Canonical Barsotti-Tate Groups of Finite Level

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $c,d\in \mathbb{N}$ be such that $h=c+d>0$. Let $H$ be a $p$-divisible group of codimension $c$ and dimension $d$ over $k$. For $m\in\mathbb{N}^\ast$ let $H[p^m]=\ker([p^m]:H\rightarrow H)$. It is a finite commutative group scheme over $k$ of $p$ power order, called a Barsotti-Tate group of level $m$. We study a particular type of $p$-divisible groups $H_\pi$, where $\pi$ is a permutation on the set $\{1,2,\dots,h\}$. Let $(M,\varphi_\pi)$ be the Dieudonn\'e module of $H_\pi$. Each $H_\pi$ is uniquely determined by $H_\pi[p]$ and by the fact that there exists a maximal torus $T$ of $GL_M$ whose Lie algebra is normalized by $\varphi_\pi$ in a natural way. Moreover, if $H$ is a $p$-divisible group of codimension $c$ and dimension $d$ over $k$, then $H[p]\cong H_\pi[p]$ for some permutation $\pi$. We call these $H_\pi$ canonical lifts of Barsotti-Tate groups of level $1$. We obtain new formulas of combinatorial nature for the dimension of $\boldsymbol{Aut}(H_\pi[p^m])$ and for the number of connected components of $\boldsymbol{End}(H_\pi[p^m])$.