自由群中的回文长度：反射、非交叉匹配与Catalan形式

中文摘要 AI 辅助

设$F=F(X)$为有限秩自由群，其回文长度相对于固定基$X$定义。我们将$F$嵌入到万有Coxeter群$W=F\rtimes_\theta\langle t\mid t^2=1\rangle$的指数为2的子群中，其中$\theta(x)=x^{-1}$对$x\in X$成立，并证明$\mathrm{pl}(g)=\min\{\ell_T(g),\ell_T(gt)\}$。Dyer的删除定理进而将反射长度等同于约化Coxeter词上非交叉等标签部分匹配中未匹配位置的最小数目。这给出了一个$O(n^3)$时间、$O(n^2)$空间的回文长度算法，同时恢复最优回文分解。匹配模型还给出了一个结构刻画。对于每个具有$k$个叶子的有序满二叉树，我们定义一个字面词模板，其叶子为回文，内部顶点携带任意词。一个约化词$w$表示回文长度至多为$k$的元素，当且仅当$w$是这些模板之一的一个字面实例。因此，$C_{k-1}$个有序二叉树形状为每个固定$k$给出了一个完备的有限族。对于$k=4$，这五个模板恰好是Frid提出的五种形式，证明了该列表的完备性。一个配套的Lean 4开发端到端地验证了每个有限秩的四回文分类，包括普通的约化词表述和字面五形式结论。

英文摘要

Let $F=F(X)$ be a free group of finite rank, with palindromic length taken with respect to the fixed basis $X$. We embed $F$ as the index-two subgroup of the universal Coxeter group $W=F\rtimes_θ\langle t\mid t^2=1\rangle$, where $θ(x)=x^{-1}$ for $x\in X$, and prove $\mathrm{pl}(g)=\min{\ell_T(g),\ell_T(gt)}$. Dyer's deletion theorem then identifies reflection length with the minimum number of unmatched positions in a noncrossing equal-label partial matching on a reduced Coxeter word. This gives an $O(n^3)$-time, $O(n^2)$-space algorithm for palindromic length, together with recovery of an optimal palindromic factorization. The matching model also gives a structural characterization. For every ordered full binary tree with $k$ leaves we define a literal word template whose leaves are palindromes and whose internal vertices carry arbitrary words. A reduced word $w$ represents an element of palindromic length at most $k$ if and only if $w$ is a literal instance of one of these templates. Hence the $C_{k-1}$ ordered binary-tree shapes give a complete finite family for each fixed $k$. For $k=4$ the five templates are exactly the five forms proposed by Frid, proving the completeness of that list. A companion Lean 4 development verifies the four-palindrome classification end to end for every finite rank, including the ordinary reduced-word formulation and the literal five-form conclusion.

发表机构

• cnu.edu.cn（首都师范大学）

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