Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 080, 26 pages      arXiv:2401.15601      https://doi.org/10.3842/SIGMA.2024.080

On $F$-Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula

Changjian Fu, Liangang Peng and Huihui Ye
Department of Mathematics, Sichuan University, Chengdu 610064, P.R. China

Received March 12, 2024, in final form August 25, 2024; Published online September 03, 2024

Abstract
We show the polynomial property of $F$-polynomials for generalized quantum cluster algebras and obtain the associated separation formulas under a mild condition. Along the way, we obtain Gupta's formulas of $F$-polynomials for generalized quantum cluster algebras. These formulas specialize to Gupta's formulas for quantum cluster algebras and cluster algebras respectively. Finally, a generalization of Gupta's formula has also been discussed in the setting of generalized cluster algebras.

Key words: $F$-polynomial; separation formula; Fock-Goncharov decomposition; generalized quantum cluster algebra; generalized cluster algebra.

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