# Branching cones and polytopes for Lie algebras

Kalmbach, Daniel; Fourier, Ghislain (Thesis advisor); Littelmann, Peter (Thesis advisor)

*Aachen : RWTH Aachen University (2022)*

Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022

Abstract

The aim of this work is to study different aspects of branchings of Lie algebras. We want topresent new results and put them in the context of already known ones. Therefore, we try toput them all in the same framework, using and extending the theory of birational sequencesof Fang, Fourier and Littelmann [10]. We will introduce branching cones and explain how to explicitly describe them for thebranchings of B_n, C-n, D-n over A-n−1 using results of Littelmann [29], [28] and Berenstein-Zelevinsky [4]. Next, we will study string branching polytopes and translate our results toLusztig’s parametrization. Both, the string and the Lusztig parametrization can be describedin the language of birational sequences. This allows us to describe our results in the same framework as results on other branchingswhich cannot be described in these parametrizations. When working with birational sequences, we need to fix a weighted order on Z^N_≥0. We choose the height-weighted opposite lexicographic order and want to justify this choice as a natural one for studying branchings. Therefore, we translate the works of other authors to the language of birational sequencesand show that - in some cases with some slight changes of the sequences - we can obtain the same results as them using our favorite order. In this context we also encounter the first embeddings which are not of Levi type. We also want to use our order for studying branchings given by foldings of Dynkin diagrams. We therefore need to generalize the definition of birational sequences. For these branchings we will only be able to give conjectures for the branching cones and polytopes, but they hold true in all examples we checked and the strategy to find them seems reasonable. We call itfolding of inequalities. In total, we will encounter three quite different approaches for computing in equalities for the branching cones and polytopes as each kind of branching has its specifics. At the end of this work, we will explain the connection of our results to branching algebras and how to get toric degenerations of them.

### Identifier

- DOI: 10.18154/RWTH-2022-03360
- RWTH PUBLICATIONS: RWTH-2022-03360