Vol. 49, No. 1, 1973

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Irreducible sums of simple multivectors

Herbert Busemann and Donald E. Glassco, II

Vol. 49 (1973), No. 1, 13–32

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=1kRi0 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.

Mathematical Subject Classification 2000
Primary: 15A75
Received: 17 July 1972
Published: 1 November 1973
Herbert Busemann
Donald E. Glassco, II