Canonicalizing with independent indices mixes up raised/lowered

Question

Indices marked as independent seem to implicitly assume a fixed upper/lower contraction when canonicalizing in some cases.

Consider the below example:

{a,b,c,d}::Indices(vector, position=independent).
$Y^{c} \Lambda_{b} \Lambda_{a} V_{c}^{a} X_{b}$;
canonicalise(_);

This returns

Y^{a} \Lambda_{b} \Lambda_{c} V_{a b} X^{c}

apparently shuffling the last indices on V and X (even though they are of independent types) because they are both contracted upon \Lambda's. This is fine if we always contract upper/lower but otherwise this is somewhat counterintuitive behavior.

Interestingly, this only happens when the additional tensor Y is present attached to the other index on V.

Comments

I confirm the behaviour!

The independent position label differs from the result from free and fixed, but is not returning the expected result.

Answer 1

The algorithm does funny things if you declare indices independent but then not stick to Einstein conventions of having only one upper and one lower index. In your example, the b index appears as subscript twice. It tried to keep those two bs as lower indices, but as a result, it had to raise the index on X.

I would classifyl this as undefined behaviour, though perhaps a warning would have been appropriate. I don't think I will spend time on 'fixing' this, especially because it isn't even clear what the right fix should do.