Maximum dimension of a subspace consisting only of non-degenerate alternating bilinear forms over finite fields.

Question

Let $V$ be a vector space over a finite field $\mathbb{F}_q$. The following result seems well-known in the literature of bilinear forms.

The maximum dimension of a subspace consisting only of non-degenerate alternating bilinear forms (except the zero form of course) on $V$ is $\text{dim}(V)/2$. Here, the above vector sub-space is contained in the vector space of all alternating bilinear forms on $V$.

In matrix language, it is the subspace consisting only of $n\times n$ invertible skew-symmetric matrices (except the zero matrix) over $\mathbb{F}_q$ of the vector space of all skew-symmetric matrices in $M_n(\mathbb{F}_q)$. It is clear that $n$ has to be even.

Can anyone suggest a proper reference where I can find this result. I have seen this result to be mentioned in many papers concerning alternating bilinear forms over finite fields, but no proper reference has been cited. I appreciate any kind of help.

Answer 1

This is a simple application of Chevalley's theorem.

Theorem (Chevalley, special case) If $\mathbb{F}$ is a finite field and $f\in\mathbb{F}[x_1,\dots,x_n]$ has no constant term, then $f=0$ has a nontrivial solution provided $n>\deg f$.

Also recall definition of Pfaffian of a skew-symmetric matrix:

Definition Let $A=(a_{ij})$ is a $2d\times 2d$ skew-symmetric matrix (over any ring). There exists a homogeneous degree-$d$ polynomial $$ \operatorname{Pf}(A)=\frac1{2^d d!}\sum_{\sigma\in S_{2d}}\operatorname{sgn}(\sigma)\prod_{i=1}^d a_{\sigma(2i-1)\sigma(2i)}\in\mathbb{Z}[\{a_{ij}:i>j\}], $$ called the Pfaffian of $A$, such that $\det(A)=(\operatorname{Pf}(A))^2$.

So suppose we have $n$ linearly independent $2d\times 2d$ skew-symmetric matrices $A_1,\dots,A_n$ such that all nontrivial linear combinations are nondegenerate. That means for all $(x_1,\dots,x_n)\in\mathbb{F}^n-\{0\}$, we have $\operatorname{Pf}(\sum_{i=1}^n x_iA_i)\neq 0$. By Chevalley's theorem, this can never happen if $n>d$ as we are working over finite field $\mathbb{F}=\mathbb{F}_q$.