We investigate the convex hulls of the eight dihedral symmetry classes of $n \times n$ alternating sign matrices, i.e., ASMs invariant under a subgroup of the symmetry group of the square. Extending the prefix-sum description of the ASM polytope, we develop a uniform core--assembly framework: each symmetry class is encoded by a set of core positions and an affine assembly map that reconstructs the full matrix from its core. This reduction transfers polyhedral questions to lower-dimensional core polytopes, which are better suited to the tool set of polyhedral combinatorics, while retaining complete information about the original symmetry class. For the vertical, vertical--horizontal, half-turn, diagonal, diagonal--antidiagonal, and total symmetry classes, we give explicit polynomial-size linear inequality descriptions of the associated polytopes. In these cases, we also determine the dimension and provide facet descriptions. The quarter-turn symmetry class behaves differently: the natural relaxation admits fractional vertices, and we need to extend the system with a structured family of parity-type Chvátal--Gomory inequalities to obtain the quarter-turn symmetric ASM polytope. Our framework leads to efficient algorithms for computing minimum-cost ASMs in each symmetry class and provides a direct link between the combinatorics of symmetric ASMs and tools from polyhedral combinatorics and combinatorial optimization.

翻译：本文研究了$n \times n$交替符号矩阵（ASM）在正方形对称群子群作用下不变的八个二面体对称类所对应的凸包。通过扩展ASM多面体的前缀和描述，我们建立了一个统一的核心-组装框架：每个对称类由一组核心位置和一个仿射组装映射编码，该映射可从核心重构完整矩阵。这种约简将多面体问题转化为低维核心多面体，后者更适用于多面体组合学的工具集，同时完整保留了原始对称类的信息。对于垂直、垂直-水平、半转、对角线、对角线-反对角线以及完全对称类，我们给出了相应多面体的显式多项式规模线性不等式描述。针对这些情形，我们还确定了其维度并提供了面描述。四分之一转对称类的行为则有所不同：其自然松弛允许分数顶点，我们需要通过引入一组具有特定结构的奇偶型Chvátal-Gomory不等式来扩展系统，从而获得四分之一转对称ASM多面体。我们的框架为计算各对称类中最小成本ASM提供了高效算法，并在对称ASM的组合学与多面体组合学及组合优化工具之间建立了直接联系。