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F50-10: Alternating sign arrays of triangular shapes

Principal Investigator: Ilse Fischer

The main objective of this part of the SFB is to solve several enumeration problems connected with alternating sign matrices (ASM) and related objects such as six-vertex configurations, monotone triangles and fully packed loop configurations. ASMs first appeared in the 1980s in the work of Robbins and Rumsey when they introduced a generalization of the determinant, and, together with Mills, they presented a conjecture on the number of ASMs that attracted a lot of attention from combinatorialists, since classical methods seemed to fail there. Zeilberger finally succeeded in proving their conjecture in 1995. Another breakthrough happenend when it was discovered that objects equivalent to ASMs had been studied in statistical physics in the context of exactly solvable models for a long time already. Since then several enumeration results on subclasses of ASMs have been established.

We aim at proving a number of conjectures proposed by the PI together with collaborators in unpublished work on the enumeration of alternating sign arrays of triangular shapes. These conjectures involve formulas that have previously appeared in the enumeration of ASMs and plane partitions, and we also aim at explaining this circumstance. We will use six-vertex model techniques, where in particular the solution of the reflection equation that has recently been employed by Behrend, Konvalinka and the PI to enumerate diagonally and antidiagonally symmetric alternating sign matrices of odd order will be an important tool. Moreover, we will also apply the PIs alternative method to deal with ASMs to establish possible connections to the Razumov–Stroganov correspondence.

Ilse Fischer