Around the Razumov-Stroganov Correspondence

Three families of combinatorial objects show remarkable relations: Alternating Sign Matrices (ASM), O(1) Dense Loops on a cylinder, and maximally-symmetric plane partitions (TSSCPP). The first two classes are related even at the level of refined enumerations, according to the "link pattern" of a configuration, a correspondence conjectured by Razumov and Stroganov in 2001 and recently proven by Cantini and the lecturer.

We review the motivations for studying ASM's (Mills, Robbins and Rumsey), the appearence of the "Laurent Phenomenon" (Fomin, Zelevinsky), the relation with the 6-Vertex Model of Statistical Mechanics, allowing for a simple proof of the enumeration (Kuperberg), and the proof of the Razumov-Stroganov original conjecture (Cantini, AS). The proof is based on the analysis of a Markov process on a Temperley-Lieb Algebra, and the generalization of a combinatorial bijection (called "gyration") due to Wieland.