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

Abstract

Let $A$ be any commutative unital ring and let $\operatorname{GL}_{2,A}$ be the general linear group scheme on $A$ of rank $2$. We study the representation theory of $\operatorname{GL}_{2,A}$ and the symmetric powers $\operatorname{Sym}^d(V)$, where $(V, \Delta)$ is the standard right comodule on $\operatorname{GL}_{2,A}$. We prove a refined Weyl character formula for $\operatorname{Sym}^d(V)$. There is for any integer $d \geq 1$ a (canonical) refined weight space decomposition $\operatorname{Sym}^d(V) \cong \oplus_i \operatorname{Sym}^d(V)^i$ where each direct summand $\operatorname{Sym}^d(V)^i$ is a comodule on $N \subseteq \operatorname{GL}_{2,A}$. Here $N$ is the schematic normalizer of the diagonal torus $T \subseteq \operatorname{GL}_{2,A}$. We prove a character formula for the direct summands of $\operatorname{Sym}^d(V)$ for any integer $d \geq 1$. 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 $\operatorname{Sym}^d(V \otimes K)$ as a comodule on $\operatorname{GL}_{2,K}$ 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 $\operatorname{GL}_{2,A}$ and $\operatorname{SL}_{2,A}$. The study may have applications to the study of groups $G$ such as $\operatorname{SL}(n,k)$ and $\operatorname{GL}(n,k)$ and quotients $G/H$ 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 $V(\lambda) \subsetneq S_{\lambda}(V)$.