A new family of tight sets in ${\mathcal {Q}}^{+}(5,q)$

Abstract

In this paper, we describe a new infinite family of $\frac{q^{2}-1}{2}$-tight sets in the hyperbolic quadrics ${\mathcal {Q}}^{+}(5,q)$, for $q \equiv 5 \text{ or } 9 \,\hbox {mod}\,{12}$. Under the Klein correspondence, these correspond to Cameron–Liebler line classes of $\mathop {\mathrm{PG}}(3,q)$ having parameter $\frac{q^{2}-1}{2}$. This is the second known infinite family of nontrivial Cameron–Liebler line classes, the first family having been described by Bruen and Drudge with parameter $\frac{q^{2}+1}{2}$ in $\mathop {\mathrm{PG}}(3,q)$ for all odd $q$. The study of Cameron–Liebler line classes is closely related to the study of symmetric tactical decompositions of $\mathop {\mathrm{PG}}(3,q)$ (those having the same number of point classes as line classes). We show that our new examples occur as line classes in such a tactical decomposition when $q \equiv 9 \,\hbox {mod}\,12$ (so $q = 3^{2e}$ for some positive integer $e$), providing an infinite family of counterexamples to a conjecture made by Cameron and Liebler (in Linear Algebra Appl 46, 91–102, 1982); the nature of these decompositions allows us to also prove the existence of a set of type $\left( \frac{1}{2}(3^{2e}-3^{e}), \frac{1}{2}(3^{2e}+3^{e}) \right)$ in the affine plane $\mathop {\mathrm{AG}}(2,3^{2e})$ for all positive integers $e$. This proves a conjecture made by Rodgers in his Ph.D. thesis.