For better understanding of this lattice, let us single out some
parts of the lattice using different colours:
Of particular interest are the following two subspaces called
D and E in the lecture (in contrast to most of the other
subspaces, these two remain invariant, when we renumber the
given subspaces U1, U2, U3).
One obtains this subspace lattice for the following subspace triple:
Let k be a field, let V = k10 (with standard basis e1,..., e10), and let
On the right, we show for some edges (they
symbolize inclusions of subspaces) some elements of the
standard basis: If the edge connects (going upwards) the
bullet V' with the bullet V" (thus V' is a subspace of V")
and if we mark this edge with the element ei, then
this means that V'+< ei > = V".
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The structure of this lattice was exhibited by Dedekind in
a paper published in the year 1900
(Über die von drei Moduln erzeugte Dualgruppe, Math. Ann.
53 (1900), 371-403.), it is the free modular lattice in 3
generators
It is not known how the corresponding free modular lattice in 4 generators (or the subspace lattices generated by 4 subspaces U1, U2, U3, U4) may look like. |
Gian-Carlo Rota wrote in 1997:
The free modular lattice with three generators (which has twenty-eight elements) is a beautiful construct that is presently exiled from textbooks in linear algebra. Too bad, because the elements of this lattice explicitly describe all projective invariants of three subspaces. |