 Polynomial representation of the general linear group

In a polynomial representation, every matrix coefficient is a polynomial in the matrix entries. The action extends to the monoid of all matrices. The scalar action splits it into homogeneous polynomial degrees; each homogeneous degree-$n$ category is the module category of the Schur algebra $S(m,n)$. Its irreducibles are the Schur modules $D_{\lambda }({\mathbb{C}}^m)$ with $\lambda ⊢n$ and at most $m$ rows.