Not an answer, rather an attempt to hijack the question...
Some time ago I have also been wondering how to wedge two vector spaces and came up with the following construction:
Let f:UtoV and g:UtoW be two vector space morphisms. We define the vector space VwedgeUW (of course, this depends not only on U, V and W, but also on f and g, but we silently leave these out of the notation - just as in the case of fibered products) as the quotient of the tensor product VotimesW by the subspace spanned by all tensors of the form fleft(uright)otimesgleft(uright) for uinU.
This is functorial, but does anyone know any use for it? Any results about the structure of VwedgeUW as a representation, if U, V and W are representations? How does this wedgeU operation "look like" in the representation ring (for instance, the usual wedge operations look like the lambda operations lambda1, lambda2, ...).
EDIT: In characteristic neq2, we have VwedgeUW=left(VotimesWright)diagupleft(left(left(fotimesgright)circleft(mathrmid+tauright)right)left(UotimesUright)right), where tau is the transposition of the two tensorands. But it is still interesting to find out what exactly is factored out in classical cases, e. g. in representation theory of Sn.
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