Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes

Christian Haase; Tyrrell B. McAllister

Home/Archive/arXiv

arXiv · 0709.4070

Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes

Christian Haase·Tyrrell B. McAllister

View original source ↗Read source document ↗Export metadata ↓

Abstract

Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi-period less than the denominator of that polytope. This phenomenon is poorly understood, and all known cases in which it occurs have been proven with ad hoc methods. In this note, we present a conjectural explanation for quasi-period collapse in rational polytopes. We show that this explanation applies to some previous cases appearing in the literature. We also exhibit examples of Ehrhart polynomials of rational polytopes that are not the Ehrhart polynomials of any integral polytope. Our approach depends on the invariance of the Ehrhart quasi-polynomial under the action of affine unimodular transformations. Motivated by the similarity of this idea to the scissors congruence problem, we explore the development of a Dehn-like invariant for rational polytopes in the lattice setting.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Christian Haase, Tyrrell B. McAllister. 2007-09-26. Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes. https://arxiv.org/abs/0709.4070

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Integral quasi-strongly regular Cayley graphs

Quasi-strongly regular graphs form a significant generalization of strongly regular graphs. We study the eigenvalues of a family of such graphs, $Γ_H(G)$, constructed from a finite group $G$ and a subgroup $H$. Our main results include a sufficient condition for $Γ_H(G)$ to be integral and an explicit computation of its entire spectrum when $H$ is normal, revealing that the spectrum in this case depends only on $|G|$ and the index $[G:H]$.

math.CO↗

arXiv2026

The exact Erdős-Ko-Rado Theorem for 3-wise $t$-intersecting uniform families

One of the central problems in extremal set theory is the determination of the maximum size of an $r$-wise $t$-intersecting $k$-graph on $n$ vertices. For $r=2$ after decades of effort, a complete solution was given by the Complete Intersection Theorem (\cite{AK0}). However the case $r\geq 3$ remains wide open. The $t$-star consisting of all $k$-sets containing a fixed $t$-set provides a lower bound. For $r=3$ and $k\geq t+2\geq44$, we determine the exact threshold for which it is optimal for $n>k$. The condition is essentially $n\geq \sqrt{t}k$. For $k\geq t+2\geq47$ and $n\geq\frac{\sqrt{4t+9}-1}{2}k$, we also establish the corresponding Hilton-Milner-type stability result.

math.CO↗

arXiv2026

A Littlewood-Richardson Rule for Forest Polynomials

The forest polynomials $\mathfrak{P}_a$ of Nadeau-Tewari form a $\mathbb{Z}$-basis of $\mathbb{Z}[x_1,x_2,\ldots]$ whose role for the cohomology of the quasisymmetric flag variety parallels that of Schubert polynomials for the classical flag variety. Nonnegativity of the structure constants $β_{a,b}^c$ in $\mathfrak{P}_a \mathfrak{P}_b = \sum_c β_{a,b}^c \mathfrak{P}_c$ is known, but no Littlewood-Richardson-style enumerative rule has been available. We give such a rule: $β_{a,b}^c$ counts pairs of forest RC graphs of forest-codes $a$ and $b$ whose lift product lands on a forest RC graph of forest-code and weight both equal to $c$. The same rule descends to the cup product on $H^\bullet(\mathrm{QFl}_n)$. The proof introduces a squash product $\boxtimes$ on RC graphs, which on full Grassmannian RC graphs of a fixed height recovers the plactic monoid, a multiplactic algebra $\mathcal{M}_n$ that models Schubert polynomial multiplication, and a lift product $*$ on RC graphs whose quasi-crystal structure, in the framework of Cain-Guilherme-Malheiro, transports the Nadeau-Tewari forest equivalence through the computation. The row-decomposition machinery on RC graphs that underlies these constructions is developed in a companion paper.

math.CO↗