Cameron-Liebler Line Classes and Two-Intersection Sets 1 Qing Xiang - PowerPoint PPT Presentation

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets Cameron-Liebler Line Classes and Two-Intersection Sets 1 Qing Xiang Department of Mathematical Sciences University of Delaware Newark, DE 19716 USA xiang@math.udel.edu Joint work with Tao Feng and Koji Momihara 1 T. Feng, K. Momihara, Q. Xiang, Cameron-Liebler line classes with parameters x = q 2 − 1 , J. Combin. Theory (A), 133 (2015), 307–338 . . . . . . 2

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets . .. Finite Classical Polar Spaces 1 . .. m -ovoids and x -tight sets 2 . .. Cameron-Liebler line classes 3 . .. Affine two-intersection sets 4 . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets Finite Classical Polar Spaces . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets Definitions Let V ( n + 1 , q ) be an ( n + 1) -dimensional vector space over F q , and let f be a non-degenerate sesquilinear or non-singular quadratic form defined on V ( n + 1 , q ) . A finite classical polar space associated with the form f is the geometry consisting of subspaces of PG( n, q ) induced by the totally isotropic subspaces with relation to f . . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets A polar space S contains the totally isotropic points, lines, planes, etc. of the ambient projective space. The generators of S are the (t.i.) subspaces of maximal dimension. The rank of S is the vector dimension of its generators For a point P , the set P ⊥ of points of S collinear with P is the intersection of the tangent hyperplane at P with S . . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets Three types of finite classical polar spaces: Orthogonal polar spaces: quadrics; symplectic polar spaces; Hermitian polar spaces. Polar Space rank form x 2 Q (2 n, q ) n 0 + x 1 x 2 + · · · + x 2 n − 1 x 2 n Q + (2 n + 1 , q ) n + 1 x 0 x 1 + x 2 x 3 + · · · + x 2 n x 2 n +1 Q − (2 n + 1 , q ) n f ( x 0 , x 1 ) + x 2 x 3 + · · · + x 2 n x 2 n +1 W (2 n + 1 , q ) n + 1 x 0 y 1 + y 0 x 1 + · · · + x 2 n y 2 n +1 + x 2 n +1 y 2 n x q +1 + · · · + x q +1 H (2 n, q 2 ) n 0 2 n x q +1 + · · · + x q +1 H (2 n + 1 , q 2 ) n + 1 0 2 n +1 . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets m -ovoids and x -tight sets . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets Let S be a finite classical polar space of rank r over the finite field F q . Denote by θ n ( q ) := q n − 1 q − 1 the number of points in PG( n − 1 , q ) . . Definition . . . An m -ovoid is a set O of points such that every generator of S meets O in exactly m points. . . . . . . Definition . . . An x -tight set is a set M of points such that { xθ r − 1 ( q ) + q r − 1 , if P ∈ M , | P ⊥ ∩ M| = xθ r − 1 ( q ) , otherwise . . . . . . . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets . Example . . . A spread of PG(3 , q ) is mapped, under the Klein correspondence, to an ovoid of the Klein quadric Q + (5 , q ) . (A spread of PG(3 , q ) is a set of q 2 + 1 lines partitioning the set of points of PG(3 , q ) .) . . . . . Ovoids of polar spaces are rare: they only exist in low rank polar spaces, such as Q (4 , q ) , Q (6 , q ) , Q + (5 , q ) , Q + (7 , q ) . . Example . . . Let S be a polar space of rank r . Then any generator M is a 1-tight set of S since |M| = q r − 1 { q − 1 = θ r − 1 ( q ) + q r − 1 , if P ∈ M , | P ⊥ ∩ M| = θ r − 1 ( q ) , otherwise . . . . . . . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets Cameron-Liebler Line Classes . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets Background Cameron-Liebler line classes were first introduced by Cameron and Liebler 2 in their study of collineation groups of PG( n, q ) , n ≥ 3 , having the same number of orbits on points as on lines. Cameron and Liebler reduced the problem to the case where n = 3 . 2P.J. Cameron, R.A. Liebler, Tactical decompositions and orbits of projective groups, Linear Algebra Appl. , 46 (1982), 91–102. . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets A collineation group of PG(3 , q ) having equally many orbits on points and lines induces a symmetric tactical decomposition on the point-line design from PG(3 , q ) , and any line class of such a tactical decomposition is a Cameron-Liebler line class. . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets A Characterization . Definition . . . Let L be a set of lines of PG(3 , q ) with |L| = x ( q 2 + q + 1) , x a nonnegative integer. We say that L is a Cameron-Liebler line class with parameter x if every spread of PG(3 , q ) contains x lines of L . . . . . . . . . 1 The complement of L in the set of all lines of PG(3 , q ) is a Cameron-Liebler line class with parameter q 2 + 1 − x . WLOG we may assume that x ≤ q 2 +1 2 . . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets “Trivial” examples Let ( P, π ) be any non-incident point-plane pair of PG(3 , q ) . . . 1 star ( P ) : the set of all lines through P , . . . . 2 line ( π ) : the set of all lines contained in the plane π . . Example . . . The following are examples of Cameron-Liebler line classes: . . 1 x = 0 : ∅ ; . . . 2 x = 1 : star ( P ) , line ( π ) ; . . . . 3 x = 2 : star ( P ) ∪ line ( π ) . . . . . . It was conjectured by Cameron and Liebler that up to taking complement these are all the examples of Cameron-Liebler line classes. . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets More Examples . 1 The first counterexample was given by Drudge 3 in PG(3 , 3) , . . and it has parameter x = 5 . . 2 Bruen and Drudge (1999) 4 generalized the above example . . into an infinite family with parameter x = q 2 +1 for all odd q . 2 . 3 Govaerts and Penttila (2005) 5 gave a sporadic example with . . parameter x = 7 in PG(3 , 4) . 3K. Drudge, On a conjecture of Cameron and Liebler, Europ. J. Combin. , 20 (1999), 263–269. 4A.A. Bruen, K. Drudge, The construction of Cameron-Liebler line classes in PG( 3 , q ), Finite Fields Appl. , 5 (1999), 35–45. 5P. Govaerts, T. Penttila, Cameron-Liebler line classes in PG( 3 , 4 ), Bull. Belg. Math. Soc. Simon Stevin , 12 (2005), 793–804. . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets I am going to talk about... . . 1 We construct a new infinite family of Cameron-Liebler line . classes with parameter x = q 2 − 1 for all q ≡ 5 or 9 (mod 12) . 2 . . . 2 In the case where q is an even power of 3 , we construct the first infinite family of affine two-intersection sets, whose existence was conjectured by Rodgers. I should remark that De Beule, Demeyer, Metsch and Rodgers also obtained the same results independently at about the same time by a more geometric approach. . . . . . .

Outline Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes Affine two-intersection sets Nonexistence . . 1 Penttila (1991): x = 3 for all q , and x = 4 for q ≥ 5 . . 2 Drudge (1999): 2 < x < √ q . . . . . . . 3 Govaerts and Storme (2004): 2 < x ≤ q , q prime. . . . 4 De Beule, Hallez and Storme (2008): 2 < x ≤ q/ 2 . . . . 5 Metsch (2010): 2 < x ≤ q . √ . . 6 Metsch (2014): 2 < x < q 3 2 − 2 q 3 q . . . 7 Gavrilyuk and Metsch (2014): A modular equality for . Cameron-Liebler line classes It seems reasonable to believe that for any fixed 0 < ϵ < 1 and constant c > 0 there are no Cameron-Liebler line classes with 2 < x < cq 2 − ϵ for sufficiently large q . . . . . . .