Year: 2026
Type: preprint
Abstract: The Bruhat order on permutation matrices extends to alternating sign matrices via corner-sum matrices, where the order is given by entrywise domination. A classical result of Lascoux and Schützenberger states that alternating sign matrices form the Dedekind-MacNeille completion of the Bruhat order on permutations. Brualdi and Dahl introduced alternating sign hypermatrices as a three-dimensional analogue of alternating sign matrices and used them to generalise Latin squares, which may be viewed as three-dimensional analogues of permutation matrices. In this paper, in analogy with the two-dimensional case, we define and study a Bruhat order $\preceq_B$ on Latin squares and alternating sign hypermatrices. We introduce the corresponding corner-sum hypermatrices $\mathcal C_n$ and prove that entrywise domination on $\mathcal C_n$ encodes this order. We show that $\mathcal C_n$ is a distributive lattice, but that, unlike in dimension two, it is not the Dedekind-MacNeille completion of the poset of Latin squares. We further characterise the covering relations for $\mathcal C_n$ and prove rank formulae generalising the classical case of alternating sign matrices. Finally, we define monotone hypertriangles, prove that they are in bijection with $\mathcal C_n$, and show that they also encode the order by entrywise domination.
Source: arXiv (Cornell University)
Authors: Carnevale, Angela, O'Brien, Cian
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162452808
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