Something that I have always admired as I’ve learnt mathematics is its amazing ability to connect disparate and seemingly distinct ideas and areas. My favourite example of this is found in the story of the alternating sign matrix conjecture, who’s proof is a journey across much of combinatorics and even leads to the realm of mathematical physics.
An alternating sign matrix is an n by n matrix, who’s entries consist of 1, 0 or -1, whose columns and rows sum to 1, and when reading down a column or row the appearance of -1 and 1 alternates. In the 1980s combinatorialists were interested in enumerating these objects; they wanted to know how many n by n alternating sign matrices were there. Mills, Robbins and Rumsey made a conjecture as to this enumeration which was later proven in 1992 by Zeilberger. The details of this proof involved expeditions into the study of plane partitions, lattice paths, symmetric and hypergeometric functions and a slew of other topics from across the expanse of combinatorics. This in itself is quite incredible, we began with what seemed like a relatively straightforward problem in its statement, however through the mathematics we were led down a path which exposed us to a wealth of other ideas and problems, rich in their own detail and intricacies.
This is not the end of the story of the alternating sign matrices. Another mathematician, Kuperberg, approached the problem from an entirely different direction; they observed that there is a bijection between the alternating sign matrices and an object of interest in statistical mechanics called the six vertex model under what is called domain wall boundary conditions. This opened up a whole new set of tools for tackling this problem. Now he could make use of the rich theory surrounding this physical model and a very simple proof of the alternating sign matrix conjecture followed in 1995. Not only that but by looking at this problem through this new statistical mechanics lens, he made progress in enumerating many subsets of alternating sign matrices under various symmetry conditions. This is the power and beauty of mathematics. It shows us how all of these ideas are in some fundamental way one in the same and lets us translate our tools and our objects across all kinds of domains and fields of inquiry.
Ainsley Nicoll
The University of Melbourne
