We prove refined enumeration results on several symmetry classes as well as related classes of alternating sign matrices with respect to classical boundary statistics, using the six-vertex model of statistical physics. More precisely, we study vertically symmetric, vertically and horizontally symmetric, vertically and horizontally perverse, off-diagonally and off-antidiagonally symmetric, vertically and off-diagonally symmetric, quarter turn symmetric as well as quasi quarter turn symmetric alternating sign matrices. Our results prove conjectures of Fischer, Duchon and Robbins.

中文翻译：

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交替符号矩阵的对称类的精细枚举

我们使用统计物理学的六顶点模型，证明了关于经典边界统计的几个对称类以及相关类的交替符号矩阵的精炼枚举结果。更准确地说，我们研究垂直对称，垂直和水平对称，垂直和水平不规则，非对角线和非对角线对称，垂直和非对角线对称，四分之一转对称以及准四分之一转对称交替符号矩阵。我们的结果证明了Fischer，Duchon和Robbins的猜想。