122 research outputs found
Reflection Groups and Polytopes over Finite Fields, III
When the standard representation of a crystallographic Coxeter group is
reduced modulo an odd prime p, one obtains a finite group G^p acting on some
orthogonal space over Z_p . If the Coxeter group has a string diagram, then G^p
will often be the automorphism group of a finite abstract regular polytope. In
parts I and II we established the basics of this construction and enumerated
the polytopes associated to groups of rank at most 4, as well as all groups of
spherical or Euclidean type. Here we extend the range of our earlier criteria
for the polytopality of G^p . Building on this we investigate the class of
3-infinity groups of general rank, and then complete a survey of those locally
toroidal polytopes which can be described by our construction.Comment: Advances in Applied Mathematics (to appear); 19 page
Reflection groups and polytopes over finite fields, II
When the standard representation of a crystallographic Coxeter group
is reduced modulo an odd prime , a finite representation in some orthogonal
space over is obtained. If has a string diagram, the
latter group will often be the automorphism group of a finite regular polytope.
In Part I we described the basics of this construction and enumerated the
polytopes associated with the groups of rank 3 and the groups of spherical or
Euclidean type. In this paper, we investigate such families of polytopes for
more general choices of , including all groups of rank 4. In
particular, we study in depth the interplay between their geometric properties
and the algebraic structure of the corresponding finite orthogonal group.Comment: 30 pages (Advances in Applied Mathematics, to appear
On the tropical Torelli map
AbstractWe construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study the fibers (tropical Torelli theorem) and the image of this map (tropical Schottky problem). Finally we determine the image of the planar tropical curves via the tropical Torelli map and we use it to give a positive answer to a question raised by Namikawa on the compactified classical Torelli map
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