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my question concerns the following: Though widely used, there does not seem to be any standard reference where the common symmetrization and anti-symmetrization identities are rigorously proven in the general setting of ##n##-dimensional pseudo-Euclidean spaces. At least I have not found any after an extensive google/literature search, but I would be happy if you can name one. An example where they are more or less just given is found in the book "General Relativity" by Wald (p. 432 sq.). Of course, I tried to prove the identities myself with some success, but there are some instances where I have trouble. I and possibly others with the same issues would greatly appreciate your help.

Now, let's get more specific:

The ##n##-dimensional Levi-Civita Symbol can be defined via the Kronecker Symbol as

$$ \varepsilon_{i_1 \dots i_n} := n! \,

\delta^{[1}_{i_1} \cdots \delta^{n]}_{i_n} \quad ,$$

which is ##1## for even permutations of ##(1, \dots, n)##, ##-1## for odd ones and ##0## otherwise.

Now, using the following definition of the determinant

$$ \det A := \varepsilon_{i_1 \dots i_n} A^{i_1}{}_1 \cdots A^{i_n}{}_n

= n! \, A^{[1}{}_1 \cdots A^{n]}{}_n \quad ,$$

one can prove the following:

$$\varepsilon^{i_1 \dots i_n}\, \varepsilon_{i_1 \dots i_n} = (-1)^s \, n! $$

Here the standard pseudo-Euclidean scalar product ##\eta## with ##s## minus-signs was used to raise the indices.

Now the problems:

1) The following identity

$$\varepsilon^{i_1 \dots i_n}\, \varepsilon_{j_1 \dots j_n} = (-1)^s \, n! \,

\delta^{[i_1}_{j_1} \cdots \delta^{i_n]}_{j_n}$$

also supposedly holds, but I got stuck in the proof. Writing everything out, pulling out minus signs and pulling fixed indices down, I was able to show that

$$ \varepsilon^{i_1 \dots i_n}\, \varepsilon_{j_1 \dots j_n}

= (-1)^s \, (n!)^2 \, \delta^{[i_1}_{1} \cdots \delta^{i_n]}_{n} \,

\delta^{1}_{[j_1} \cdots \delta^{n}_{j_n]} \quad , $$

but what to do now?

2) Using the above identity, I would like to obtain a nice expression for

$$\varepsilon^{i_1 \dots i_k i_{k+1} \dots i_n}\, \varepsilon_{i_1 \dots i_k j_{k+1}

\dots j_n} $$

in terms of the Kronecker. I figured that I need to partially "dissolve" the anti-symmetrization, so I am looking for an identity along the lines of

$$ T_{[i_1 \dots i_k i_{k+1} \dots i _l ]} = f (n,k,l) \, \sum_{\sigma \in S^l} \text{sgn} ( \sigma ) \, T _{[ \sigma (i_1) \dots \sigma (i_k) ] \sigma( i_{k+1}) \dots

\sigma(i _l) }

$$

where ##f (n,k,l)## is some normalization factor. How do I prove this (algebraically)?

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# A Algebraic Proofs of Levi-Civita Symbol Identities

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