Standard Jordan partitions with three parameters

Abstract

We extend previous work on standard two-parameter Jordan partitions by Barry (Commun Algebra 43:4231–4246, 2015) to three parameters. Let $J_r$ denote an $r \times r$ matrix with minimal polynomial $(t-1)^r$ over a field F of characteristic p. For positive integers $n_1$, $n_2$, and $n_3$ satisfying $n_1 \le n_2 \le n_3$, the Jordan canonical form of the $n_1 n_2 n_3 \times n_1 n_2 n_3$ matrix $J_{n_1} \otimes J_{n_2} \otimes J_{n_3}$ has the form $J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _m}$ where $\lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _m>0$ and $\sum _{i=1}^m \lambda _i=n_1 n_2 n_3$. The partition $\lambda (n_1,n_2,n_3:p)=(\lambda _1, \lambda _2,\ldots , \lambda _m)$ of $n_1 n_2 n_3$, which depends on $n_1$, $n_2$, $n_3$, and p, will be called a Jordan partition. We will define what we mean by a standard Jordan partition and give necessary and sufficient conditions for its existence.