Square ice, alternating sign matrices and classical orthogonal polynomials

Filippo Colomo

The six-vertex model, or square ice, with Domain Wall Boundary Condition, is considered in connection with the problem of Alternating Sign Matrices (ASM) enumerations. Using standard techniques from the theory of classical orthogonal polynomials, we provide a unified and simplified treatment of all ASM enumerations. In particular, previously known results can be recovered in a very simple and straightforward way, while the long standing problems of refined 3-enumeration, and of all doubly refined enumerations are completely solved.