= Polynomial representation of the general linear group
{title2=$\rho_{ij}\in\mathbb C[g_{ab}]$}
= Polynomial representation
{synonym}
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 $\lambda\vdash n$ and at most $m$ rows.
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