|The many faces of alternating-sign matrices|
Alternating-sign matrices (more specifically, "compatible
pairs" of alternating-sign matrices) arose from the work of Dave Robbins
and his collaborators on Dodgson condensation. Unbeknowst to anyone at
the time, there was a natural combinatorial context for such pairs,
namely, the dimer model studied by statistical physicists. It is
perhaps fortunate that this connection was not discovered prematurely,
or the marvelous formula conjectured by Mills, Robbins, and Rumsey might
not have been noticed.
Known proofs of this conjecture required that the enumeration of alternating-sign matrices be situated in a wider context, namely, either numerical arrays (Zeilberger) or the square ice model of statistical physics (Kuperberg). More recently, a third context has received attention: the fully-packed loop (FPL) model. One striking feature of the FPL model is that it leads to many (still conjectural) enumerative formulas that do not translate over to the other two contexts. It is tempting to speculate that studies of the loop model may give rise to a completely different proof of the 1,2,7,42,... theorem.
The first half of my talk will fill in the key details of the sketch given above. The second half will treat a variety of topics related to alternating-sign matrices, which may include: the asymptotic limiting shape of random ASMs; applications of Robbins' and Rumsey's lambda-determinant to the enumeration of domino-tilings of squares; asynchronous Dodgson condensation and the octahedron recurrence; and a new analogue of Dodgson condensation (the cube recurrence) whose true significance is still unclear.
Postscript slides used in the talk may be downloaded from www.math.wisc.edu/~propp/DR.ps. Follow along while listening to the mp3 audio file of Jim Propp's talk.