Sách Enumeration of alternating sign matrices of even size (quasi-)invariant under a quarter-turn rotatio

An alternating sign matrix is a square matrix with entries in {ư1, 0, 1} and such that in any row and column: the non-zero entries alternate in sign, and their sum is equal to.Their enumeration formula was conjectured by Mills, Robbins and Rumsey [8], and proved years later by Zeilberger [16], and almost simultaneously by Kuperberg [6]. Kuperberg’s proof is based on the study of the partition function of a square ice model whose states are in bijection with ASM’s. Kuperberg was able to get an explicit formula for the partition function for some special values of the spectral parameter. To do this, he used a determinant representation for the partition function, that was obtained by Izergin [4]. Izergin’s proof is based on the Yang-Baxter equations, and on recursive relations discovered by Korepin [5].