118 research outputs found
h-vectors of Gorenstein polytopes
We show that the Ehrhart h-vector of an integer Gorenstein polytope with a
regular unimodular triangulation satisfies McMullen's g-theorem; in particular,
it is unimodal. This result generalizes a recent theorem of Athanasiadis
(conjectured by Stanley) for compressed polytopes. It is derived from a more
general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a
field) by a "long" regular sequence in such a way that the quotient is still a
normal affine monoid algebra. This technique reduces all questions about the
Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenstein polytope Q with
exactly one interior lattice point, provided each lattice point in a multiple
cP, c in N, can be written as the sum of n lattice points in P. (Up to a
translation, the polytope Q belongs to the class of reflexive polytopes
considered in connection with mirror symmetry.) If P has a regular unimodular
triangulation, then it follows readily that the Ehrhart h-vector of P coincides
with the combinatorial h-vector of the boundary complex of a simplicial
polytope, and the g-theorem applies.Comment: 12 pages; besides minor modifications the main result needs the
additional assumption that the polytope P has a regular unimodular
triangulation. The extra hypothesis has been included as well as the crucial
construction where it is use
Normaliz: Algorithms for Affine Monoids and Rational Cones
Normaliz is a program for solving linear systems of inequalities. In this
paper we present the algorithms implemented in the program, starting with
version 2.0
Partitions of single exterior type
We characterize the irreducible representations of the general linear group
GL(V) that have multiplicity 1 in the direct sum of all Schur modules of a
given exterior power of V. These have come up in connection with the relations
of the lower order minors of a generic matrix. We show that the minimal
relations conjectured by Bruns, Conca and Varbaro are exactly those coming from
partitions of single exterior type
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