顶点的坐标为$1,-1,0,0$的全排列的多面体为截半立方体

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在$x_1 + x_2 + x_3 + x_4 = 0$内，顶点的坐标为 $1,-1,0,0$ 的全排列的多面体为截半立方体
gr.inc/question/the-subgroup-nt-subset-g-acts … adjoint-representat/

The maximal torus $\mathfrak{t} \subset \mathfrak{su}(4)$ is 3-dimensional, with roots $x_i - x_j$ ($i \neq j$) in the hyperplane $x_1 + x_2 + x_3 + x_4 = 0$. These roots correspond to the vertices of a cuboctahedron, a polyhedron with 8 triangular and 6 square faces. The Weyl group $ W = S_4$ permutes $ x_1, x_2, x_3, x_4$, which is equivalent to the rotational symmetries of a cube (or octahedron). The rotational symmetry group of a cube is $ S_4$, as it permutes the four space diagonals. Thus, $ S_4$ embeds into $ O(3)$ as the subgroup preserving the cuboctahedron’s structure, acting via orthogonal transformations in 3D space.

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该多面体共有 12 个顶点：
(1, -1, 0, 0), (-1, 1, 0, 0)
(1, 0, -1, 0), (-1, 0, 1, 0)
(1, 0, 0, -1), (-1, 0, 0, 1)
(0, 1, -1, 0), (0, -1, 1, 0)
(0, 1, 0, -1), (0, -1, 0, 1)
(0, 0, 1, -1), (0, 0, -1, 1)

每个点都与四个点相连。例如，$(1, -1, 0, 0)$ 与 $(1, 0, -1, 0),(1, 0, 0, -1),(0, -1, 1, 0),(0, -1, 0, 1)$ 相连。

共有 8 个三角形面。例如，$(1, -1, 0, 0),(1, 0, -1, 0),(0, -1, -1, 0)$ 组成一个三角形面。

共有 6 个正方形面。例如，$(1, -1, 0, 0),(1, 0, -1, 0),(0, 0, -1, 1),(0, -1, 0, 1)$ 组成一个正方形面。