Canonical Barsotti-Tate Groups of Finite Level

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $c,din mathbb{N}$ be such that $h=c+d>0$. Let $H$ be a $p$-divisible group of codimension $c$ and dimension $d$ over $k$. For $minmathbb{N}^ast$ let $H[p^m]=ker([p^m]:Hrightarrow H)$. 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 Dieudonne 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 $boldsymbol{End}(H_pi[p^m])$.