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

Gruber, Jonathan

doi:10.1090/ert/599

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

Represent. Theory 26 (2022), 68-93

DOI: https://doi.org/10.1090/ert/599

Published electronically: March 2, 2022

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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.

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