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

TL;DR

This paper proves a conjecture about the weighted projections of alternating sign matrices (ASMs), showing they correspond exactly to vectors majorized by a specific vector, and explores implications for ASM polytopes and related structures.

Contribution

It provides a proof that all vectors majorized by a certain vector are weighted projections of ASMs, advancing understanding of ASM structure and their geometric properties.

Findings

Confirmed the conjecture relating weighted projections and majorization.

Introduced row-increasing triangles and established their relation to monotone triangles.

Analyzed the connection between ASM elements and the ASM polytope, including permutohedron relationships.

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 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.