Duality between Final-Seed and Initial-Seed Mutations in Cluster Algebras
Abstract
We study the duality between the mutations and the initial-seed mutations in cluster algebras, where the initial-seed mutations are the transformations of rational expressions of cluster variables in terms of the initial cluster under the change of the initial cluster. In particular, we define the maximal degree matrices of the $F$-polynomials called the $F$-matrices and show that the $F$-matrices have the self-duality which is analogous to the duality between the $C$- and $G$-matrices.
- Publication:
-
SIGMA
- Pub Date:
- May 2019
- DOI:
- 10.3842/SIGMA.2019.040
- arXiv:
- arXiv:1808.02156
- Bibcode:
- 2019SIGMA..15..040F
- Keywords:
-
- cluster algebra; mutation; duality;
- Mathematics - Rings and Algebras;
- Mathematics - Combinatorics;
- Mathematics - Representation Theory;
- 13F60
- E-Print:
- SIGMA 15 (2019), 040, 24 pages