white@brahms.BERKELEY.EDU (Samuel P. White) (02/07/86)
I have a couple of problems I have been working on. If you
have anything useful to say on either I would be quite interested.
Let M(n) be the nxn matrices over a field k. (I do not
think it matters if you assume k is the complex numbers but
I have been told it does not hurt to generalize.) Let V be
a k-vector subspace which consists entirely of nilpotent elements
of M(n). Can V generate all of M(n) as an algebra?
The next problem is a little more complicated, but if you
object to the problem because it is technical and demand
a reason for its existence, the problem does have an intimate
connection with the rank of an abelian variety. Let G be
a group that acts on the set X = { 1, 2,..., n }, i.e. G is a
subgroup of the permutation group on n letters. Assume G acts
transitively on the set X and let V be an n-dimensional vector
space over the rationals where G gets a natural representation
on V by permuting the basis elements of V in accordance with
G's action on X. Does there exist a subset S of V with the
following properties:
1.) S is closed under the action of G.
2.) S consists of elements whose coordinates are 0's and 1's.
3.) There do not exist two disjoint subsets A and B of X
satisfying the following two properties:
a.) A and B have the same number of elements.
b.) The number of 1's occuring at the coordinates
of V listed in A is the same as the number of
1's occuring at the coordinates corresponding
to the elements of B for every x in S. In other
words, A and B are disjoint subsets of the coordinates
of V and I am counting the number of 1's that
occur in each subset and seeing if they are equal.
4.) S does not span V.
ucbvax!brahms!white Samuel P. White/UCB Math Dept/Berkeley CA 94720