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Perturbative gauge theory and 2+2=4

The group $S_4$ of permutations on four elements has an irreducible representation corresponding to the partition 4=2+2. This representation appears in several different mathematical contexts: the Jacobi identity of Lie algebras; the Schouten identity for spinors; differences and the cross ratio in the projective plane; the Grassmannian space of 2-planes in linear algebra; the Riemann tensor in differential geometry; and more. We observe that all these occurrences are connected through perturbative non-abelian gauge theory, thereby acting as a leitmotif.