关于粘合格的置换不变构造
On permutation-invariant construction of glued lattices
浏览论文内容
中文总结 AI 辅助
本文研究由向量在置换作用下的轨道生成的置换不变格，给出达到秩界的充分条件，证明其粘合结构并导出行列式公式，同时推广至代数整格并加强相关结果。
中文摘要 AI 辅助
给定一个在 $n$ 个字母上的置换 $\tau$，我们考虑由向量 $\boldsymbol x \in \mathbb R^n$ 在 $\tau$ 通过坐标置换作用下的轨道所张成的格。这类格推广了重要的循环格类，并已在文献~\cite{perm} 中被研究，其中建立了其秩的一个界。我们证明了 $\boldsymbol x$ 达到此界的一个充分条件。我们进一步研究了此类置换不变格的结构，证明它们是由置换向量从正交循环块粘合而成，并给出了一个以这些块的行列式和置换向量的范数表示的格的行列式公式。当 $\boldsymbol x$ 为整数向量时，这些块是各自维度中根格 $A_k$ 的子格，且根格本身及其粘合直和也可由此构造实现。我们还展示了从循环数域集合构造置换不变代数整格的粘合构造。最后，我们证明了文献~\cite{lf_ek} 中关于由代数共轭集合张成的良圆格的相关构造的一个先前结果的加强版本。
英文摘要
Given a permutation $τ$ on $n$ letters, we consider lattices spanned by an orbit of one vector $\boldsymbol x$ in $\mathbb R^n$ under the action of $τ$ by permutation of the coordinates. Such lattices generalize the important class of cyclic lattices and have previously been studied in~\cite{perm}, where a bound on their rank was established. We prove a sufficient condition on $\boldsymbol x$ for this bound to be achieved. We further investigate the structure of such permutation-invariant lattices, proving that they are glued by the permuted vector from the orthogonal cyclic blocks and giving a determinant formula for the lattice in terms of determinants of these blocks and the norm of the permuted vector. In the case $\boldsymbol x$ is an integer vector, these blocks are sublattices of the root lattices $A_k$ in respective dimensions with root lattices themselves and their glued direct sums also realizable by this construction. We also exhibit a glued construction of permutation-invariant algebraic integral lattices from collections of cyclic number fields. Finally, we prove a strengthened version of a previous result of~\cite{lf_ek} on a related construction of well-rounded lattices spanned by sets of algebraic conjugates.
发表机构
- São Paulo State University（圣保罗州立大学）
- Claremont McKenna College（克莱蒙特麦肯纳学院）
机构由 AI 辅助整理，请以论文原文为准。
