1,974 research outputs found
Contains and Inside relationships within combinatorial Pyramids
Irregular pyramids are made of a stack of successively reduced graphs
embedded in the plane. Such pyramids are used within the segmentation framework
to encode a hierarchy of partitions. The different graph models used within the
irregular pyramid framework encode different types of relationships between
regions. This paper compares different graph models used within the irregular
pyramid framework according to a set of relationships between regions. We also
define a new algorithm based on a pyramid of combinatorial maps which allows to
determine if one region contains the other using only local calculus.Comment: 35 page
Antiprismless, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes
This article exhibits a 4-dimensional combinatorial polytope that has no
antiprism, answering a question posed by Bernt Lindst\"om. As a consequence,
any realization of this combinatorial polytope has a face that it cannot rest
upon without toppling over. To this end, we provide a general method for
solving a broad class of realizability problems. Specifically, we show that for
any semialgebraic property that faces inherit, the given property holds for
some realization of every combinatorial polytope if and only if the property
holds from some projective copy of every polytope. The proof uses the following
result by Below. Given any polytope with vertices having algebraic coordinates,
there is a combinatorial "stamp" polytope with a specified face that is
projectively equivalent to the given polytope in all realizations. Here we
construct a new stamp polytope that is closely related to Richter-Gebert's
proof of universality for 4-dimensional polytopes, and we generalize several
tools from that proof
Geometric and combinatorial realizations of crystal graphs
For irreducible integrable highest weight modules of the finite and affine
Lie algebras of type A and D, we define an isomorphism between the geometric
realization of the crystal graphs in terms of irreducible components of
Nakajima quiver varieties and the combinatorial realizations in terms of Young
tableaux and Young walls. For affine type A, we extend the Young wall
construction to arbitrary level, describing a combinatorial realization of the
crystals in terms of new objects which we call Young pyramids.Comment: 34 pages, 17 figures; v2: minor typos corrected; v3: corrections to
section 8; v4: minor typos correcte
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