The ring of character values depends only on the segment: SU(n), Sp(2m) and GL(n) at a torsion element

Let gq = exp(2πiρ/q) be the principal element attached to q in a compact simple group, and let A be the ring generated by all character values at gq, an order in the integers O of its field. A division lemma — multiplying a monic polynomial by a monic integer factor does not change the ring generated by its coefficients — gives three operations on the segment of exponents that leave A unchanged: adjoining an eigenvalue 1, which makes the ring of SU(2m+1) equal to that of Sp(2m); the complement n ↦ q−n, level-rank duality as an equality of orders; and the period n ↦ n+q. So A depends only on the segment. At p, the index of a stable segment is the sum of its layer defects, and the relative class number governs the first layer. A prime p divides [O:A] only if the prime-to-p part r of q divides n−1, n or n+1, and inside these families it does whenever p is an odd prime dividing q and φ(r) ≥ 4. For the non-real ring of GL(n), with n = Nr or Nr+1, 2 ≤ n < q, r ≥ 5 odd and p odd, the order is locally a free quadratic extension of the ring of the recentred segment.