Let $R=\Lambda_{\mathbb{F}_p}[e_1,...,e_n]$ be the exterior algebra on $n$ generators over the finite field with $p$ elements (this arises naturally as the mod-p cohomology ring of the $n$-dimensional torus). This is a graded-commutative algebra over $\mathbb{F}_p$ with $|e_i|=1$. Let $R_m\subseteq R$ denote the subspace of homogeneous degree $m$ elements (so in particular $R=\bigoplus_{m=0}^{n}R_m$).
For some reason, given $m,r \in \mathbb{N}$, I need to calculate the numbers
$$ \#\{\alpha\in R_m \mid \alpha^r=0 \} $$ i.e. how many degree $m$ elements $\alpha$ vanish when raised to the $r$-th power. More precisely, I am interested in the behavior of this sequence as a function of $n$ (for fixed $m$ and $r$). The question is
Is there anything known about this counting problem? Is it related to something known?
I would be happy to know even some very general facts about the behavior of this sequence.
What I have so far is this:
Trivial Observations:
- When $p=2$, we have $\alpha^2=0$ for all $\alpha$.
- For $m$ odd, we have $\alpha^2=0$ for all $\alpha$ (we get $\alpha \wedge\alpha=-\alpha \wedge\alpha$ and for $p\ne 2$, this implies $\alpha \wedge\alpha=0$).
- For $m$ even, $\alpha^p=0$ for all $\alpha$ (binomial theorem).
It follows that the interesting cases are when $p\ne 2$, $m$ even and $1<r<p$.
For $m=2$
When $m=2$ (and $p\ne2$) we can identify the elements $\alpha$ of $R_2$ with skew-symmetric $n\times n$ matrices $A$ over $\mathbb{F}_p$. Using the canonical form for such matrices it is easy to see that the condition $\alpha^r=0$ is equivalent to $rank(A) < 2r$. The formula for the number of skew-symmetric matrices of rank exactly $k$ over $\mathbb{F}_p$ is
$$ p^{k\left(k-1\right)}\left(1-p\right)^{r}\frac{\left[n\right]_{p}!}{\left[n-2k\right]_{p}!\left[2k\right]_{p}!!} $$
This uses the "q-factorial" notation and can be found for example in this paper (by solving some recursion formulas on the rank/dimension).