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

TL;DR

This paper develops a refined Weyl character formula for symmetric powers of the standard comodule on the general linear group scheme over any commutative ring, generalizing classical results and providing explicit examples.

Contribution

It introduces a canonical refined weight space decomposition for symmetric powers of comodules on -group schemes, leading to a new character formula that extends classical Weyl formulas.

Findings

Refined weight space decomposition exists for  and  group schemes.

Derived a character formula for summands of symmetric powers.

Connected the refined decomposition to irreducible modules in positive characteristic.

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 $V(\lambda) \subsetneq S_{\lambda}(V)$.