263 research outputs found
Base subsets of polar Grassmannians
Let be a thick building of type
. Let also be
the Grassmannian of -dimensional singular subspaces of the associated polar
space (of rank ). We write for the corresponding
shadow space of type . Every bijective transformation of
sending base subsets to base subsets (the shadows of
apartments) is a collineation of , and it is induced by a
collineation of if or
Metric characterization of apartments in dual polar spaces
Let be a polar space of rank and let , be the polar Grassmannian formed by -dimensional singular
subspaces of . The corresponding Grassmann graph will be denoted by
. We consider the polar Grassmannian
formed by maximal singular subspaces of and show that the image of every
isometric embedding of the -dimensional hypercube graph in
is an apartment of . This follows
from a more general result (Theorem 2) concerning isometric embeddings of
, in . As an application, we classify all
isometric embeddings of in , where
is a polar space of rank (Theorem 3)
Schubert decompositions for quiver Grassmannians of tree modules
Let be a quiver, a representation of with an ordered basis \cB
and \ue a dimension vector for . In this note we extend the methods of
\cite{L12} to establish Schubert decompositions of quiver Grassmannians
\Gr_\ue(M) into affine spaces to the ramified case, i.e.\ the canonical
morphism from the coefficient quiver of w.r.t.\ \cB is not
necessarily unramified.
In particular, we determine the Euler characteristic of \Gr_\ue(M) as the
number of \emph{extremal successor closed subsets of }, which extends the
results of Cerulli Irelli (\cite{Cerulli11}) and Haupt (\cite{Haupt12}) (under
certain additional assumptions on \cB).Comment: 22 page
Isometric embeddings of Johnson graphs in Grassmann graphs
Let be an -dimensional vector space () and let
be the Grassmannian formed by all -dimensional
subspaces of . The corresponding Grassmann graph will be denoted by
. We describe all isometric embeddings of Johnson graphs
, in , (Theorem 4). As a
consequence, we get the following: the image of every isometric embedding of
in is an apartment of if and
only if . Our second result (Theorem 5) is a classification of rigid
isometric embeddings of Johnson graphs in , .Comment: New version -- 14 pages accepted to Journal of Algebraic
Combinatoric
The generating rank of the unitary and symplectic Grassmannians
We prove that the Grassmannian of totally isotropic -spaces of the polar
space associated to the unitary group () has generating rank when . We also reprove the main result of Blok [Blok2007], namely that
the Grassmannian of totally isotropic -spaces associated to the symplectic
group has generating rank , when
Splitting Polytopes
A split of a polytope is a (regular) subdivision with exactly two maximal
cells. It turns out that each weight function on the vertices of admits a
unique decomposition as a linear combination of weight functions corresponding
to the splits of (with a split prime remainder). This generalizes a result
of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite
metric spaces.
Introducing the concept of compatibility of splits gives rise to a finite
simplicial complex associated with any polytope , the split complex of .
Complete descriptions of the split complexes of all hypersimplices are
obtained. Moreover, it is shown that these complexes arise as subcomplexes of
the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].Comment: 25 pages, 7 figures; minor corrections and change
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