Alternating sign matrices with reflective symmetry and plane partitions: $n+3$ pairs of equivalent statistics and a Cauchy-type identity

TL;DR

This paper extends the understanding of symmetric alternating sign matrices and related plane partitions by introducing parameters, establishing generating function identities, and connecting to symplectic characters, advancing towards explicit bijections.

Contribution

It generalizes known results by adding parameters, equates multivariate generating functions, and links these to symplectic characters, moving closer to explicit bijective proofs.

Findings

Multivariate generating functions coincide for extended objects.

Equinumeracy of VSASMs and lozenge tilings follows from generating function identities.

Expansion into symplectic characters relates to symmetric plane partitions.

Abstract

Vertically symmetric alternating sign matrices (VSASMs) of order $2n+1$ are known to be equinumerous with lozenge tilings of a hexagon with side lengths $2n+2$, $2n$, $2n+2$, $2n$, $2n+2$, $2n$ and a central triangular hole of size $2$ that exhibit a cyclical as well as a vertical symmetry, but finding an explicit bijection proving this belongs to the most difficult problems in bijective combinatorics. Towards constructing such a bijection, we generalize the result by introducing certain natural extensions for both objects along with $n+3$ parameters and show that the multivariate generating functions with respect to these parameters coincide. This is a significant step from a constant number of equidistributed statistics to a linear number of statistics in $n$. The equinumeracy of VSASMs and the lozenge tilings is then an easy consequence of this result, which is obtained by specializing the generating functions to signed enumerations for both types of objects and then applying certain sign-reversing involutions. Another main result concerns the expansion of the multivariate generating function into symplectic characters as a sum over totally symmetric self-complementary plane partitions, which is in perfect analogy to the situation for ordinary ASMs where the Schur expansion can be written as a sum over totally symmetric plane partitions. This is exciting as it is reminiscent of the well-known Cauchy identity, and the Cauchy identity does have a bijective proof using the Robinson-Schensted-Knuth correspondence, and thus the result raises the question of whether there is a variation of the Robinson-Schensted-Knuth correspondence that does eventually lead to a bijective proof.