Symmetries of the universal karyon tilings

Universal karyon tilings $\mathcal{T}(v,\mu, \rho)$ are generated by the parallelepipeds $T_{0}, T_{1}, \ldots, T_{d}$ dividing the real space $\mathbb{R}^{d}$. The tilings $\mathcal{T}(v,\mu, \rho)$ are parameterized by triples $(v, \mu, \rho)$ running through the infinite cylinder $\triangle \times \triangle \times \mathbb{R}$ with the base $\triangle \times \triangle$ that is the direct product of two simplices $\triangle$ of dimension $d$. The parameter $v$ defines the geometry of the parallelepipeds $T_{k}$ and the two others $\mu, \rho$ define the symmetry of the karyon tiling \break $\mathcal{T}(v,\mu, \rho)$. We consider the usual and generalized symmetries of tilings $\mathcal{T}(v,\mu, 0)$. The generalized symmetries are quasi-symmetries that map the tilings $\mathcal{T}(v,\mu, 0)$ to their dual tilings $\mathcal{T}^{*}(v,\mu, 0)$.