scholarly journals Higher Spin Alternating Sign Matrices

10.37236/1001 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Roger E. Behrend ◽  
Vincent A. Knight
Keyword(s):  
Convex Polytope ◽  
Ehrhart Polynomial ◽  
Alternating Sign Matrices ◽  
Integer Points ◽  
Wall Boundary Conditions ◽  
Alternating Sign Matrix ◽  
Higher Spin ◽  
Statistical Mechanical ◽  
Sign Matrix ◽  
Integral Convex Polytope

We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer $r$, all complete row and column sums are $r$, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin $r/2$ statistical mechanical vertex models with domain-wall boundary conditions. The case $r=1$ gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size $n$ are the integer points of the $r$-th dilate of an integral convex polytope of dimension $(n{-}1)^2$ whose vertices are the standard alternating sign matrices of size $n$. It then follows that, for fixed $n$, these matrices are enumerated by an Ehrhart polynomial in $r$.

Projective toric codes

2021 ◽  
pp. 1-26
Author(s):  
Jade Nardi
Keyword(s):  
Error Correcting Code ◽  
Convex Polytope ◽  
Global Section ◽  
Explicit Construction ◽  
Geometric Error ◽  
Integral Points ◽  
Toric Codes ◽  
Algorithmic Technique ◽  
The One ◽  
Integral Convex Polytope

Any integral convex polytope [Formula: see text] in [Formula: see text] provides an [Formula: see text]-dimensional toric variety [Formula: see text] and an ample divisor [Formula: see text] on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on [Formula: see text], obtained by evaluating global section of the line bundle corresponding to [Formula: see text] on every rational point of [Formula: see text]. This work presents an extension of toric codes analogous to the one of Reed–Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with nonzero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope [Formula: see text] and an algorithmic technique to get a lower bound on the minimum distance is described.

scholarly journals The $h$-Vector of a Gorenstein Toric Ring of a Compressed Polytope

10.37236/1891 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Hidefumi Ohsugi ◽  
Takayuki Hibi
Keyword(s):  
Convex Polytope ◽  
Integral Convex Polytope

A compressed polytope is an integral convex polytope all of whose pulling triangulations are unimodular. A $(q - 1)$-simplex $\Sigma$ each of whose vertices is a vertex of a convex polytope ${\cal P}$ is said to be a special simplex in ${\cal P}$ if each facet of ${\cal P}$ contains exactly $q - 1$ of the vertices of $\Sigma$. It will be proved that there is a special simplex in a compressed polytope ${\cal P}$ if (and only if) its toric ring $K[{\cal P}]$ is Gorenstein. In consequence it follows that the $h$-vector of a Gorenstein toric ring $K[{\cal P}]$ is unimodal if ${\cal P}$ is compressed.

scholarly journals On Flow Polytopes, Order Polytopes, and Certain Faces of the Alternating Sign Matrix Polytope

2019 ◽  
Vol 62 (1) ◽  
pp. 128-163 ◽  
Author(s):  
Karola Mészáros ◽  
Alejandro H. Morales ◽  
Jessica Striker
Keyword(s):  
Flow Polytopes ◽  
Alternating Sign Matrix ◽  
Sign Matrix

Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture

2001 ◽  
Vol 108 (1) ◽  
pp. 85 ◽  
Author(s):  
Doron Zeilberger ◽  
David M. Bressoud
Keyword(s):  
Alternating Sign Matrix ◽  
Sign Matrix

scholarly journals Non-Normal Very Ample Polytopes and their Holes

10.37236/3656 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Akihiro Higashitani
Keyword(s):  
Convex Polytope ◽  
Integral Convex Polytope

In this paper, we show that for given integers $h$ and $d$ with $h \geq 1$ and $d \geq 3$, there exists a non-normal very ample integral convex polytope of dimension $d$ which has exactly $h$ holes.

scholarly journals The Alternating Sign Matrix Polytope

10.37236/130 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Jessica Striker
Keyword(s):  
Convex Hull ◽  
Complete Characterization ◽  
Stochastic Matrices ◽  
Von Neumann ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Alternating Sign Matrix ◽  
Sign Matrix ◽  
Equivalent Description

We define the alternating sign matrix polytope as the convex hull of $n\times n$ alternating sign matrices and prove its equivalent description in terms of inequalities. This is analogous to the well known result of Birkhoff and von Neumann that the convex hull of the permutation matrices equals the set of all nonnegative doubly stochastic matrices. We count the facets and vertices of the alternating sign matrix polytope and describe its projection to the permutohedron as well as give a complete characterization of its face lattice in terms of modified square ice configurations. Furthermore we prove that the dimension of any face can be easily determined from this characterization.

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