Let $R = \Lambda \mathbb{C}^n$ be the exterior algebra on $\mathbb{C}^n$ for some positive integer $n$. It is an associative (graded-commutative) algebra of $\mathbb{C}$-dimension $2^n$.


Suppose we have an injective ring homomorphism $R\rightarrow M\_N(\mathbb{C})$ into the ring $M\_N(\mathbb{C})$ of $N\times N$ matrices over $\mathbb{C}$, for some positive integer $N$. (I.e., a faithful matrix representation of $R$.)


**What's the smallest $N$ can be, and what's the explicit ring map? In particular, can $N$ be smaller than $2^n$?**


I know that $N$ can be taken to be $\leq 2^n$, because we can embed $R$ in $M\_{2^n}(\mathbb{C})$ by considering its left action on itself (i.e., its regular representation). Explicitly we get the map $R\rightarrow M\_{2^n}(\mathbb{C})$ by choosing a $\mathbb{C}$-basis for $R$ and then sending an element of $R$ to the matrix that represents its left action on that basis.


Meanwhile, I also know $N\geq 2^{n/2}$, because the $\mathbb{C}$-dimension of $M\_N(\mathbb{C})$ is $N^2$, so injectivity of $R\rightarrow M\_N(\mathbb{C})$ implies immediately that $2^n\leq N^2$.


So $N$ has to be exponential in $n$. But specifically, are there matrix representations smaller than the regular representation? If so, what are they? If not, how can we see this?


(I have verified by hand in the simplest nontrivial case $n=2$ that there's no faithful matrix representation smaller than the regular representation.)


**Aside:** A brief comment on motivation. The question is not motivated by my own research. A colleague asked me it a few months ago. The reason he asked has been rendered moot by subsequent developments in his work, so afaik it's not motivated by his research anymore either. But the question got stuck in my mind because it feels like a very fundamental representation-theoretic question, but I didn't know the answer and I wasn't sure where to look it up, and I asked a few folks who didn't know either.



Smallest faithful matrix representation of the exterior algebra
Let $M$ be a faithful module for $R$, and let $a\in R$ be the product of the exterior generators, so that $a$ generates the socle of $R$. Then there an element $m\in M$ such that $am\ne 0$. The map $r\mapsto rm$ is then an injective homomorphism from the regular representation of $R$ to $M$, so the dimension of $M$ is at least $2^n$.


