Symmetry Classes of Alternating-Sign Matrices under One Roof

Abstract

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