The nineteen-vertex model and alternating sign matrices

Affine Alternating Sign Matrices

An Alternating sign matrix is a square matrix of 0’s, 1’s, and −1’s in which the sum of the entries in each row or column is 1 and the signs of the nonzero entries in each row or column alternate. This paper attempts to define an analogue to alternating sign matrices which is infinite and periodic. After showing the analogue we define shares desirable cahracteristics with alternating sign matri...

Alternating sign matrices

Symmetric alternating sign matrices

In this note we consider completions of n×n symmetric (0,−1)-matrices to symmetric alternating sign matrices by replacing certain 0s with +1s. In particular, we prove that any n×n symmetric (0,−1)-matrix that can be completed to an alternating sign matrix by replacing some 0s with +1s can be completed to a symmetric alternating sign matrix. Similarly, any n × n symmetric (0,+1)-matrix that can ...

Alternating sign matrices and tournaments

Keys and Alternating Sign Matrices

In [12], Lascoux and Schützenberger introduced a notion of key associated to any Young tableau. More recently Lascoux defined the key of an alternating sign matrix by recursively removing all −1’s in such matrices. But alternating sign matrices are in bijection with monotone triangles, which form a subclass of Young tableaux. We show that in this case these two notions of keys coincide. Moreove...