Symmetric polynomials and (arrowed) monotone triangles

In the research project Symmetric polynomials and (arrowed) monotone triangles we intend to study certain symmetric polynomials which are connected to combinatorics. The symmetric polynomials of our interest are defined to encode certain properties of combinatorial objects; this allows us to obtain a better understanding of the underlying objects by studying their corresponding polynomials. The combinatorial objects we are interested in are semistandard Young tableaux (SSYTs), alternating sign matrices (ASMs) and plane partitions. Alternating sign matrices and plane partitions are two families of 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 certain symmetry classes of ASMs and of plane partitions are equinumerous, however a combinatorial proof is known only for one instance of this equinuerousity. This is a rather untypical and hence mysterious situation. It is still an open problem to find an explicit bijection between most of these classes of objects, i.e., a map which translates objects of one class into objects of another class in a one-to-one correspondence. The third family of combinatorial objects appearing in this research project are semistandard Young tableaux (SSYTs). These objects are very well understood since they can be seen as the defining objects for the Schur polynomials, an important and long studied family of symmetric polynomials. For this project however, we intend to study Macdonald polynomials which generalise Schur polynomials and therefore encode more structure of the SSYTs. In particular we intend to prove a generalisation of the so called dual RSK algorithm for Macdonald polynomials which was conjectured by Gabriel Frieden and myself. The classical dual RSK algorithm is defined for Schur polynomials and has many applications in combinatorics, representation theory, probability theory or algebraic geometry. The involved researchers include beside Florian Schreier-Aigner and prospective employees also the cooperation partners Ilse Fischer and Gabriel Frieden.