13 research outputs found
Some simple modules for classical groups and -ranks of orthogonal and Hermitian geometries
We determine the characters of the simple composition factors and the
submodule lattices of certain Weyl modules for classical groups. The results
have several applications. The simple modules arise in the study of incidence
systems in finite geometries and knowledge of their dimensions yields the
-ranks of these incidence systems.Comment: 33 pages, corrected parity of u in statement of Theorem 1.6(b
Proof of the Kakeya set conjecture over rings of integers modulo square-free
A Kakeya set is a set containing a
line in each direction. We show that, when is any square-free integer, the
size of the smallest Kakeya set in is at least
for any -- resolving a special
case of a conjecture of Hickman and Wright. Previously, such bounds were only
known for the case of prime . We also show that the case of general can
be reduced to lower bounding the rank of the incidence matrix of
points and hyperplanes over