenburg, form a
basis for Schur functors of finite free modules over arbitrary commutative
rings. When the ring is the complex numbers, this produces a new basis for the
irreducible polynomial representations of $\operatorname{GL}_n(\mathbb{C})$.
Moreover, in this case it also produces new basis for the Pl\"ucker algebra, a
subalgebra of the polynomial ring over $\mathbb{C}$ in $n^2$ variables, which
is of independent combinatorial and geometric interests. As an aside we also
show that these results hold for other combinatorial objects called reverse row
strict tableau.
