| Summary: | Abstract We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of n-particle amplitudes in planar N $$ \mathcal{N} $$ = 4 Yang-Mills theory and certain polytopes associated to the Grassmannian Gr(4, n). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of n ≤ 8, and find that the polytope C † 4 9 $$ {\mathcal{C}}^{\dagger}\left(4,9\right) $$ encodes all rational letters, and all square roots of the algebraic letters, of known nine-particle amplitudes.
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