Efficiently Store Arrays with Symmetry 

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X SymmetrizedArray[{{1, 2} -> a, {1, 3} -> b, {2, 3} -> c}, {3, 3}, Antisymmetric[{1, 2}]]; % // Normal // MatrixForm

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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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X normal = Array[Signature[{##}] &, ConstantArray[8, 8]]; sparse = SparseArray@normal; symmetrized = SymmetrizedArray[Range[8] -> 1, ConstantArray[8, 8], Antisymmetric[All]];

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X normal == sparse == symmetrized ByteCount /@ {normal, sparse, symmetrized}

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This allows exterior algebra computations in large dimensions—in this case a 4-form and 20-form in 25 dimensions.

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X form1 = SymmetrizedArray[_ :> RandomInteger[], ConstantArray[25, 4], Antisymmetric[All]]; form2 = SymmetrizedArray[_ :> RandomInteger[], ConstantArray[25, 20], Antisymmetric[All]];

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X HodgeDual[form1 \[TensorWedge] form2] // Normal

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