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

In this paper, we describe a new infinite family of $$\frac{q^{2}-1}{2}$$q2-12-tight sets in the hyperbolic quadrics $${\mathcal {Q}}^{+}(5,q)$$Q+(5,q), for $$q \equiv 5 \text{ or } 9 \,\hbox {mod}\,{12}$$q≡5or9mod12. Under the Klein correspondence, these correspond to Cameron–Liebler line classes of $$\mathop {\mathrm{PG}}(3,q)$$PG(3,q) having parameter $$\frac{q^{2}-1}{2}$$q2-12. 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 $$\frac{q^{2}+1}{2}$$q2+12 in $$\mathop {\mathrm{PG}}(3,q)$$PG(3,q) for all odd $$q$$q. The study of Cameron–Liebler line classes is closely related to the study of symmetric tactical decompositions of $$\mathop {\mathrm{PG}}(3,q)$$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 $$q \equiv 9 \,\hbox {mod}\,12$$q≡9mod12 (so $$q = 3^{2e}$$q=32e for some positive integer $$e$$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 $$\left( \frac{1}{2}(3^{2e}-3^{e}), \frac{1}{2}(3^{2e}+3^{e}) \right) $$12(32e-3e),12(32e+3e) in the affine plane $$\mathop {\mathrm{AG}}(2,3^{2e})$$AG(2,32e) for all positive integers $$e$$e. This proves a conjecture made by Rodgers in his Ph.D. thesis.