Symmetry classes of alternating-sign matrices under one roof

TL;DR

This paper extends the enumeration of various symmetry classes of alternating-sign matrices (ASMs) using determinant and Pfaffian formulas, revealing new classes and generalizing previous results.

Contribution

It introduces several new ASM classes, generalizes existing symmetry classes, and provides determinant-based enumeration formulas for these classes.

Findings

Enumerated new ASM symmetry classes such as UASMs, UUASMs, OSASMs, OOSASMs, and UOSASMs.

Derived determinant and Pfaffian formulas for counting these classes.

Provided explicit enumerations and generalizations of previous ASM enumeration results.

Abstract

In a previous article [math.CO/9712207], we derived the alternating-sign matrix (ASM) theorem from the Izergin-Korepin determinant for a partition function for square ice with domain wall boundary. Here we show that the same argument enumerates three other symmetry classes of alternating-sign matrices: VSASMs (vertically symmetric ASMs), even HTSASMs (half-turn-symmetric ASMs), and even QTSASMs (quarter-turn-symmetric ASMs). The VSASM enumeration was conjectured by Mills; the others by Robbins [math.CO/0008045]. We introduce several new types of ASMs: UASMs (ASMs with a U-turn side), UUASMs (two U-turn sides), OSASMs (off-diagonally symmetric ASMs), OOSASMs (off-diagonally, off-antidiagonally symmetric), and UOSASMs (off-diagonally symmetric with U-turn sides). UASMs generalize VSASMs, while UUASMs generalize VHSASMs (vertically and horizontally symmetric ASMs) and another new class, VHPASMs (vertically and horizontally perverse). OSASMs, OOSASMs, and UOSASMs are related to the remaining symmetry classes of ASMs, namely DSASMs (diagonally symmetric), DASASMs (diagonally, anti-diagonally symmetric), and TSASMs (totally symmetric ASMs). We enumerate several of these new classes, and we provide several 2-enumerations and 3-enumerations. Our main technical tool is a set of multi-parameter determinant and Pfaffian formulas generalizing the Izergin-Korepin determinant for ASMs and the Tsuchiya determinant for UASMs [solv-int/9804010]. We evaluate specializations of the determinants and Pfaffians using the factor exhaustion method.