Alternating Sign Arrays and Plane Partitions

Alternating sign arrays and plane partitions are classical objects in enumerative combinatorics with various connections to other areas such as algebra and geometry as well as to statistical physics. Our focus will be on unravelling unexplained relations between several classes of such objects. Such relations were first discovered conjecturally about 40 years ago. Concretely, it was observed that apparently for certain pairs of classes of objects there is the same number of objects. A particularly transparent and satisfying explanation of such a relation is an assignment between the two classes such that each object in one class gets assigned precisely one object in the other class and vice versa. However, such assignments seem to be extremely difficult to construct for the objects under consideration. Indeed such problems belong to the most difficult in the field, and it is the goal of the project to solve some of the open problems. After Mills, Robbins and Rumsey had formulated their first groundbreaking conjectures in the 1980s, Zeilberger and Kuperberg succeeded in proving some of their deep findings in the 1990s. For this purpose, new methods needed to developed, in fact, in the case of Kuperberg it turned out that methods that had been introduced by physicist earlier can be applied. Since then, the methods have been further developed, which allows for a systematical approach to attack various open problems now. More concretely, we will explore whether we can introduce new parameters in existing calculations and, in the case of success, draw the combinatorial conclusions. It is expected that in certain cases totally new calculations will be necessary and this will lead to an extension of our methods. Then we will also investigate whether existing combinatorial algorithms can be modified so that they can be applied to our classes of objects, and also here it will be necessary to construct also new algorithms. Finally, we will also consider a recent connection to algebraic geometry in the form of Grothendieck polynomials.