Coxeter combinatorics for sum formulas in the representation theory of algebraic groups

TL;DR

This paper introduces new recursive and duality formulas for the Jantzen sum formula in the representation theory of algebraic groups, providing computational tools and bounds for Weyl modules in positive characteristic.

Contribution

It presents a simple recursion formula and a duality relation for the Jantzen sum formula, along with representation-theoretic explanations and bounds on filtration lengths.

Findings

Recursion formula for Jantzen sum in Weyl modules

Duality formula relating Jantzen and Andersen's sum formulas

Upper bounds on the length of Jantzen filtrations

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $\mathbb{F}$ of characteristic $p\geq h$, the Coxeter number of $G$. We observe an easy `recursion formula' for computing the Jantzen sum formula of a Weyl module with $p$-regular highest weight. We also discuss a `duality formula' that relates the Jantzen sum formula to Andersen's sum formula for tilting filtrations and we give two different representation theoretic explanations of the recursion formula. As a corollary, we also obtain an upper bound on the length of the Jantzen filtration of a Weyl module with $p$-regular highest weight in terms of the length of the Jantzen filtration of a Weyl module with highest weight in an adjacent alcove.