# On character formulas for simple and tilting modules

## TL;DR

This paper demonstrates how characters of tilting modules can be explicitly used to derive simple module characters for connected reductive algebraic groups over fields of any characteristic, linking tilting and simple module character formulas.

## Contribution

It establishes a concrete method to obtain simple module characters from tilting module characters for all primes, extending previous results to all characteristics.

## Key findings

- Provides a formula for simple module characters from tilting module characters.
- Shows the approach applies for all primes, not just large p.
- Connects tilting module character formulas with simple module character formulas.

## Abstract

We show that the characters of tilting modules can be used, in a concrete and explicit way, to obtain the simple characters of a connected reductive algebraic group $G$ over an algebraically closed field $\Bbbk$ of characteristic $p$, for all $p$. Thus, once a formula for the characters of the indecomposable tilting $G$-modules has been found, a formula for the simple modules has been also. An immediate implication is that the work of Achar, Makisumi, Riche, and Williamson in \cite{AMRW} provides a character formula for simple $G$-modules when $p>h$, the Coxeter number of $G$.