Alternating Sign Matrices: Extensions, König-Properties, and Primary Sum-Sequences

Abstract

This paper is concerned with properties of permutation matrices and alternating sign matrices (ASMs). An ASM is a square $$(0,\pm 1)$$-matrix such that, ignoring 0’s, the 1’s and $$-1$$’s in each row and column alternate, beginning and ending with a 1. We study extensions of permutation matrices into ASMs by changing some zeros to $$+1$$ or $$-1$$. Furthermore, several properties concerning the term rank and line covering of ASMs are shown. An ASM A is determined by a sum-matrix $$\varSigma (A)$$ whose entries are the sums of the entries of its leading submatrices (so determined by the entries of A). We show that those sums corresponding to the nonzero entries of a permutation matrix determine all the entries of the sum-matrix and investigate some of the properties of the resulting sequence of numbers. Finally, we investigate the lattice-properties of the set of ASMs (of order n), where the partial order comes from the Bruhat order for permutation matrices.