A refined Weyl character formula for comodules on ${\mathrm{GL}}_{2,A}$

Let $A$ be any commutative unital ring and let ${\mathrm{GL}}_{2,A}$ be the general linear group scheme on $A$ of rank $2$. We study the representation theory of ${\mathrm{GL}}_{2,A}$ and the symmetric powers ${\mathrm{Sym}}^d(V)$, where $(V,\mathrm{\Delta })$ is the standard right comodule on ${\mathrm{GL}}_{2,A}$. We prove a refined Weyl character formula for ${\mathrm{Sym}}^d(V)$. There is for any integer $d\ge 1$ a (canonical) refined weight space decomposition ${\mathrm{Sym}}^d(V)\cong {\oplus }_i{\mathrm{Sym}}^d(V{)}^i$ where each direct summand ${\mathrm{Sym}}^d(V{)}^i$ is a comodule on $N\subseteq {\mathrm{GL}}_{2,A}$. Here $N$ is the schematic normalizer of the diagonal torus $T\subseteq {\mathrm{GL}}_{2,A}$. We prove a character formula for the direct summands of ${\mathrm{Sym}}^d(V)$ for any integer $d\ge 1$. This refined Weyl character formula implies the classical Weyl character formula. As a Corollary we get a refined Weyl character formula for the pull back ${\mathrm{Sym}}^d(V\otimes K)$ as a comodule on ${\mathrm{GL}}_{2,K}$ where $K$ is any field. We also calculate explicit examples involving the symmetric powers, symmetric tensors and their duals. The refined weight space decomposition exists in general for group schemes such as ${\mathrm{GL}}_{2,A}$ and ${\mathrm{SL}}_{2,A}$. The study may have applications to the study of groups $G$ such as $\mathrm{SL}(n,k)$ and $\mathrm{GL}(n,k)$ and quotients $G\mathrm{/}H$ where $k$ is an arbitrary field (or a Dedekind domain) and $H\subseteq G$ is a closed subgroup. The refined weight space decomposition of $S_{\lambda }(V)$ has a relation with irreducible module over a field of positive characteristic. In an example I prove it recovers the irreducible module $V(\lambda )⊊S_{\lambda }(V)$.