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We explore the relationship between holoraumy and Hodge duality beyond fourdimensions. We find this relationship to be ephemeral beyond six dimensions: itis not demanded by the structure of such supersymmetrical theories. In fourdimensions for the case of the vector-tensor $\cal N$ = 4 multiplet, however,we show that such a linkage is present. Reduction to 1D theories presentsevidence for a linkage from higher-dimensional supersymmetry to aninfinite-dimensional algebra extending $\mathfrak{Spin}(N)$.

Infinite-Dimensional Algebraic $\mathfrak{Spin}$($N$) Structure in  Extended/Higher Dimensional SUSY Holoraumy for Valise and On-Shell  Supermultiplet Representations