12:45
Scattering amplitudes in N=4 super-Yang-Mills theory are known to be functions of cluster variables of Gr(4,n) and certain algebraic functions of cluster variables. In this talk we give an overview of the known cluster algebraic structure of both tree amplitudes and the symbol of loop amplitudes. We suggest 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). Finally we discuss a property of the symbol called cluster adjacency.