Greub, Multilinear Algebra, Notation

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Does any one know what he means with $\tilde E = \otimes_\alpha E_\alpha$ and $\tilde F = \otimes_\beta F_\beta$ ? That is on pages 14 and 15.

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This is an error in Greub's book. Those should be direct sums over the families, not tensor products. He's intending to construct the tensor product of two direct sums.

This errata listing may be helpful to you.