Alternating Sign Matrices (ASMs) are famously difficult to enumerate. Even more elusive—perhaps impossible—are satisfying bijective proofs of the numerous equinumerosity results involving them. In the first half of this course, we will review several foundational combinatorial constructions for ASMs and related objects that are key to bijective proofs of related results.
These constructions include the spider move (used to enumerate domino tilings of the Aztec diamond), Wieland gyration (central to the ingenious proof of the Razumov–Stroganov conjecture by Cantini and Sportiello), graphical condensation (a key tool behind many beautiful results on the enumeration of lozenge tilings), and the Robinson–Schensted–Knuth (RSK) correspondence.
In the second half of the course, we will introduce Robbins polynomials, generalizations of Schur polynomials that serve as multivariate generating functions for ASMs. These polynomials satisfy Cauchy- and Littlewood-type identities that extend the classical ones. This is particularly exciting because Littlewood-type identities play a central role in several non-bijective proofs of the equinumerosity results mentioned above, and the classical Cauchy and Littlewood identities admit elegant bijective proofs via RSK. We will also see that Robbins polynomials arise as special cases of the fully inhomogeneous spin Hall–Littlewood symmetric rational functions introduced by Borodin and Petrov, and we will explain how Littlewood-type identities for Robbins polynomials have been lifted to this much more general framework.
The second part of the course is based on joint work with Moritz Gangl, Hans Höngesberg, and Florian Schreier-Aigner.