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I consider the permutations [itex]\sigma_i[/itex], where [itex]i\in \{1,\ldots,n\}[/itex], of the following kind:

[itex]\sigma_i[/itex] is obtained by choosing thei-th element from(1,..,n)and by shifting it to the first position; for instance [itex]\sigma_3 = (3,1,2,\ldots,n)[/itex]. The parity of [itex]\sigma_i[/itex] is clearly [itex](-1)^{i-1}[/itex].

Forn=3, I would like to express the following sum based on [itex]\sigma_i[/itex]-permutations in a compact form, using the Levi-Civita [itex]\varepsilon[/itex] and Kronecker [itex]\delta[/itex] symbols:

[tex]a_{123} - a_{213} + a_{312}[/tex]

The first index is 1,2,3 so that would be easy to obtain with a repeated index, but I find the remaining two indices difficult to pull out by manipulations with the [itex]\varepsilon[/itex] and [itex]\delta[/itex] symbols.

Any help?

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# Problem with Einstein notation and Levi-Civita symbol

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