Cameron–Liebler k-sets in subspaces and non-existence conditions

Abstract

In this article we generalize the concepts that were used in the PhD thesis of Drudge to classify Cameron–Liebler line classes in PG$(n,q), n\ge 3$, to Cameron–Liebler sets of k-spaces in ${{\,\mathrm{\mathrm {PG}}\,}}(n,q)$ and ${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$. In his PhD thesis, Drudge proved that every Cameron–Liebler line class in ${{\,\mathrm{\mathrm {PG}}\,}}(n,q)$ intersects every 3-dimensional subspace in a Cameron–Liebler line class in that subspace. We are using the generalization of this result for sets of k-spaces in ${{\,\mathrm{\mathrm {PG}}\,}}(n,q)$ and ${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$. Together with a basic counting argument this gives a very strong non-existence condition, $n\ge 3k+3$. This condition can also be improved for k-sets in ${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$, with $n\ge 2k+2$.

Appendix

This section gives an alternative proof of Theorem 6.5 for $k=1$. This is in fact interesting because this proof is based on similar arguments as the proof of Theorem 6.2. It is in our view nice to see that we can obtain this result by arguments that are not based on exploiting the connection between Cameron–Liebler k-sets in ${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$ and ${{\,\mathrm{\mathrm {PG}}\,}}(n,q)$.

Keep in mind that this proof only works for $k=1$, but if there is an equivalent of Theorem 2.15 for k-spaces, we could probably use the same technique as in Theorem 6.2. Our guess is that by using this technique the result stays the same as Theorem 6.5. yet it would be interesting to see these arguments, and we could of course be wrong.

1.1 For Cameron–Liebler line classes in ${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$

Theorem 8.1

[5, Theorems 6.5 and 6.8] Suppose that ${\mathcal {L}}$ is a Cameron–Liebler line class of parameter x in ${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$, for $n\ge 3$, then the following statements are true.

• If $x=1$, then ${\mathcal {L}}$ is a point-pencil.

• The parameter x cannot be 2.

Here we state a stronger result for Cameron–Liebler line classes in ${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$.

Theorem 8.2

Suppose that ${\mathcal {L}}$ is a Cameron–Liebler line class of parameter x in ${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$, with $n\ge 4$, then $x\in \{0, 1\}$ or $x\ge 2\left( \frac{q^{n-1}-1}{q^{2}-1}\right) +1.$

Proof

Suppose that ${\mathcal {L}}$ is a Cameron–Liebler line class in ${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$, which is not empty nor a point-pencil. Hence, by Theorem 8.1, $x>2$. Choose a line $\ell \in {\mathcal {L}}$. Then, by Lemma 4.2, for $3\le t \le n-1$,

$$\begin{aligned} x=\frac{q^{t-2}-1}{q^{n-2}-1}\frac{\sum _{\ell \in \pi _i} x_{\pi _i}}{\genfrac[]{0.0pt}{}{n-3}{t-3}_q}-\frac{q^{n-1}-1}{q^{t-1}-1}+1. \end{aligned}$$

(11)

We now have the following facts:

• 1. Since $\ell \in {\mathcal {L}}$, every Cameron–Liebler line class in every t-dimensional subspace $\pi _i$ has parameter $x_{\pi _i}\ge 1$.

• 2. If there exists a t-space where ${\mathcal {L}}\cap [\pi _i]_1$ has parameter $x_{\pi _i}=1$, then, by Theorem 8.1, it is a point-pencil. Hence, by Theorem 3.7, it follows that ${\mathcal {L}}$ is a point-pencil. So we may suppose that $x_{\pi _i}>1$.

• 3. Using Theorem 8.1, we know that $x_{\pi _i}>2$.

So, we conclude that for every t-space $\pi _i$ through $l\in {\mathcal {L}}$, it holds that $x_{\pi _i}\ge 3$. So, using Eq. (11), we obtain that

$$\begin{aligned} \begin{aligned} x&=\frac{q^{t-2}-1}{q^{n-2}-1}\frac{\sum _{\ell \in \pi _i} x_{\pi _i}}{\genfrac[]{0.0pt}{}{n-3}{t-3}_q}-\frac{q^{n-1}-1}{q^{t-1}-1}+1 \\&\ge \frac{q^{t-2}-1}{q^{n-2}-1}\frac{3\genfrac[]{0.0pt}{}{n-1}{t-1}_q}{\genfrac[]{0.0pt}{}{n-3}{t-3}_q}-\frac{q^{n-1}-1}{q^{t-1}-1}+1 \\&\ge 3 \left( \frac{q^{n-1}-1}{q^{t-1}-1} \right) -\frac{q^{n-1}-1}{q^{t-1}-1}+1\\&\ge 2\left( \frac{q^{n-1}-1}{q^{t-1}-1}\right) +1\\&\ge 2\left( \frac{q^{n-1}-1}{q^{2}-1}\right) +1, \end{aligned} \end{aligned}$$

where in the last line, we have chosen $t=3$. $\square$