Bilinear forms on finite abelian groups and group-invariant Butson Hadamard matrices

Abstract

Let K be a finite abelian group and let exp⁡(K) denote the least common multiple of the orders of the elements of K. A BH(K,h) matrix is a K-invariant |K|×|K| matrix H whose entries are complex hth roots of unity such that HH⁎=|K|I, where H⁎ denotes the complex conjugate transpose of H, and I is the identity matrix of order |K|. Let νp(x) denote the p-adic valuation of the integer x. Using bilinear forms on K, we show that a BH(K,h) exists whenever(i)νp(h)≥⌈νp(exp⁡(K))/2⌉ for every prime divisor p of |K| and(ii)ν2(h)≥2 if ν2(|K|) is odd and K has a direct factor Z2. Employing the field descent method, we prove that these conditions are necessary for the existence of a BH(K,h) matrix in the case where K is cyclic of prime power order.