Synopsis:
We will discuss combinatorial topics related to cluster algebras and
total positivity.
The course will be loosely based on several chapters of the forthcoming textbook
"Introduction to Cluster Algebras"
by Sergey Fomin, Lauren Williams, and Andrei Zelevinsky;
as well as other sources.
W 09/05/2018.
Forerunners of cluster algebras:
Coxeter's frieze patterns.
Glide reflections and Laurent phenomenon.
[Coxeter], [Conway-Coxeter].
F 09/07/2018.
Ptolemy's relation
and triangulations of an n-gon.
M 09/10/2018. Grassmannian Gr(2,n). Proofs of n-periodicity,
glide reflection symmetry, and the positive Laurent phenomenon for
Ptolemy's algebra and frieze patterns.
W 09/12/2018. Matchings in graphs. A combinatorial formula
for Laurent polynomials. Somos sequences.
[Carroll-Price], [Propp], [Schiffler].
Bibliography:
[Carroll-Price]
G. D. Carroll, G. Price:
Two new combinatorial models for the Ptolemy recurrence,
2003: paper
[Conway-Coxeter]
J. H. Conway and H. S. M. Coxeter. Triangulated
polygons and frieze patterns. Math. Gaz.,
57(400):87-94, 175-183, 1973.
[Coxeter] H. S. M. Coxeter, Frieze patterns,
Acta Arithmetica XVII (1971), 297-310:
paper.
[Fomin-Williams-Zelevinsky] S. Fomin, L. Williams, A. Zelevinsky:
Introduction to Cluster Algebras:
arXiv:1608.05735
(chapters 1-3),
arXiv:1707.07190
(chapters 4-5).
[Propp] J. Propp: The combinatorics of frieze patterns and Markoff numbers:
arXiv:0511633, 2005.
[Schiffler]
R. Schiffler: A cluster expansion formula (A_n case):
Electronic Journal of Combinatorics,
arXiv:0611956.
This webpage will be updated periodically. All information related to the
course (list of lectures, references, problem sets, etc.) will be posted
here.