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, ±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 Σ(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.

Multi-alternating sign matrices

“…In the ASM case, i.e., when R and S are all ones vectors of length n, then every ASM of order n is an extreme point of the ASM polytope conv (A n ) [3]. For more general R and S, this may not be the case, as the following example shows.…”