Universal karyon tilings

V. G. Zhuravlev

Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 3, Tome 490 (2020), pp. 49-93

Résumé

Universal karyon tilings $\mathcal{T}^{d}(v,\mu)$ of the real $d$-dimensional space $\mathbb{R}^{d}$ are constructed. These tilings depend on two free parameters: the star $v=\{ v_0, \ldots, v_d \}$ formed by $d + 1$ vectors $v_0, \ldots, v_d\in\mathbb{R}^{d}$, and the weight vector $\mu=( \mu_0,\mu_1, \ldots,\mu_d)\in\mathbb{R}^{d+1}$ with $\mu_k>0$ satisfying $\mu_0+\mu_1+ \ldots + \mu_d=1$. The tiling $\mathcal{T}^{d}(v,\mu)$ contains the karyon $\mathrm{Kr}=T_{0}\cup T_{1}\cup \ldots \cup T_{d} \subset\mathcal{T}(v,\mu) $ consisting of all types of parallelepipeds $T_{0},T_{1},\ldots,T_{d}$ from which the tiling $\mathcal{T}^{d}(v,\mu)$ is formed. The karyon $\mathrm{Kr}$ is a convex parallelohedron uniquely determined by the star $v$. Coordinates $\mu_k$ of the weight vector $\mu$ set the frequency of occurrence of parallelepipeds $T_{k} \in \mathrm {Kr}$ in the karyon tiling $\mathcal{T}^{d}(v,\mu)$.