Web and tableau representations of rank two cluster variables for Gr (5, 9)

Tymoczko and Russell provide a bijection between semi-standard tableaux of rectangular shapes and $s{l}_r$-webs when $r\le 3$. Due to Chang, Duan, Fraser, and Li, cluster variables in the Grassmannian cluster algebra correspond to certain semi-standard tableaux of rectangular shapes. When $r=2$ or $r=3$, the cluster variables in the Grassmannian cluster algebra can be represented both by tableaux and webs. Recently, Elkin, Musiker, and Wright refined the twist map, providing a method to connect webs and cluster polynomials through the compatibility defined by Lam in 2015, thereby also connecting cluster variables and webs. In this paper, we study the webs corresponding to rank 2 Plücker polynomials, particularly cluster variables in the Grassmannian cluster algebra $\mathbb{C}[\mathrm{G}\mathrm{r}(5,9)]$. Additionally, we examine the webs corresponding to rank 2 cluster variables in $\mathrm{Gr}(5,9)$, and we find 11 distinct webs (up to dihedral translation) arising from these variables. Consequently, we propose a conjecture that for all cluster variables the webs obtained through the diagram method and the compatibility method coincide.