arXiv 1504.021572015-04-09math.CO

对称群上块转置的组合性质

Combinatorial properties of block transpositions on Symmetric groups

Annachiara Korchmaros

备注：: 92 pages, 5 figures, PhD thesis

摘要：

在研究置换群的组合方面的一个主要问题是确定对称群$\Sym_n$中关于生成元集的距离。一个众所周知的这样的情况是当生成器集合$S_n$由块换位组成时。需要说明的是，“置换的块转置距离”是该置换与Cayley图$\Cay$中的单位置换的距离，“通过块转置对置换进行排序”等价于在$\Cay$中寻找最短路径。本文的原始结果涉及$\Sym_n$关于$S_n$和自同构群$\Aut(\Cay)$的块转置直径的上下界。一个显着的贡献是显示如何从环面等价可以获得双射映射$\Sym_n$，我们称之为\emph{toric maps}。利用复曲面映射的性质，讨论了复曲面类内块转置距离不变性原理在证明Eriksson界中的作用。进一步证明了$\Aut(\Cay)$是右平移群与$\textsf{N}\rtimes\textsf{D}_{n+1}$的乘积，其中$\textsf{N}$是按元素固定$S_n$的子群，$\textsf{D}_{n+1}$是极大循环子群由环面映射生成的二面体群.计算机辅助计算支持我们的猜想，$\textsf{N}$是平凡的。证明了顶点集为$S_n$的子图$Γ$是$2(n-2)$ -正则图，其自同构群为$\textsf{D}_{n+1}$ .我们证明了$\Cay$的某些方面，特别是$Γ$有多达$n+1$的最大团的大小$2$，它的子图$Γ(V)$的顶点是那些在这些团是一个$3$ -正则哈密尔顿图，和$\textsf{D}_{n+1}$忠实于$V$作为一个顶点正则自同构群。

英文摘要：

A major problem in the study of combinatorial aspects of permutation groups is to determine the distances in the symmetric group $\Sym_n$ with respect to a generator set. One well-known such a case is when the generator set $S_n$ consists of block transpositions. It should be noted that "the block transposition distance of a permutation" is the distance of the permutation from the identity permutation in the Cayley graph $\Cay$, and "sorting a permutation by block transpositions" is equivalent to finding shortest paths in $\Cay$. The original results in our thesis concern the lower and upper bounds on the block transpositions diameter of $\Sym_n$ with respect to $S_n$ and the automorphism group $\Aut(\Cay)$. A significant contribution is to show how from the toric equivalence can be obtained bijective maps on $\Sym_n$ that we call \emph{toric maps}. Using the properties of the toric maps, we discuss the role of the invariance principle of the block transposition distance within toric classes in the proof of the Eriksson bound. Furthermore, we prove that $\Aut(\Cay)$ is the product of the right translation group by $\textsf{N}\rtimes\textsf{D}_{n+1}$, where $\textsf{N}$ is the subgroup fixing $S_n$ elementwise, and $\textsf{D}_{n+1}$ is a dihedral group whose maximal cyclic subgroup is generated by the toric maps. Computer aided computation supports our conjecture that $\textsf{N}$ is trivial. Also, we prove that the subgraph $Γ$ with vertex-set $S_n$ is a $2(n-2)$-regular graph whose automorphism group is $\textsf{D}_{n+1}$. We show some aspects of $\Cay$, notably $Γ$ has as many as $n+1$ maximal cliques of size $2$, its subgraph $Γ(V)$ whose vertices are those in these cliques is a $3$-regular Hamiltonian graph, and $\textsf{D}_{n+1}$ acts faithfully on $V$ as a vertex regular automorphism group.

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英文摘要：

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