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 Jr denote an r×r matrix with minimal polynomial (t−1)r over a field F of characteristic p. For positive integers n1, n2, and n3 satisfying n1 ≤ n2 ≤ n3, the Jordan canonical form of the n1n2n3 × n1n2n3 matrix Jn1 ⊗ Jn2 ⊗ Jn3 has the form Jλ1 ⊕ Jλ2 ⊕···⊕ Jλm where λ1 ≥ λ2 ≥ ··· ≥ λm > 0 and m i=1 λi = n1n2n3. The partition λ(n1, n2, n3 : p)=(λ1, λ2,...,λm) of n1n2n3, which depends on n1, n2, n3, 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.