Geometric realizations of substitutions

Geometric Realizations of Substitutions

Geometric Realizations of Polyhedral Complexes

We discuss two problems in combinatorial geometry. First, given a geometric polyhedral complex in R (a family of 3-polytopes attached face-to-face), can it always be realized over Q? We give a negative answer to this question, by presenting an irrational polyhedral complex with 1278 convex polyhedra. We then present a universality theorem on the space of realizations of such complexes. Second, ...

Geometric Realizations of Hermitian Curvature Models

We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact be realized by a Hermitian manifold of constant scalar curvature and constant ⋆-scalar curvature which satisfies the Kaehler condition at the point in question.

Geometric Realizations of Representations of Finite Length

Let G = H × R be a semidirect product Lie group. We reduce the problem of deciding which indecomposable representations of G may be realized in subquotients of spaces of sections of vector bundles over infinitesimal neighborhoods of orbits of H in the dual of R, to a problem involving only representations of the H-stabilizers of the orbits.

Geometric and Combinatorial Realizations of Crystals of Enveloping Algebras

Kashiwara and Saito have defined a crystal structure on the set of irreducible components of Lusztig’s quiver varieties. This gives a geometric realization of the crystal graph of the lower half of the quantum group associated to a simply-laced Kac-Moody algebra. Using an enumeration of the irreducible components of Lusztig’s quiver varieties in finite and affine type A by combinatorial data, w...