S-Characters of Dihedral Groups of Order 2p: A Classification via Circulant Gram Matrices and Root Systems

Let p be an odd prime and let D2p be the dihedral group of order 2p. We give an explicit description of the ordinary and non-ordinary S-characters of D2p. For p≥5, the number of ordinary S-characters is (p+9)/2. For p≥5, the total number is p+5, except for p=7, where three additional configurations associated with the E7 root lattice give a total of 15. The proof converts nonnegativity on rotations into positive semidefiniteness of an integral circulant Gram matrix. The resulting norm-2 vectors are then classified by means of finite simply laced root systems. We also determine all zero sets and provide exact certificates for the exceptional p=7 calculation.