Visualize a Rank-4 Array
Rank-4 arrays can be visualized by projecting them down to matrices. In the following, warm-colored squares represent independent components, with like-colored squares equal. Cold-colored squares are the negatives of the independent components, and white squares are forced to be zero by the symmetry.
symmetries = {Symmetric[All], Symmetric[{1, 2}], {{2, 3, 4, 1}, -1}, Antisymmetric[{1, 2}], {Symmetric[{1, 2}], Antisymmetric[{3, 4}]}, {Antisymmetric[{1, 2}], Antisymmetric[{3, 4}], {Cycles[{{1, 3}, {2, 4}}], 1}}, Antisymmetric[All]};labels = Column /@ {{"Complete ", "Symmetry"}, {"Symmetric ", "in 1 pair"}, {"Anticyclic", "Symmetry"}, {"Antisymmetric ", "in 1 pair"}, {"1 Symmetric and ", "1 Antisymmetric Pair"}, {"Riemann", "Symmetry"}, {"Complete", "Antisymmetry "}};Manipulate[MatrixPlot@ArrayFlatten@Normal@Block[{c = 1}, SymmetrizedArray[_ :> c++, ConstantArray[dimension, 4], symmetry]], {{symmetry, Symmetric[All]}, Thread[symmetries -> labels], ControlType -> SetterBar}, {{dimension, 5}, Range[2, 8], ControlType -> SetterBar}]