Off-diagonally symmetric alternating sign matrices
Nishu Kumari
TL;DR
This work advances the exact enumeration of off-diagonally symmetric alternating sign matrices (OSASMs). It develops generating and partition function formalisms for diagonally symmetric ASMs (DSASMs) and OSASMs via a six-vertex model and Pfaffian techniques, then derives a complete product formula for the number of odd-order OSASMs by connecting to symplectic characters under a specialized parameter regime. It also proves a symmetry property for even-order OSASMs, showing a reflective invariance in the column-position statistic, and thereby resolves the remaining Behrend–Fischer–Koutschan conjectures on OSASM enumeration. The approach blends combinatorial model bijections, Pfaffian identities, and representation-theoretic objects (symplectic characters) to obtain exact counts and structural symmetries with potential broader implications for symmetry classes of ASMs.
Abstract
A diagonally symmetric alternating sign matrix (DSASM) is a symmetric matrix with entries $-1$, $0$ and $1$, where the nonzero entries alternate in sign along each row and column, and the sum of the entries in each row and column equals $1$. An off-diagonally symmetric alternating sign matrix (OSASM) is a DSASM, where the number of nonzero diagonal entries is 0 for even-order matrices and 1 for odd-order matrices. Kuperberg (Ann. Math., 2002) studied even-order OSASMs and derived a product formula for counting the number of OSASMs of any fixed even order. In this work, we provide a product formula for the number of odd-order OSASMs of any fixed order. Additionally, we present an algebraic proof of a symmetry property for even-order OSASMs. This resolves all the three conjectures of Behrend, Fischer, and Koutschan (arXiv, 2023) regarding the exact enumeration of OSASMs.