Weighted projections of alternating sign matrices: Latin-like squares and the ASM polytope

Abstract

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 defined using alternating sign hypermatrices. Given row-vector z_n=(n,…,2,1), the weighted projection of an ASM A is equal to z_nA. Brualdi and Dahl proved that the weighted projection of an n×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 ASMn is examined, and a characterisation is given for the relationship between elements of ASMn corresponding to the same point in the regular n-permutohedron. Finally, the limitations of the main result for characterising alternating sign hypermatrix Latin-like squares are considered.