Symmetry classes of alternating-sign matrices under one roof
Symmetry classes of alternating-sign matrices under one roof
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DOI：
10.2307/3597283
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发表时间：
2000-08
影响因子：
4.9
通讯作者：
G. Kuperberg
中科院分区：
文献类型：
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作者：
G. Kuperberg
在以前的文章[数学。CO/9712207]中，我们从Izergin-Korepin行列式中推导出具有畴壁边界的方形冰的配分函数的交替符号矩阵（ASM）定理。在这里，我们表明，相同的参数列举了其他三个对称类的交替符号矩阵：VSASM（垂直对称ASM），甚至HTSASM（半圈对称ASM），甚至QTSASM（四分之一圈对称ASM）。VSASM枚举是由米尔斯完成的;其他枚举是由罗宾斯完成的[math.CO/0008045]。我们介绍了几种新类型的ASM：UASM（带U形转弯侧的ASM），UUASM（两个U形转弯侧），OSASM（非对角对称ASM），OOSASM（非对角，非反对角对称）和UOSASM（非对角对称U形转弯侧）。UASMs概括了VSASM，而UUASMs概括了VHSASM（垂直和水平对称的ASMs）和另一个新的类别，VHPASM（垂直和水平反常）。OSASM、OOSASM和UOSASM与ASM的其余对称类相关，即DSASM（对角对称）、DASASM（对角、反对角对称）和TSASM（完全对称ASM）。我们列举了其中的几个新类，并提供了几个2-枚举和3-枚举。我们的主要技术工具是一组多参数行列式和Pfidian公式，将Izergin-Korepin行列式推广为ASMs和Tsuchiya行列式推广为UASMs [solv-int/9804010]。我们评估专业化的决定因素和Pfidians使用的因素穷举法。
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.