Subspace Triples

The largest subspace lattice generated by 3 subspaces has the following shape:

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

  • U1 = < e2, e5, e8,  e9, e10  >
  • U2 = < e3, e6, e7,  e9, e10  >
  • U3 = < e4, e5+e6,  e7,  e8, e10  >

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".



BIREP
Last modified: Nov 8, 2010