Polynomial representations of GL_r and their highest weights

Assume the Axiom of Choice. Let $V$ be a finite-dimensional complex vector space of dimension $r\ge 1$ and put $G=\mathrm{GL}(V)$; fixing a basis, identify $G$ with ${\mathrm{GL}}_r(\mathbb{C})$ and let $g_{ij}$ denote the matrix entries of $g\in G$. A finite-dimensional representation $\rho \text{\hspace{0.17em}⁣}:G\to \mathrm{GL}(W)$ is polynomial if in some pair of bases (equivalently, in every pair of bases) the matrix coefficients of $\rho$ are polynomial functions of the $g_{ij}$; it is rational if these coefficients are rational functions defined on all of $G$.

For the diagonal torus $T=\{\mathrm{diag}(t_1,\dots ,t_r):t_i\in {\mathbb{C}}^{\times }\}$ a polynomial representation is a direct sum of weight spaces $W_{\alpha }=\{w\in W:\rho (t)w=t^{\alpha }w\text{ for all }t\in T\},\alpha =({\alpha }_1,\dots ,{\alpha }_r)\in {\mathbb{Z}}^r,t^{\alpha }=t_1^{{\alpha }_1}\cdots t_r^{{\alpha }_r},$ the eigenvalues $\alpha$ with $W_{\alpha }\mathrm{\ne }0$ being the weights of $W$ (Etingof §27.3; Goodman--Wallach Ch. 8 §8.1.2). A weight of a polynomial representation has nonnegative entries, $\alpha \in {\mathbb{Z}}_{\ge 0}^r$: the matrix coefficients are polynomial and $t\mapsto t^{\alpha }$ occurs as a polynomial character, so ${\alpha }_i<0$ is impossible. Consequently the highest weight of a polynomial irreducible representation of $G$ (with respect to the Borel subgroup of upper triangular matrices) is a partition $\lambda =({\lambda }_1\ge \cdots \ge {\lambda }_r\ge 0)$ padded by zeros, that is, a partition with at most $r$ parts (Partitions, English diagrams, and conjugation).

For every partition $\lambda$ with $\mathrm{\ell }(\lambda )\le r$ put $n=\mathrm{\mid }\lambda \mathrm{\mid }$ and let $S_{\lambda }(V):={\mathrm{Hom}}_{S_n}(S^{\lambda },V^{\otimes n})$ be the Schur--Weyl module of Schur-Weyl decomposition and highest weights, where $S^{\lambda }$ is the complex Specht module (Column antisymmetrizers, polytabloids, and Specht modules) and $S_n$ acts on $V^{\otimes n}$ by place permutations (Commuting symmetric-group and linear actions on a tensor power). Then $S_{\lambda }(V)$ is a nonzero polynomial irreducible representation of $G$ of highest weight $\lambda$, and distinct partitions with at most $r$ parts give non-isomorphic modules (Schur-Weyl decomposition and highest weights parts (1)--(4)). Conversely every polynomial irreducible representation of $G$ is isomorphic to $S_{\lambda }(V)$ for exactly one partition $\lambda$ with $\mathrm{\ell }(\lambda )\le r$: this is the classical type-$A$ highest-weight classification of polynomial representations (Etingof §27.3--27.4; Goodman--Wallach Theorem 5.5.22 for the corresponding rational classification; see Schur modules and their characters for the module notation used below).