Two-basepoint Terwilliger algebras and the quantum symmetry of prime-order circulants
Abstract
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>6φ(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 C31(2,4,8,15) and C41(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≤250, settling the Paley graphs Pp with p≤241, the first beyond P17. What remains is the capture of a single explicit vector: the midpoint 2−1 of the two basepoints.
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BibTeX
@misc{marashdeh2026twobasepoint,
title = {{Two-basepoint Terwilliger algebras and the quantum symmetry of prime-order circulants}},
author = {Mohammad F. Marashdeh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00462},
eprint = {2609.00462},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00462},
}
RIS
TY - UNPB
TI - Two-basepoint Terwilliger algebras and the quantum symmetry of prime-order circulants
AU - Mohammad F. Marashdeh
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00462
UR - https://arxiv.org/abs/2609.00462
AB - 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≤250, settling the Paley graphs P_p with p≤241, the first beyond P_17. What remains is the capture of a single explicit vector: the midpoint 2^−1 of the two basepoints.
ER -