🧄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.

Proof of Covariant Index Anti-Symmetrisation | Proof and Derivationviadean on Notion