$g$-vectors and $DT$-$F$-polynomials for Grassmannians

TL;DR

This paper reviews Frobenius categorification of cluster algebras and applies it to determine g-vectors of Plücker coordinates and express F-polynomials via 3D Young diagrams, offering new proofs for existing theorems.

Contribution

It provides a Frobenius categorification approach to compute g-vectors and F-polynomials for Grassmannians, connecting cluster algebra invariants with combinatorial models.

Findings

Determined g-vectors of Plücker coordinates relative to the initial seed.

Expressed F-polynomials associated with the Donaldson-Thomas transformation using 3D Young diagrams.

Provided a new proof for a theorem of Daping Weng.

Abstract

We review $\mathrm{Hom}$-infinite Frobenius categorification of cluster algebras with coefficients and use it to give two applications of Jensen--King--Su's Frobenius categorification of the Grassmannian: 1) we determine the $g$-vectors of the Pl\"ucker coordinates with respect to the triangular initial seed and 2) we express the $F$-polynomials associated with the Donaldson--Thomas transformation in terms of $3$-dimensional Young diagrams thus providing a new proof for a theorem of Daping Weng.