"Alternating sign matrices"
Abstract:Alternating sign matrices (ASMs) were discovered by Robbins and Rumsey in the early 1980's in connection with their study of Charles Dodgson's 19th-century condensation algorithm for computing matrix determinants. They have since been found to have fascinating properties and connections to representation theory, statistical physics and various combinatorial objects such as domino tilings and plane partitions. Two early conjectures of Robbins, Mills and Rumsey about the enumeration of ASMs of order N - both the total enumeration and the so-called refined enumeration with respect to a natural parameter indexing the first row of the matrix - became famous open problems and were resolved in the mid-1990's by Zeilberger.
In this talk I will give an overview of this fascinating field and describe some recent results, obtained in joint works with Ilse Fischer and with Matan Karklinsky, on a more detailed "doubly-refined" enumeration of ASMs with respect to the first two rows of the matrix. I will conclude with some speculation on how an extension of these results might be used to attack some famous open problems on the behavior of randomly chosen ASMs of large order.