双基点Terwilliger代数与素数阶循环图的量子对称性

中文摘要 AI 辅助

素数阶的顶点传递图具有怎样的量子对称性？Banica、Bichon和Chenevier提出的这个问题在Paley图的稠密区域仍未解决，而相干代数方法在该区域无法提供信息。对于每一个此类图，我们附加其分圆方案的双基点Terwilliger代数，并研究该代数从基点生成的模：完备性要求量子置换代数是交换的，且该模不允许中间状态，仅包含恰好两个点质量或全部p个点质量。因此，在任意深度捕获的单个点质量就足够，Chassaniol的轨道准则是深度一的情况。由此得出三个结论：1. 精确计数论证将Banica--Bichon--Chenevier阈值p>6^φ(k)替换为二次界p>(k-1)(k-2)+2，其中k为类型；2. 四个证书（每个是模p的短加法列表）解决了Chassaniol留下的两个未决图C₃₁(2,4,8,15)和C₄₁(4,10,16,18)，并在无需机器辅助的情况下完成了类型不超过10的分类；3. 精确计算将二分法“量子对称性当且仅当图是完全图或空图”扩展到所有素数阶p≤250，解决了p≤241的Paley图Pₚ，这是首个超出P₁₇的结果。剩余工作是捕获单个显式向量：两个基点的中点2⁻¹。

英文摘要

Which vertex-transitive graphs of prime order have quantum symmetry? The question of Banica, Bichon and Chenevier is open in the dense regime of Paley graphs, where coherent-algebra methods give no information. To each such graph we attach a two-basepoint Terwilliger algebra of its cyclotomic scheme and study the module it generates from the basepoints: fullness forces the quantum permutation algebra to be commutative, and the module admits no intermediate state, containing either exactly two point masses or all $p$ of them. One point mass, captured at any depth, therefore suffices, and Chassaniol's orbital criterion is the depth-one case. Three consequences follow. A sharp counting argument replaces the Banica--Bichon--Chenevier threshold $p>6^{φ(k)}$ by the quadratic bound $p>(k-1)(k-2)+2$, where $k$ is the type. Four certificates, each a short list of additions modulo $p$, settle $C_{31}(2,4,8,15)$ and $C_{41}(4,10,16,18)$, the two graphs left open by Chassaniol, and complete the classification for type at most $10$ without machine assistance. An exact computation extends the dichotomy ``quantum symmetry if and only if complete or empty'' to all prime orders $p\le250$, settling the Paley graphs $P_{p}$ with $p\le241$, the first beyond $P_{17}$. What remains is the capture of a single explicit vector: the midpoint $2^{-1}$ of the two basepoints.

发表机构

• Mutah University（穆塔大学）

机构由 AI 辅助整理，请以论文原文为准。