交替符号矩阵与超矩阵及拉丁方的推广
Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Square
摘要：
交替符号矩阵（ASM）是一个$(0, \pm 1)$矩阵，其中每行和每列中的非零元素的符号交替。我们将这个概念推广到超矩阵：一个$n\times n\times n$超矩阵$A=[a_{ijk}]$是一个{ \em交替符号超矩阵}，或者ASHM，如果它的每个平面，通过固定三个索引之一，是一个ASM。几个有关ASHM的结果，如找到一个$n\times n\times n$ ASHM的非零的最大数目，和相关的拉丁方的属性。此外，我们调查完成问题，其中一个问，如果一个子超矩阵可以完成（扩展）到ASHM。我们证明了几个这类定理。
英文摘要：
An alternating sign matrix, or ASM, is a $(0, \pm 1)$-matrix where the nonzero entries in each row and column alternate in sign. We generalize this notion to hypermatrices: an $n\times n\times n$ hypermatrix $A=[a_{ijk}]$ is an {\em alternating sign hypermatrix}, or ASHM, if each of its planes, obtained by fixing one of the three indices, is an ASM. Several results concerning ASHMs are shown, such as finding the maximum number of nonzeros of an $n\times n\times n$ ASHM, and properties related to Latin squares. Moreover, we investigate completion problems, in which one asks if a subhypermatrix can be completed (extended) into an ASHM. We show several theorems of this type.
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