论文标题
交替符号矩阵的加权投影：拉丁正方形和ASM Polytope
Weighted projections of alternating sign matrices: Latin-like squares and the ASM polytope
论文作者
论文摘要
Brualdi和Dahl（2018）引入了交替符号矩阵（ASM）的加权投影，这是迈向使用交替的符号超级通道引入的拉丁正方形概括的一步。如果$ z_n =（n，\ dots，2,1）$，则ASM $ a $的加权投影等于$ z_n^ta $。 Brualdi和Dahl证明了$ n \ times n $ asm的加权投影由向量$ z_n $进行了主要的成绩，并推测了任何由$ z_n $批准的正整数矢量都是某些ASM的加权投影。本文的主要结果通过单调三角形提供了这种猜想的证明。引入了单调三角形的放松，称为排行三角形。结果表明，对于任何排列三角形$ t $，都存在一个单调三角$ m $，因此$ m $的每个条目的发生次数与$ t $相同的次数。还概述了具有给定加权投影的ASM的构造。检查了主要结果与有关ASM polytope $ asm_n $的现有结果的关系，并为$ n $ permutoheDron中的$ asm_n $的元素之间的关系给出了$ asm_n $之间的关系的特征。最后，考虑了表征交替标志超硫酸拉丁正方形的主要结果的局限性。
The weighted projection of an alternating sign matrix (ASM) was introduced by Brualdi and Dahl (2018) as a step towards characterising a generalisation of Latin squares they introduced using alternating sign hypermatrices. If $z_n = (n,\dots,2,1)$, then the weighted projection of an ASM $A$ is equal to $z_n^TA$. Brualdi and Dahl proved that the weighted projection of an $n \times n$ ASM is majorized by the vector $z_n$, and conjectured that any positive integer vector majorized by $z_n$ is the weighted projection of some ASM. The main result of this paper presents a proof of this conjecture, via monotone triangles. A relaxation of a monotone triangle, called a row-increasing triangle, is introduced. It is shown that for any row-increasing triangle $T$, there exists a monotone triangle $M$ such that each entry of $M$ occurs the same number of times as in $T$. A construction is also outlined for an ASM with given weighted projection. The relationship of the main result to existing results concerning the ASM polytope $ASM_n$ is examined, and a characterisation is given for the relationship between elements of $ASM_n$ corresponding to the same point in the permutohedron of order $n$. Finally, the limitations of the main result for characterising alternating sign hypermatrix Latin-like squares is considered.
