Composition Tableaux basis for Schur functors and the Pl\"ucker algebra

Abstract

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.