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The Grassmannian of k-planes in n-dimensional space is a well-loved variety that is the keeper of many interesting combinatorial problems. Its coordinate ring can be endowed with something called a cluster structure; that is, some combinatorially generated algebra that is of interest to many mathematicians and physicists alike. In this talk, we will discuss joint work with Moriah Elkin and Gregg Musiker about a combinatorial model for these Grassmannian cluster algebras. The generators of these algebras become rather complicated for k>2 and in our work, we aim to give a graph theoretic description of these generators. More specifically, we give a formula for these generators in terms of higher dimer models, a generalization of perfect matchings on a graph. This formula employs SL_k web combinatorics and we conjecture these webs are the key ingredient to understanding Grassmannian cluster algebras.