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Journal of High Energy Physics

We review various aspects of cluster algebras and the ways in which they appear in the study of loop-level amplitudes in planar ${\cal N} = 4$ supersymmetric Yang-Mills theory. In particular, we highlight the different forms of cluster-algebraic structure that appear in this theory's two-loop MHV amplitudes---considered as functions, symbols, and at the level of their Lie cobracket---and recount how the `nonclassical' part of these amplitudes can be decomposed into specific functions evaluated on the $A_2$ or $A_3$ subalgebras of Gr$(4,n)$. We then extend this line of inquiry by searching for other subalgebras over which these amplitudes can be decomposed. We focus on the case of seven-particle kinematics, where we show that the nonclassical part of the two-loop MHV amplitude is also constructible out of functions evaluated on the $D_5$ and $A_5$ subalgebras of Gr$(4,7)$, and that these decompositions are themselves decomposable in terms of the same $A_4$ function. These nested decompositions take an especially canonical form, which is dictated in each case by constraints arising from the automorphism group of the parent algebra.

Cluster algebras and the subalgebra constructibility of the seven-particle remainder function