Typical ranks for m × n × (m - 1) n tensors with m ≤ n

In various application fields, tensor type data are used recently and then a typical rank is important. There may be more than one typical ranks over the real number field. It is well known that the set of 2×n×n tensors has two typical ranks n,n+1 for n≥2, that the set of 3×4×8 tensors has two typical ranks 8,9, and that the set of 4×4×12 tensors has two typical ranks 12,13. In this paper, we show that the set of m×n×(m-1)n tensors with m≤n has two typical ranks (m-1)n,(m-1)n+1 if m≤ρ(n), where ρ is the Hurwitz-Radon function defined as ρ(n)=2^b+8c for nonnegative integers a,b,c such that n=(2a+1)2^b+4c and 0≤b<4.