Is there good examples of collection of nilpotent operators that commute with themselves?
Is there a good reference for a collection commutative nilpotent operators that commute with themselves or commutative zero divisors?
Specifically for large enough $m\in\Bbb N$ I am looking for $t=\alpha m$ for some fixed $\alpha>0$ matrices $A_1,\dots,A_t$ that are commutative ( $\forall i,j\in\{1,\dots,t\}$ $A_iA_j=A_jA_i$ holds) and satisfy one of following two:
(1) For any $i\in\{1,\dots,t\}$ $A_i^2=0$. What is smallest size of matrices (can they be $\beta m\times\beta m$ for some fixed $\beta>0$)?
(2) $\forall i_1,i_2,i_3,\dots\in\{1,\dots,t\}$ we have $Tr(A_{i_1}A_{i_2}A_{i_3}\dots)\neq0\iff {i_1}\neq {i_2}\wedge {i_2}\neq {i_3}\wedge {i_3}\neq {i_1}\dots$ holds. Can matrices also be of size $\beta m\times\beta m$ for some fixed $\beta>0$?
Here is a theorem that is relevant to your question: A Lie sub algebra of $End(V)$ (V finite dimensional) consisting of nilpotent operators is simultaneously strictly upper triangularizable. I mean that you can pick a basis so that they all are zero on and below the diagonal. (This is Engels theorem. A nice reference is Serre's Lie algebras and Lie groups.)
(The condition on commutativity means that these form a Lie sub algebra.)
I'm still a little confused about what specifically you are asking, but you can probably play around with this a bit to produce a result.
Concrete translation of a specific case:
Let $A_i$ be a collection of commuting, nilpotent operators on a finite dimensional vector space $V$. Then there is a basis $v_i$ for $V$ in which each $A_i$ is represented by a strictly upper triangular matrix.