Alternating Sign Matrices and BeyondJames ProppUniversity of Wisconsin
September 17,

ABSTRACT


The solution of the alternating sign matrix by Zeilberger is likely to be only the end of the first chapter of a very long and interesting story. Here are excerpts from three later chapters, with an emphasis on accessible open problems. One chapter concerns enumeration of ASMs that are invariant under various symmetry groups, and more generally, the enumeration of constrained alternating sign patterns. I'll discuss experimental work on halved ASMs (largely inspired by some stillobscure results of the physicist Tsuchiya). Another chapter concerns the denselypacked loop representation of ASMs, and a discovery made by Harvard undergraduates Carl Bosley and Lukasz Fidkowski and later proved by M.I.T. undergraduate Ben Wieland: there is a sense in which alternating sign matrices of order $n$ are governed by the symmetry group $D(n)$ (the dihedral group of order $2n$). A third chapter concerns the matrix condensation process (due to Dodgson) that inspired Mills, Robbins, and Rumsey to invent ASMs in the first place. Variant forms of condensation give rise to new analogues of ASMs. There are no obvious counterparts of the MillsRobbinsRumsey conjecture for these analogues, but there are still interesting enumerative phenomena that await explanation. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

