Structure of seeds in generalized cluster algebras

Tomoki Nakanishi

Vol. 277 (2015), No. 1, 201–218

DOI: 10.2140/pjm.2015.277.201

Abstract

We study generalized cluster algebras, introduced by Chekhov and Shapiro. When the coefficients satisfy the normalization and quasireciprocity conditions, one can naturally extend the structure theory of seeds in the ordinary cluster algebras by Fomin and Zelevinsky to generalized cluster algebras. As the main result, we obtain formulas expressing cluster variables and coefficients in terms of $c$-vectors, $g$-vectors, and $F$-polynomials.

Keywords

cluster algebra

Mathematical Subject Classification 2010

Primary: 13F60

Milestones

Received: 27 November 2014

Revised: 21 January 2015

Accepted: 22 January 2015

Published: 6 August 2015