MSI Weekly Bulletin - Week starting Monday 4 September, 2006

In 2001 I found myself looking with Jan de Gier and Bernard Nienhuis at the components of the groundstate eigenvector of a loop model in statistical mechanics built on the Temperley-Lieb algebra. This algebra, sometimes called the Temperley-Lieb-Jones algebra, is a quotient of the Hecke algebra. Using exact results from small systems we discovered that the eigenvector components count objects known as alternating sign matrices. This led us to make a number of surprising conjectures, some of which have recently been proved. In this seminar I will give an elementary introduction to these results and review some of the recent progress in this field.