128 research outputs found
Points and hyperplanes of the universal embedding space of the dual polar space DW(5,q), q odd
It was proved earlier that there are 6 isomorphism classes of hyperplanes in the dual polar space (5,q) even, which arise from its Grassmann-embedding. In the present paper, we extend these results to the case that , which affords the universal embedding of (5,q)$
Minimum distance of Symplectic Grassmann codes
We introduce the Symplectic Grassmann codes as projective codes defined by
symplectic Grassmannians, in analogy with the orthogonal Grassmann codes
introduced in [4]. Note that the Lagrangian-Grassmannian codes are a special
class of Symplectic Grassmann codes. We describe the weight enumerator of the
Lagrangian--Grassmannian codes of rank and and we determine the minimum
distance of the line Symplectic Grassmann codes.Comment: Revised contents and biblograph
Quasiflats in hierarchically hyperbolic spaces
The rank of a hierarchically hyperbolic space is the maximal number of
unbounded factors in a standard product region. For hierarchically hyperbolic
groups, this coincides with the maximal dimension of a quasiflat. Examples for
which the rank coincides with familiar quantities include: the dimension of
maximal Dehn twist flats for mapping class groups, the maximal rank of a free
abelian subgroup for right-angled Coxeter and Artin groups, and, for the
Weil--Petersson metric, the rank is the integer part of half the complex
dimension of Teichm\"{u}ller space.
We prove that any quasiflat of dimension equal to the rank lies within finite
distance of a union of standard orthants (under a mild condition satisfied by
all natural examples). This resolves outstanding conjectures when applied to
various examples. For mapping class group, we verify a conjecture of Farb; for
Teichm\"{u}ller space we answer a question of Brock; for CAT(0) cubical groups,
we handle special cases including right-angled Coxeter groups. An important
ingredient in the proof is that the hull of any finite set in an HHS is
quasi-isometric to a CAT(0) cube complex of dimension bounded by the rank.
We deduce a number of applications. For instance, we show that any
quasi-isometry between HHSs induces a quasi-isometry between certain simpler
HHSs. This allows one, for example, to distinguish quasi-isometry classes of
right-angled Artin/Coxeter groups. Another application is to quasi-isometric
rigidity. Our tools in many cases allow one to reduce the problem of
quasi-isometric rigidity for a given hierarchically hyperbolic group to a
combinatorial problem. We give a new proof of quasi-isometric rigidity of
mapping class groups, which, given our general quasiflats theorem, uses simpler
combinatorial arguments than in previous proofs.Comment: 58 pages, 6 figures. Revised according to referee comments. This is
the final pre-publication version; to appear in Duke Math. Jou
Ad-nilpotent ideals of a Borel subalgebra: generators and duality
It was shown by Cellini and Papi that an ad-nilpotent ideal determines
certain element of the affine Weyl group, and that there is a bijection between
the ad-nilpotent ideals and the integral points of a simplex with rational
vertices. We give a description of the generators of ad-nilpotent ideals in
terms of these elements, and show that an ideal has generators if and only
it lies on the face of this simplex of codimension . We also consider two
combinatorial statistics on the set of ad-nilpotent ideals: the number of
simple roots in the ideal and the number of generators. Considering the first
statistic reveals some relations with the theory of clusters
(Fomin-Zelevinsky). The distribution of the second statistic suggests that
there should exist a natural involution (duality) on the set of ad-nilpotent
ideals. Such an involution is constructed for the series A,B,C.Comment: LaTeX2e, 23 page
Pseudo-embeddings and pseudo-hyperplanes
We generalize some known results regarding hyperplanes and projective embeddings of point-line geometries with three points per line to geometries with an arbitrary but finite number of points on each line. In order to generalize these results, we need to introduce the new notions of pseudo-hyperplane, (universal) pseudo-embedding, pseudo-embedding rank and pseudo-generating rank. We prove several connections between these notions and address the problem of the existence of (certain) pseudo-embeddings. We apply this to several classes of point-line geometries. We also determine the pseudo-embedding rank and the pseudo-generating rank of the projective space PG (n,4) and the affine space AG (n,4
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