2023/11/06 by Germain, Antoine de Saint
#13F60 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2311.03073
Motivated by cluster ensembles, we introduce a new variant of frieze patterns associated to acyclic cluster algebras, which we call \bf Y-frieze patterns. Using the mutation rules for \bf Y-variables, we define a large class of \bf Y-frieze patterns called unitary \bf Y-frieze patterns, and show that the ensemble map induces a map from (unitary) frieze patterns to (unitary) \bf Y-frieze patterns. In rank 2, we show that \bf Y-frieze patterns are (associated to) friezes of generalised cluster algebras. In finite type (not necessarily rank 2), we show that \bf Y-frieze patterns share the same symmetries as frieze patterns, and prove that their number is finite.