Gorenstein单形及其有限阿贝尔群
Gorenstein simplices and the associated finite abelian groups
摘要：
已知维数为$d$的格单形对应于$(\mathbb{R}/\mathbb{Z})^{d+1}$的有限阿贝尔子群。反之，给定$(\mathbb{R}/\mathbb{Z})^{d+1}$的有限阿贝尔子群，使得每个元素的所有元素之和是整数，我们可以得到维数为$d$的格单形。本文讨论了Gorenstein单形的有限交换群阿贝尔群的一个刻画。特别是，我们提出了完整的Gorenstein单形的归一化体积等于$p,p^2$和$pq$，其中$p$和$q$是素数与$p \neq q$。此外，我们还计算了Gorenstein单形的对偶单形的体积。
英文摘要：
It is known that a lattice simplex of dimension $d$ corresponds a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$. Conversely, given a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$ such that the sum of all entries of each element is an integer, we can obtain a lattice simplex of dimension $d$. In this paper, we discuss a characterization of Gorenstein simplices in terms of the associated finite abelian groups. In particular, we present complete characterizations of Gorenstein simplices whose normalized volume equals $p,p^2$ and $pq$, where $p$ and $q$ are prime numbers with $p \neq q$. Moreover, we compute the volume of the dual simplices of Gorenstein simplices.
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