AI 中文总结
本文证明阶为2的魔面超立方体在任意维数下存在且唯一（在特定变换下），并应用于推广卡朱拉霍幻方、回答Coxeter问题，证明最完美对象在Weyl群作用下为单轨道。
AI 中文摘要
一个$k$阶、$n$维的\textit{幻线超立方体}（或简称为\textit{幻超立方体}）是将数字$1,\dots,k^n$排列在一个$k\times\cdots\times k$（$n$重）网格中，使得每条平行于坐标轴的$k$个数字的线上的幻和都相同。虽然对于每个阶数$k\ge 3$和每个维数$n$都存在这样的超立方体，但在任何维数$n\ge 2$中都不存在阶数为$2$的幻线超立方体。对于阶数为$2$的超立方体，我们将幻条件从线放宽到二维面或平面。如果在一个$2\times\cdots\times2$（$n$重）网格中排列数字$1,\dots,2^n$，使得每个$2\times2$的面具有相同的幻和，则称其为\textit{魔面}的。我们证明，在每一个维数$n\ge0$中，都存在阶数为$2$的魔面超立方体，并且当$n$为偶数时，它在大小为$2^n(n+1)!$的某个自然变换集合下是唯一的；当$n$为奇数时，在大小为$2^n n\cdot n!$的变换集合下是唯一的。作为一个应用，我们恢复并推广了经典的$4\times4$卡朱拉霍幻方，回答了Coxeter关于作用于$384$个“最完美”$4\times4$幻方的群的问题，并将该图景扩展到更高维数。特别地，我们证明，在维数$n$中，这些最完美对象在Weyl群$W(B_{2n})$的某个自然作用下形成单个轨道。
英文摘要
A $\textit{magic-lined hypercube}$ (or, simply, $\textit{magic hypercube}$) of order $k$ and dimension $n$ is an arrangement of the numbers $1,\dots,k^n$ in a $k\times\cdots\times k$ ($n$-fold) grid such that every line of $k$ numbers parallel to a coordinate axis has the same magic sum. While such hypercubes exist for every order $k\ge 3$ and every dimension $n$, no magic-lined hypercube of order $2$ exists in any dimension $n\ge 2$. For hypercubes of order $2$, we thus relax the magic condition from lines to two-dimensional faces or planes. We call an arrangement of the numbers $1,\dots,2^n$ in a $2\times\cdots\times2$ ($n$-fold) grid \emph{magic-faced} if every $2\times2$ face has the same magic sum. We prove that a magic-faced hypercube of order $2$ exists in every dimension $n\ge0$, and that it is unique up to a certain natural set of transformations of size $2^n(n+1)!$ when $n$ is even and $2^n n\cdot n!$ when $n$ is odd. As an application, we recover and generalize the classical $4\times4$ Khajuraho magic square, answer a question of Coxeter on the group acting on the $384$ ``most-perfect" $4\times4$ magic squares, and extend the picture to higher dimensions. In particular, we prove that, in dimension $n$, these most-perfect objects form a single orbit under a certain natural action of the Weyl group $W(B_{2n})$.
Comments31 pages
