In this talk we will study scattering amplitudes N=4 super-Yang-Mills theory. In this theory, scattering amplitudes are known to be functions of cluster variables of Gr(4,n) and certain algebraic functions of cluster variables. We will give an overview of how this cluster algebraic structure manifests, and will exploit it in an algorithm for computing symbol alphabets by solving matrix equations of the form C.Z = 0 associated with plabic graphs. These matrix equations associate functions on Gr(m,n) to parameterizations of certain cells of Gr_+ (k,n) indexed by plabic graphs. We are able to reproduce all known algebraic functions of cluster variables appearing in known symbol alphabets. We further show that it is possible to obtain all rational symbol letters (in fact all cluster variables) by solving C.Z = 0 if one allows C to be an arbitrary cluster parameterization of the top cell of Gr_+ (n-4,n).
