Symmetric functions and alternating sign matrices

Alternating sign matrices (ASMs) and plane partitions are combinatorial objects with an intriguing but mysterious relation. ASMs were introduced in the early 80ies by David Robbins and Howard Rumsey and are in correspondence to configurations of the six-vertex model, a well-studied model in statistical physics. Plane partitions were introduced in the late 19th century by Major Percy Alexander MacMahon and turned out to be connected among others to the theory of symmetric functions as well as to statistical physics. In the course of the last four decades, it was proven that ASMs are equinumerous to three further combinatorial families; one of these families is that of descending plane partitions (DPPs). However, and this makes their relation very mysterious, it is still an open problem to find an explicit bijection between any of these four families of combinatorial objects, i.e., a map which translates objects of one family into objects of another family in a one-to-one correspondence. In the following research project Symmetric functions and alternating sign matrices, our aim is to study a family of symmetric functions which is linked to ASMs and d-DPPs, a generalisation of DPPs and cyclically symmetric plane partitions. In literature, symmetric functions associated to ASMs are obtained by using the six-vertex model approach. In this research project, we use the operator formula approach, which is less studied in literature, to obtain a family of symmetric functions. For these functions, I found in collaboration with François Bergeron a very combinatorial description and a surprising link to the enumeration of d-DPPs, which we want to prove in the course of this project. In particular we will study a family of symmetric functions which can be expressed as a sum of Schur functions, where each summand is associated to a totally symmetric plane partitions (TSPPs), another symmetry class of plane partitions. A conjecture of Bergeron and me is that certain evaluations of these symmetric functions are equal to a weighted enumeration of ASMs or the enumeration of d-DPPs respectively. In order to prove these conjectures we start with certain special cases, which can be found in literature. In a next step we plan to transfer typical methods and techniques used in the six-vertex approach to our setting. To conclude, the presented research project is centred about a novel family of symmetric functions which was not regarded in literature yet. This family stands out due to its very combinatorial description involving TSPPs, which were not connected to ASMs, and for being connected to a weighted enumeration of ASMs and the enumeration of d-DPPs.

The research project ``Symmetric functions and alternating sign matrices'' achieved new results within two areas of combinatorics: on the one hand the relation between ``alternating sign matrices'' (ASMs) and ``plane partitions'' was studied, on the other hand a generalisation of the Robinson-Schensted correspondence for Macdonald polynomials was found. ASMs and plane partitions are combinatorial objects with an intriguing but mysterious relation. They were introduced in the early 80ies by David Robbins and Howard Rumsey as follows: An ASM is a configuration of the numbers 1, 0, and -1 in a square such that the sum of the entries in each row or column is equal to 1 and such that the non-zero entries alternate in each row or column respectively. Plane partitions were introduced in the late 19th century by Major Percy Alexander MacMahon and are configurations of natural integers on a grid which are not increasing along rows or columns respectively. In the course of the last four decades, it was proven that ASMs are equinumerous among others to a certain symmetry class of plane partitions which are called ``descending plane partitions'', or DPPs for short. It is a very unusual that no simple combinatorial proof, i.e., bijection, is known which ``translates'' ASMs into DPPs. In the first part of the project, I introduced together with Ilse Fischer two families of objects generalising ASMs and DPPs respectively. One of our main results states that these objects are equinumerous as well. Furthermore we describe a linear number of properties (so called statistics), which are equidistributed for both sets of objects. This is an important step in understanding the relation between ASMs and DPPs since we enlarged the number of statistics which can be considered simultaneously from 4 to n+3, where n is the ``size'' of the regarded objects. In the second part of the project, I studied together with Gabriel Frieden a generalisation of the Robinson-Schensted correspondence, or RS for short. The RS correspondence is a bijection, i.e., a ``translation'', between two different combinatorial objects and has many applications in various fields of mathematics. The Macdonald polynomials which were introduced in the 1980s can be used to define additional properties for these objects, usually called weights. Together with Frieden, I found a generalisation of RS for those weighted objects, the qRSt correspondence. Contrary to the classical RS correspondence, the qRSt correspondence is probabilistic, i.e., each step of the ``translating algorithm'' between the objects is chosen with respect to a certain probability.