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

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- by Jonathan Gruber PDF
- Represent. Theory
**26**(2022), 68-93 Request permission

## 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.## References

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## Additional Information

**Jonathan Gruber**- Affiliation: École Polytechnique Federale de Lausanne, 1015 Lausanne, Switzerland
- MR Author ID: 1425592
- ORCID: 0000-0001-5975-8041
- Email: jonathan.gruber@epfl.ch
- Received by editor(s): July 7, 2021
- Received by editor(s) in revised form: August 20, 2021
- Published electronically: March 2, 2022
- Additional Notes: This work was supported by the Swiss National Science Foundation under grant number FNS 200020_175571.
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory
**26**(2022), 68-93 - MSC (2020): Primary 20G05
- DOI: https://doi.org/10.1090/ert/599
- MathSciNet review: 4388551