Represented Coxeter matroids of types Cnβ and Dnβ, that is, symplectic
and orthogonal matroids arising from totally isotropic subspaces of symplectic
or (even-dimensional) orthogonal spaces, may also be represented in buildings
of type Cnβ and Dnβ, respectively. Indeed, the particular buildings
involved are those arising from the flags or oriflammes, respectively, of
totally isotropic subspaces. There are also buildings of type Bnβ arising
from flags of totally isotropic subspaces in odd-dimensional orthogonal space.
Coxeter matroids of type Bnβ are the same as those of type Cnβ (since they
depend only upon the reflection group, not the root system). However, buildings
of type Bnβ are distinct from those of the other types. The matroids
representable in odd dimensional orthogonal space (and therefore in the
building of type Bnβ) turn out to be a special case of symplectic (flag)
matroids, those whose top component, or Lagrangian matroid, is a union of two
Lagrangian orthogonal matroids. These two matroids are called a Lagrangian
pair, and they are the combinatorial manifestation of the ``fork'' at the top
of an oriflamme (or of the fork at the end of the Coxeter diagram of Dnβ).
Here we give a number of equivalent characterizations of Lagrangian pairs,
and prove some rather strong properties of them.Comment: Requires amssymb.sty; 12 pages, 2 LaTeX figure