Schur modules and their characters

Assume the Axiom of Choice. Let $V={\mathbb{C}}^r$ with $r\ge 1$, and let $\lambda$ be a partition with at most $r$ parts (Partitions, English diagrams, and conjugation). Put $n=\mathrm{\mid }\lambda \mathrm{\mid }$ and define the Schur module $S_{\lambda }(V):={\mathrm{Hom}}_{S_n}(S^{\lambda },V^{\otimes n}),$ 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); the $\mathrm{GL}(V)$-action is by postcomposition, $(g\cdot \phi )(s)=g^{\otimes n}\phi (s)$. By Schur-Weyl decomposition and highest weights the space $S_{\lambda }(V)$ is a nonzero irreducible polynomial $\mathrm{GL}(V)$-module of highest weight $\lambda$ in the sense of Polynomial representations of GL_r and their highest weights; for a partition with $\mathrm{\ell }(\lambda )>r$ we define $S_{\lambda }(V):=0$.

Let $T=\{\mathrm{diag}(t_1,\dots ,t_r)\}\subseteq \mathrm{GL}(V)$ be the diagonal torus as in Polynomial representations of GL_r and their highest weights. The character of a finite-dimensional polynomial $\mathrm{GL}(V)$-module $W$ is the polynomial $\mathrm{ch}W={\sum }_{\alpha \in {\mathbb{Z}}^r}\dim W_{\alpha }{x}^{\alpha }\in \mathbb{Z}[x_1,\dots ,x_r],x^{\alpha }=x_1^{{\alpha }_1}\cdots x_r^{{\alpha }_r},$ where $W_{\alpha }$ is the weight space of $T$. Characters are additive over direct sums, and $\mathrm{ch}(W\otimes W^{\mathrm{\prime }})=\mathrm{ch}W\cdot \mathrm{ch}{W}^{\mathrm{\prime }}$ for finite-dimensional polynomial modules, because $(W\otimes W^{\mathrm{\prime }}{)}_{\alpha }={⨁}_{\beta +\gamma =\alpha }{W}_{\beta }\otimes W_{\gamma }^{\mathrm{\prime }}$. The character of a polynomial module is a symmetric polynomial: conjugation by a permutation matrix $g_{\sigma }$ carries $\mathrm{diag}(t_1,\dots ,t_r)$ to $\mathrm{diag}(t_{\sigma (1)},\dots ,t_{\sigma (r)})$, and characters of representations of a group are class functions, so $\mathrm{ch}W$ is invariant under permuting $x_1,\dots ,x_r$. In particular $\mathrm{ch}{S}_{\lambda }(V)$ is a symmetric polynomial for $\mathrm{\ell }(\lambda )\le r$, homogeneous of degree $n$ because the Schur--Weyl decomposition of Schur-Weyl decomposition and highest weights exhibits $S_{\lambda }(V)$ as a direct summand of $V^{\otimes n}$, all of whose weights have total degree $n$; it is $0$ for $\mathrm{\ell }(\lambda )>r$.

Remarks. The module $S_{\lambda }(V)$ is the multiplicity space $M_{\lambda }$ of Schur-Weyl decomposition and highest weights; the two notations denote the same object. The character $\mathrm{ch}{S}_{\lambda }(V)$ is computed explicitly in Semistandard tableaux expand Schur characters as the sum over semistandard tableaux of shape $\lambda$ with entries in $\{1,\dots ,r\}$, and it is the rank-$r$ Schur polynomial $s_{\lambda }(x_1,\dots ,x_r)$ of Stable Schur functions from bialternants.