Alternating Sign Arrays: Symmetry & Symmetric Functions

Establishing an enumeration formula for alternating sign matrices presented a challenging open problem for more than a decade. Widely known as the alternating sign matrix conjecture, the problem was eventually resolved through the independent breakthroughs of Zeilberger and Kuperberg in 1996. Since then, alternating sign arrays have become classical structures in combinatorics. They are renowned not only for their profound connections to other areas of mathematics but also for the notoriously difficult problems they provide. One noteworthy connection is the relation between alternating sign arrays and the theory of symmetric functions which lie at the core of algebraic combinatorics. How the combinatorics of alternating sign arrays is linked to symmetric functions, remains far from fully understood and has received only little attention thus far, particularly concerning symmetry classes of alternating sign matrices. In this project, we investigate both old and new problems surrounding symmetry classes of alternating sign arrays and their connection to the theory of symmetric functions. We focus on arrowed monotone triangles, a recent generalisation of alternating sign matrices due to Ilse Fischer and Florian Schreier-Aigner. One primary objective is to enhance this novel and innovative perspective by lifting existing results, such as enumerative results on monotone triangles (without arrows) and results connecting symmetry classes of alternating sign matrices with symmetric functions, to the level of arrowed monotone triangles. Additionally, we study halved arrowed monotone triangles to establish new connections between vertically symmetric alternating sign arrays and symplectic Schur polynomials. A crucial aspect of the project involves exploring the Schur expansion of a generating function associated with arrowed monotone triangles. This expansion appears to have surprising relations to several symmetry classes of alternating sign matrices, contributing to a deeper understanding of the intricate interplay between these combinatorial objects.