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Denoting by V n(F) the
n-dimensional vector space over the field F of characteristic 0, let V rn(F) be the
linear space of all r-vectors R over V n(F) and Grn(F) the Grassmann cone of the
simple r-vectors R in V rn(F). The sum R = ∑
i=1kRi(Ri ∈ Grn(F)) is irreducible
if R is not the sum of fewer than k elements of Grn(F). (Duality reduces
the interesting cases to 2 ≦ r ≦ n∕2.) Such sums are trivial only for r = 2,
because ∧
l=1kRi≠0 while always sufficient for irreducibility is then also
necessary. Extension of F does not influence irreducibility if r = 2 but it can for
r > 2.
The sets Wrn(F,k) of those R in V rn(F) which are irreducible sums of k terms
behave as expected when r = 2, but have the most surprising properties for larger r.
Although V 36(F) = ⋃
k=1sW36(F,k) and W36(F,3)≠ϕ, the sets W36(R or C,2) have
interior points as sets in V 36 ( R resp. C) and so does W36(R,3) but W36(C,3) does
not.
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