Triangular Ice Combinatorics

P. Di Francesco, University of Illinois and IPhT Saclay

Alternating Sign Matrices (ASM) are at the confluent of many interesting
combinatorial/algebraic problems: Laurent phenomenon for the octahedron
equation, configurations of the Square Ice (Six Vertex model), Descending
Plane Partitions (DPP), etc. Here we consider the Triangular Lattice version
of the Ice model with suitable boundary conditions leading to an
integrable 20 Vertex model. Configurations give rise to generalizations
of ASM, which we coin Alternating Phase Matrices (APM). We generalize the
ASM-DPP correspondence by showing that APM are equinumerous to
the quarter-turn symmetric domino tilings of a quasi-Aztec square with a central
cross-shaped hole, and obtain a compact determinant formula for their enumeration.
We also present conjectures for triangular Ice with other types of boundary conditions,
and results on the limit shape of large APM.

(joint work with E. Guitter, IPhT Saclay, France).

Presentation (PDF File)

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