134,923 research outputs found
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)
Tits Geometry and Positive Curvature
There is a well known link between (maximal) polar representations and
isotropy representations of symmetric spaces provided by Dadok. Moreover, the
theory by Tits and Burns-Spatzier provides a link between irreducible symmetric
spaces of non-compact type of rank at least three and irreducible topological
spherical buildings of rank at least three.
We discover and exploit a rich structure of a (connected) chamber system of
finite (Coxeter) type M associated with any polar action of cohomogeneity at
least two on any simply connected closed positively curved manifold. Although
this chamber system is typically not a Tits geometry of type M, we prove that
in all cases but two that its universal Tits cover indeed is a building. We
construct a topology on this universal cover making it into a compact spherical
building in the sense of Burns and Spatzier. Using this structure we classify
up to equivariant diffeomorphism all polar actions on (simply connected)
positively curved manifolds of cohomogeneity at least two.Comment: 43 pages, to appear in Acta Mathematic
The Dirichlet space: A Survey
In this paper we survey many results on the Dirichlet space of analytic
functions. Our focus is more on the classical Dirichlet space on the disc and
not the potential generalizations to other domains or several variables.
Additionally, we focus mainly on certain function theoretic properties of the
Dirichlet space and omit covering the interesting connections between this
space and operator theory. The results discussed in this survey show what is
known about the Dirichlet space and compares it with the related results for
the Hardy space.Comment: 35 pages, typoes corrected, some open problems adde
On compression of Bruhat-Tits buildings
We obtain an analog of the compression of angles theorem in symmetric spaces
for Bruhat--Tits buildings of the type .
More precisely, consider a -adic linear space and the set of
all lattices in . The complex distance in is a complete system of
invariants of a pair of points of under the action of the complete
linear group. An element of a Nazarov semigroup is a lattice in the duplicated
linear space . We investigate behavior of the complex distance under
the action of the Nazarov semigroup on the set .Comment: 6 page
Ground-State Spaces of Frustration-Free Hamiltonians
We study the ground-state space properties for frustration-free Hamiltonians.
We introduce a concept of `reduced spaces' to characterize local structures of
ground-state spaces. For a many-body system, we characterize mathematical
structures for the set of all the -particle reduced spaces, which
with a binary operation called join forms a semilattice that can be interpreted
as an abstract convex structure. The smallest nonzero elements in ,
called atoms, are analogs of extreme points. We study the properties of atoms
in and discuss its relationship with ground states of -local
frustration-free Hamiltonians. For spin-1/2 systems, we show that all the atoms
in are unique ground states of some 2-local frustration-free
Hamiltonians. Moreover, we show that the elements in may not be the
join of atoms, indicating a richer structure for beyond the convex
structure. Our study of deepens the understanding of ground-state
space properties for frustration-free Hamiltonians, from a new angle of reduced
spaces.Comment: 23 pages, no figur
On the varieties of the second row of the split Freudenthal-Tits Magic Square
Our main aim is to provide a uniform geometric characterization of the
analogues over arbitrary fields of the four complex Severi varieties, i.e.~the
quadric Veronese varieties in 5-dimensional projective spaces, the Segre
varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in
14-dimensional projective spaces, and the exceptional varieties of type
in 26-dimensional projective space. Our theorem can be
regarded as a far-reaching generalization of Mazzocca and Melone's approach to
finite quadric Veronesean varieties. This approach takes projective properties
of complex Severi varieties as smooth varieties as axioms.Comment: Small updates, will be published in Annales de l'institut Fourie
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