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Tensor ordering

asked 2019-06-23 01:57:27 -0600

joshualin gravatar image

I've been trying to figure out how tensor indices work with sage and I have a really simple question - how are the indices ordered after contracting two tensors? For example, if I have two tensors S,T or type (s_1,s_2) and (t_1,t_2) and I contract them, how will the indices of the resulting tensor be ordered? e.g. if S and T are both of type (3,3), then:

$$ S.\text{contract}(1,T,4) = S^{abc}_{\quad def} {\color{white}*} T^{ghi}_{\quad jbk}$$

how would the resulting tensors indicices be ordered?

$$R^{ac\quad ghi}_{\quad def \quad jk}$$

or

$$ R^{acghi}_{\quad \quad defjk}$$

I tried looking on the page for tensor indices but I couldn't figure it out; experimentation seemed to suggest the second but I wanted to be sure. Thanks; and sorry if this is a silly question whose explanation I missed in the docs

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answered 2019-06-23 10:54:57 -0600

eric_g gravatar image

Yes this is the second form, namely $$ S^{abc}_{\quad def} T^{ghi}_{\quad jbk} = R^{acghi}_{\quad \ \ defjk} $$ This is so because in SageMath, the contravariant indices come always before the covariant ones.

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Asked: 2019-06-23 01:57:27 -0600

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Last updated: Jun 23