# What is the name of this tensor?

A matrix M is usually called a hollow matrix if all of its diagonal elements are zero:

$$M_{pp} = 0, \quad \forall \: p.$$

We can generalize this to an $$n$$-way tensor T, such that:

$$T_{p_1 \cdots p_n} = 0, \quad \mbox{if p_i = p_j for some i \ne j}.$$

I call these types of tensors hollow. Are there other names for these?

Edit: made the question more precise.

• Do you mean an element is zero if all indices $p_i$ are equal, or if at least two of them are equal? May 13, 2021 at 18:12
• At least two of them are equal. May 14, 2021 at 19:07
• a special case, if it is also antisymmetric under exchange of any two indices, is called a completely antisymmetric tensor. May 14, 2021 at 20:27
• This hollow condition is not invariant under conjugation by a linear transformation, so it doesn't seem very natural to me; it is basis dependent. It would be nice if you could include a sentence explaining where it might naturally arise, or why it is important. May 15, 2021 at 9:47
• 1. It may occur in many-fermion wavefunctions due to the Pauli exclusion principle. Usually, it is also antisymmetric as @CarloBeenakker pointed out. May 16, 2021 at 11:06