A new type of array allows you to specify and store only the independent components.
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For matrices, the only nontrivial symmetries are symmetry and antisymmetry, but for higher-rank arrays there are many more possibilities. The following figure illustrates some of them for a rank-4 array. 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.
Storing only the independent components can produce enormous space savings.
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This allows exterior algebra computations in large dimensions—in this case a 4-form and 20-form in 25 dimensions.