Multiple rational normal forms in Lie theory

2025/06/02 by Dmitriy Voloshyn, Voloshyn, Dmitriy
Mathematics · Physics and Astronomy · #13F60 (Secondary) #20G07 (Primary) 20F55 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2506.01530

openalex publication_date 2025/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

We study the decomposition of a generic element g ∈ G of a connected reductive complex algebraic group G in the form g = N(g) B(g) u N(g)^-1 where N: G \dashrightarrow N_- and B : G \dashrightarrow B₊ are rational maps onto a unipotent subgroup N_- and a Borel subgroup B₊ opposite to N_-, and u is a representative of a Weyl group element u. We introduce a class of rational Weyl group elements that give rise to such decompositions, and study their various properties.

Citations

• Introduction to Cluster Algebras. Chapters 1-3
• Cluster algebras: an introduction
• Total positivity and cluster algebras
• Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups
• Moduli spaces of local systems and higher Teichmuller theory
• Cluster algebras and Poisson geometry
• Cluster algebras I: Foundations
• Double Bruhat cells and total positivity

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