The order generated by a column of the SU(n)_k S-matrix is locally a numerical semigroup ring (Part I)

Fix a column μ of the modular S-matrix of SU(n)k and let Aμ be the ring generated by the ratios Sλμ/S0μ: the character values of SU(n) at the element of finite order that labels μ. Its field is known; we determine the ring when the height p = n+k is prime. Aμ is the tensor product of the integers of a cyclotomic field Q(ζm), fixed by the charge of μ under the centre, with an order CT that depends only on the reduced spectrum T ⊂ Fp of the element. Locally at p, CT is a monomial ring: its valuations form a numerical semigroup ST, written down exactly by Gauss sums and Stickelberger's theorem. Every property of the order at p is then a property of ST. The index [Oμ : Aμ] is pφ(m)δ with δ the number of gaps of ST; Aμ is Gorenstein if and only if ST is symmetric; p Oμ ⊆ Aμ when the conductor is at most the ramification index; level–rank duality replaces T by its negated complement and keeps ST; the vacuum column has ST = N0 and is maximal. When p > (n−1)(n−2), and for every p when n ≤ 12, ST is read in Fp: it is generated by the ramification index e = (p−1)/h and the j < e with Mhj(T) ≠ 0, where Mr(T) = ∑t∈T tr and h is the order of the multiplicative stabiliser of T. At prime height, the first non-Gorenstein column, by n and then by level, is at SU(5)12.