A SWISS CHEESE THEOREM FOR LINEAR OPERATORS WITH TWO INVARIANT SUBSPACES

We study systems (V, T, U-1, U-2) consisting of a finite dimensional vector space V, a nilpotent k-linear operator T : V -> V and two T-invariant subspaces U-1 subset of U-2 subset of V. Let S(n) be the category of such systems where the operator T acts with nilpotency index at most n. We determine the dimension types (dim U-1, dim U-2/U-1, dim V/U-2) of indecomposable systems in S(n) for n <= 4. It turns out that in the case where n = 4 there are infinitely many such triples (x, y, z), they all lie in the cylinder given by vertical bar x - y vertical bar, vertical bar y - z vertical bar, vertical bar z - x vertical bar <= 4. But not each dimension type in the cylinder can be realized by an indecomposable system. In particular, there are holes in the cylinder. Namely, no triple in (x, y, z) is an element of (3, 1, 3) + N(2, 2, 2) can be realized, while each neighbor (x +/- 1, y, z), (x, y +/- 1, z), (x, y, z +/- 1) can. Compare this with Bongartz' No-Gap Theorem, which states that for an associative algebra A over an algebraically closed field, there is no gap in the lengths of the indecomposable A-modules of finite dimension.