## Projektdetails

### Abstract

Wider Research Context.

Alternating sign matrices and plane partitions are classical objects in enumerative combinatorics with several fascinating features: Various classes of such objects are counted by simple formulas, but the proofs of these formulas are usually complicated and not insightful. There are also mysterious relations between different classes of such objects, as indicated by the fact that they have the same numbers, but so far such relations are not well-explained. It is a characteristic of this field that conjectures on such formulas and relations are usually true as there does not seem to be a case where such a conjecture turned out to be wrong in the end. Alternating sign matrices and plane partitions have also rich connections to other fields such as representation theory, symmetric functions, integrable models, and, as revealed recently, also to Schubert calculus: Schubert polynomials and Grothendieck polynomials are generating functions of alternating sign matrices.

Research Questions.

Our goals are grouped into five intertwined topics. We plan to attack the oldest conjectures in the field on the relation between GOGs and MAGOGs. Then, in our recent work relating alternating sign triangles to two classes of plane partitions, a formula that is reminiscent of the Cauchy identity plays a crucial role. This led us to new conjectures that we intend to study, in particular we will explore whether a Robinson-Schensted-Knuth type algorithm is lurking behind these identities. Another topic concerns a systematic approach to transfer statistics and other parameters that are known for certain classes to other classes. Similarly, we will do this for (symmetry) operations, for which there exist some old conjectures of Mills, Robbins and Rumsey. Lastly, we will explore the connection between alternating sign matrices and Grothendieck polynomials.

Approach.

In the last 15 years, the PI has connected all classes that are known to be equinumerous with alternating sign matrices by (complicated) computations. Transfers of statistics, parameters and operations will be tackled through refining, generalizing and modifying these computations. As for the conjectures on GOGs and MAGOGs, we have already derived formulas for both types of objects, and we need to connect them through similar computations, possibly combined with certain techniques related to a proof of the Bender-Knuth (ex-)conjecture by the PI. As for the Cauchy type identities, we will also take inspiration from the classical approaches.

Level of Originality.

We plan to study long-standing conjectures in a systematic way by means of modifying existing linking computations. At the same time, the observations around the Cauchy type identities and the Grothendieck polynomials are new. Would it turn out that a variation of the RSK algorithm is behind some connections, this would be particularly pleasing.

Primary Researchers Involved.

1 Postdoc, 1 Predoc, and the PI

Alternating sign matrices and plane partitions are classical objects in enumerative combinatorics with several fascinating features: Various classes of such objects are counted by simple formulas, but the proofs of these formulas are usually complicated and not insightful. There are also mysterious relations between different classes of such objects, as indicated by the fact that they have the same numbers, but so far such relations are not well-explained. It is a characteristic of this field that conjectures on such formulas and relations are usually true as there does not seem to be a case where such a conjecture turned out to be wrong in the end. Alternating sign matrices and plane partitions have also rich connections to other fields such as representation theory, symmetric functions, integrable models, and, as revealed recently, also to Schubert calculus: Schubert polynomials and Grothendieck polynomials are generating functions of alternating sign matrices.

Research Questions.

Our goals are grouped into five intertwined topics. We plan to attack the oldest conjectures in the field on the relation between GOGs and MAGOGs. Then, in our recent work relating alternating sign triangles to two classes of plane partitions, a formula that is reminiscent of the Cauchy identity plays a crucial role. This led us to new conjectures that we intend to study, in particular we will explore whether a Robinson-Schensted-Knuth type algorithm is lurking behind these identities. Another topic concerns a systematic approach to transfer statistics and other parameters that are known for certain classes to other classes. Similarly, we will do this for (symmetry) operations, for which there exist some old conjectures of Mills, Robbins and Rumsey. Lastly, we will explore the connection between alternating sign matrices and Grothendieck polynomials.

Approach.

In the last 15 years, the PI has connected all classes that are known to be equinumerous with alternating sign matrices by (complicated) computations. Transfers of statistics, parameters and operations will be tackled through refining, generalizing and modifying these computations. As for the conjectures on GOGs and MAGOGs, we have already derived formulas for both types of objects, and we need to connect them through similar computations, possibly combined with certain techniques related to a proof of the Bender-Knuth (ex-)conjecture by the PI. As for the Cauchy type identities, we will also take inspiration from the classical approaches.

Level of Originality.

We plan to study long-standing conjectures in a systematic way by means of modifying existing linking computations. At the same time, the observations around the Cauchy type identities and the Grothendieck polynomials are new. Would it turn out that a variation of the RSK algorithm is behind some connections, this would be particularly pleasing.

Primary Researchers Involved.

1 Postdoc, 1 Predoc, and the PI

Status | Laufend |
---|---|

Tatsächlicher Beginn/ -es Ende | 21/06/21 → 20/06/25 |