# Proof of Covariant Index Anti-Symmetrisation

This is the demonstration of the dual anti-symmetry of the generalised Kronecker delta $$\delta\_{b\_1 . . b\_n}^{a\_1 . . a\_n}$$. Defined as a determinant, it is completely anti-symmetric in both its contravariant (upper) indices and its covariant (lower) indices. The derived relation, $$\delta\_{b\_1 \ldots b\_n}^{a\_1 \ldots a\_n}= n!\delta\_{\left\[b\_1\right.}^{a\_1} \ldots \delta\_{b\_n}^{a\_n}$$, confirms that anti-symmetrising the covariant indices of the simple Kronecker delta product $$\delta\_{b\_1}^{a\_1} \ldots \delta\_{b\_n}^{a\_n}$$ yields the generalised delta, mirroring the given result for contravariant index anti-symmetrisation. Crucially, the factor of $$n!$$ appears because it is needed to cancel the $$\frac{1}{n!}$$factor inherently present in the definition of the anti-symmetrisation operator, resulting in the determinant definition.

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