The characterization of an Alternating Sign Matrix using a triplet

Abstract

Alternating Sign Matrix(ASM for short) is a square matrix which is consist of 0, 1 and -1. In this paper, we characterize an ASM by showing a bijection between alternating sign matrix and six vertex model, and a bijection between six vertex model and height function. In order to show these bijections, we define a triplet $({a}_{i, j}, {c}_{i, j}, {r}_{i, j})$ for each entry of an ASM. We also define a \textsl{track} for each index of height function, and state more properties of height function.