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

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.