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 $|K|\times |K|$ matrix $H$ whose entries are complex $h$th roots of unity such that $H{H}^{\ast }=|K|I$, where $H^{\ast }$ denotes the complex conjugate transpose of $H$, and $I$ is the identity matrix of order $|K|$. Let ${\nu }_p(x)$ denote the $p$-adic valuation of the integer $x$. Using bilinear forms on $K$, we show that a $BH(K,h)$ exists whenever (i) ${\nu }_p(h)\ge \lceil {\nu }_p(\exp (K))/2\rceil$ for every prime divisor $p$ of $|K|$ and (ii) ${\nu }_2(h)\ge 2$ if ${\nu }_2(|K|)$ is odd and $K$ has a direct factor ${\mathbb{Z}}_2$. 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.