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Given $k \geq 1$, what is the minimal $n$ (if it exists) such that there're matrices $A_1, \cdots A_k \in M_n(\mathbb{C}) $ satisfying $\forall i,j \quad A_i A_j = - A_j A_i$ and $A_1 \times A_2 \cdots \times A_k \neq 0$ ? In particular each $A_i ^2$ must be zero.

Motivation : make Ramyslov's elementary proof of the Amitsur–Levitzki identity look even more elementary.

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Such matrices define a representation of the exterior algebra $\Lambda ^ *(V)$ where $V$ is a vector space of dimension $k$. Your non-vanishing condition holds exactly when the representation is faithful, because $\Lambda^*(V)$ has a unique minimal nonzero ideal equal to $\Lambda^k (V)$.

So a free module of rank one is the smallest faithful representation, with dimension $2^k$.

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    $\begingroup$ The motivating objective of the question was already obtained by Shmuel Rosset in "A new proof of the Amitsur-Levitski identity" in 1976. $\endgroup$ – Jan Jitse Venselaar Jun 27 '12 at 10:37

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