MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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 $2^k$.

share|cite|improve this answer
The motivating objective of the question was already obtained by Shmuel Rosset in "A new proof of the Amitsur-Levitski identity" in 1976. – Jan Jitse Venselaar Jun 27 '12 at 10:37

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.