34 research outputs found
On the rank of incidence matrices in projective Hjelmslev spaces
Let be a finite chain ring with , , and let . Let be an integer sequence satisfying . We consider the incidence matrix of all shape versus all shape subspaces of with . We prove that the rank of over is equal to the number of shape subspaces. This is a partial analog of Kantor's result about the rank of the incidence matrix of all dimensional versus all dimensional subspaces of . We construct an example for shapes and for which the rank of is not maximal
The Projective Line Over the Finite Quotient Ring GF(2)[]/ and Quantum Entanglement I. Theoretical Background
The paper deals with the projective line over the finite factor ring
GF(2)[]/. The line is endowed with 18
points, spanning the neighbourhoods of three pairwise distant points. As
is not a local ring, the neighbour (or parallel) relation is
not an equivalence relation so that the sets of neighbour points to two distant
points overlap. There are nine neighbour points to any point of the line,
forming three disjoint families under the reduction modulo either of two
maximal ideals of the ring. Two of the families contain four points each and
they swap their roles when switching from one ideal to the other; the points of
the one family merge with (the image of) the point in question, while the
points of the other family go in pairs into the remaining two points of the
associated ordinary projective line of order two. The single point of the
remaining family is sent to the reference point under both the mappings and its
existence stems from a non-trivial character of the Jacobson radical, , of the ring. The factor ring is isomorphic to GF(2)
GF(2). The projective line over features nine
points, each of them being surrounded by four neighbour and the same number of
distant points, and any two distant points share two neighbours. These
remarkable ring geometries are surmised to be of relevance for modelling
entangled qubit states, to be discussed in detail in Part II of the paper.Comment: 8 pages, 2 figure
A geometric construction of panel-regular lattices in buildings of types ~A_2 and ~C_2
Using Singer polygons, we construct locally finite affine buildings of types
~A_2 and ~C_2 which admit uniform lattices acting regularly on panels. This
construction produces very explicit descriptions of these buildings as well as
very short presentations of the lattices. All but one of the ~C_2-buildings are
necessarily exotic. To the knowledge of the author, these are the first
presentations of lattices in buildings of type ~C_2. Integral and rational
group homology for the lattices is also calculated.Comment: 42 pages, small corrections and cleanup. Results are unchanged