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Definitions like this one are common in books:
We obviously want the term "tensor" to apply to multilinear maps with the variable slots in any weird order. What I'm wondering is if there's a terminology for this sort of thing, that makes it easier to talk about it. Something like "Two tensors of type ##(k,l)## are said to be equiflubbery if they have their variable slots in the same order".
Note that if we choose to use that first definition I mentioned (without the supplementary comment), so that the variable slots must be in a specific order, then we can't even say that the tensor product of a tensor of type (m,n) with a tensor of type (m',n') is a tensor of type (m+m',n+n'). It would just be some multilinear map that isn't a tensor, or at least isn't a tensor that has a type.
For all ##k,l\in\mathbb N##, a multilinear map $$T:\underbrace{V^*\times\cdots\times V^*}_{k\text{ factors}}\times \underbrace{V\times\cdots\times V}_{l\text{ factors}}\to\mathbb R$$ is said to be a tensor of type ##(k,l)## on ##V##.
Lee calls this a tensor of type ##l\choose k## instead of type ##(k,l)##, but that's not the issue I want to talk about. These definitions are often followed by a comment that says that even if we change the order of the variable slots, we would still consider the map a tensor of type ##(k,l)##. In other words, For all ##k,l\in\mathbb N##, a multilinear map ##T:W_1\times\cdots \times W_{k+l}\to\mathbb R## is said to be a tensor of type ##(k,l)## if
(a) ##W_i=V^*## for ##k## different values of ##i##, all of which are in ##\{1,\dots,k+l\}##.
(b) ##W_i=V## for ##l## different values of ##i##, all of which are in ##\{1,\dots,k+l\}##.
This creates a problem. If we had assigned the term "tensor of type ##(k,l)##" only to multilinear maps with all the V* variable slots to the left of all the V variable slots, then we could have defined a vector space structure on the set of tensors of type ##(k,l)##, and used such vector spaces to define vector bundles over a manifold. Then we could have defined tensor fields as sections of such bundles. But when we assign the term "tensor of type ##(k,l)##" to multilinear maps with the slots in any weird order, we can't define a vector space structure on the set of tensors on a given type. (a) ##W_i=V^*## for ##k## different values of ##i##, all of which are in ##\{1,\dots,k+l\}##.
(b) ##W_i=V## for ##l## different values of ##i##, all of which are in ##\{1,\dots,k+l\}##.
We obviously want the term "tensor" to apply to multilinear maps with the variable slots in any weird order. What I'm wondering is if there's a terminology for this sort of thing, that makes it easier to talk about it. Something like "Two tensors of type ##(k,l)## are said to be equiflubbery if they have their variable slots in the same order".
Note that if we choose to use that first definition I mentioned (without the supplementary comment), so that the variable slots must be in a specific order, then we can't even say that the tensor product of a tensor of type (m,n) with a tensor of type (m',n') is a tensor of type (m+m',n+n'). It would just be some multilinear map that isn't a tensor, or at least isn't a tensor that has a type.