Real mutually unbiased bases and representations of groups of odd order by real scaled Hadamard matrices of 2-power size

Abstract: We prove the following two results relating real mutually unbiased bases and representations of finite groups of odd order. Let $q$ be a power of 2 and $r$ a positive integer. Then we can find a $q^{2r}\times q^{2r}$ real orthogonal matrix $D$, say, of multiplicative order $q^{2r-1}+1$, whose $q^{2r-1}+1$ powers $D$, \dots, $D^{q^{2r-1}+1}=I$ define $q^{2r-1}+1$ mutually unbiased bases in $\mathbb{R}^{q^{2r}}$. Thus the scaled matrices $q^rD$, \dots, $q^rD^{q^{2r-1}}$ are $q^{2r-1}$ different Hadamard matrices. When we take $q=2$, we achieve the maximum number of real mutually unbiased bases in dimension $2^{2r}$ using the elements of a cyclic group. We also prove the following. Let $G$ be an arbitrary finite group of odd order $2k+1$, where $k\geq 3$. Then $G$ has a real representation $R$, say, of degree $2^{2^{k-1}}$ such that the elements $R(\sigma)$, $\sigma\in G$, define $|G|$ mutually unbiased bases in $\mathbb{R}^{d}$, where $d= 2^{2^{k-1}}$. In addition, a group of order 5 defines five real mutually unbiased bases in $\mathbb{R}^{16}$ and a group of order 3 defines three real mutually unbiased bases in $\mathbb{R}^{4}$. Thus, an arbitrary group of odd order has a faithful representation by real scaled Hadamard matrices of 2-power size.