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Feijtel_58761500_2022.pdf
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- Abstract
- We examine two conjectures regarding alternating sign matrices, both of which were stated in the 1980’s and proven in the 1990’s. First, we investigate the ASM Conjecture for which we give two different proofs. The first follows Kuperberg’s work using Cauchy determinants and the second combines results from Okada, Gorin and Panova and Stroganov relating to Schur polynomials. Secondly, we study the refined ASM Conjecture and give a proof thereof that builds upon our findings with respect to Schur polynomials. All three proofs rely on one crucial observation: there is a one-to-one correspondence between ASMs and states of the six-vertex model on the square lattice with domain-wall boundary conditions. In light of this bijection, it becomes clear that we need to compute the partition function of the model. We thus convince ourselves—through a rigorous proof—that given the properties of this function, the formula found by Izergin and Korepin is in fact the right one. It is at this stage that the Yang-Baxter equation for the six-vertex model is decisively used.