Affine quantum Bruhat graphs and double affine Demazure products

Dean, Lewis (2026) Affine quantum Bruhat graphs and double affine Demazure products. PhD thesis, University of Glasgow.

Abstract

In recent work, Muthiah and Puskás give a conjectural definition of a Demazure product for a double affine Weyl semigroup WT by considering its associated Kac-Moody affine Hecke algebra. Motivated by this, we generalise work by Schremmer, who gave a formula for the single affine Demazure product using the quantum Bruhat graph. We focus on type cSL2, where we prove that the corresponding affine quantum Bruhat graph has a well-defined weight function, and satisfies nice properties involving length posivite sets, allowing us to construct a well-defined Demazure product for 𝑊𝑇 of this type. We go on to show that this product is associative for level greater than one, as well as linking it to the single affine Demazure product.

Furthermore, assuming that our double affine Demazure product is well-defined, we then give specific results connecting this product with Muthiah and Orr’s length function on 𝑊𝑇 in general type, verifying that it matches examples computed by Muthiah and Puskás in the Kac-Moody affine Hecke algebra. We also discuss the semi-infinite Bruhat order and its link to the quantum Bruhat graph, giving a candidate definition for the double affine semi-infinite Bruhat order, and obtaining some preliminary results.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Supported by funding from the Engineering and Physical Sciences Research Council (EPSRC).
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Funder's Name: Engineering and Physical Sciences Research Council (EPSRC)
• Supervisor's Name: Muthiah, Dr. Dinakar and Feigin, Professor Misha
• Date of Award: 2026
• Depositing User: Theses Team
• Unique ID: glathesis:2026-86179
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 14 Aug 2026 08:44
• Last Modified: 14 Aug 2026 08:46
• Thesis DOI: 10.5525/gla.thesis.86179
• URI: https://theses.gla.ac.uk/id/eprint/86179