Combinatorics and Algebra Seminar- SLk-Tilings and Paths in  Zk

Title:  SL_k-Tilings and Paths in  Z^k

Speaker: Zachery Peterson, William and Mary
Date and time: Wednesday 10/22, 4–5pm
Place: Phillips 108

 

Abstract: An SL_k-frieze is a bi-infinite array of integers where adjacent entries satisfy a certain diamond rule. SL₂-friezes were introduced and studied by Conway and Coxeter. Later, these were generalized to infinite matrix-like structures called tilings as well as higher values of k. A recent paper by Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and SL₂-tilings. We extend this result to higher k by constructing a bijection between SL_k-tilings and certain pairs of bi-infinite strips of vectors in Z^k called paths. The key ingredient in the proof is the relation to Plucker friezes and Grassmannian cluster algebras. As an application, we obtain results about periodicity, duality, and positivity for tilings