Séminaire Lotharingien de Combinatoire, B92b (2025), 18 pp.
Sam Armon
Extremal Subsets and Atom-Positivity
Abstract.
Demazure crystals give a combinatorial framework for the study of Demazure modules. They are extremal, in that they satisfy Kashiwara's string property, and they are Demazure atom-positive, in that they decompose naturally into subsets called crystal Demazure atoms. The properties of extremality and atom-positivity are further linked by a conjecture of Polo, which concerns the structure of the tensor product of two Demazure modules. We study these two properties in isolation - first providing a recursive formula for generating crystal Demazure atoms, which generalizes a result of Lascoux and Schützenberger, and then examining the structure of extremal subsets and their characters - before commenting on the connection (or lack thereof) between them.
Received: September 19, 2024.
Revised: August 16, 2025.
Accepted: August 16, 2025.
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