Indecomposable Modules in the Grassmannian Cluster Category $\mathrm{C}\mathrm{M}(B_{5,10})$

In this paper we study indecomposable rank 2 modules in the Grassmannian cluster category $\mathrm{C}\mathrm{M}(B_{5,10})$. This is the smallest wild case containing modules whose profile layers are $5$-interlacing. We construct all rank 2 indecomposable modules with filtration ${i,i+2,i+4,i+6,i+8}\mid {i+1,i+3,i+5,i+7,i+9}$, classify them up to isomorphism, and parameterize all infinite families of non-isomorphic rank 2 modules.