Cameron Liebler line classes in P G(n, 4)
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1 Cameron Liebler line classes in P G(n, 4) Ivan Mogilnykh Sobolev Institute of Mathematics, Novosibirsk joint work with Alexander Gavrilyuk, Krasovski Institute of Mathematics and Mechanics, Ekaterinburg Finite Fields and their Applications, Magdeburg, July 23, 2013.
2 For a point P of P G(3, q) define: Star(P ) = {l : l is a line incident to P },
3 For a point P of P G(3, q) define: Star(P ) = {l : l is a line incident to P }, for a plane H in P G(3, q): line(h) = {l : l is a line in H}.
4 For a point P of P G(3, q) define: Star(P ) = {l : l is a line incident to P }, for a plane H in P G(3, q): line(h) = {l : l is a line in H}. for an incident point-hyperplane pair (P, H): pen(p, H) = Star(P ) line(h).
5 Cameron-Liebler classes in P G(3, q) Let χ L be the characteristic function of a set of lines L in P G(3, q). A set of lines L in P G(3, q) is a Cameron-Liebler line class with parameter x in P G(3, q) if one of the following holds: (i) For every line l, {m L \ {l} : m meets l} = (q + 1)x + (q 2 1)χ L (l).
6 Cameron-Liebler classes in P G(3, q) Let χ L be the characteristic function of a set of lines L in P G(3, q). A set of lines L in P G(3, q) is a Cameron-Liebler line class with parameter x in P G(3, q) if one of the following holds: (i) For every line l, {m L \ {l} : m meets l} = (q + 1)x + (q 2 1)χ L (l). (ii) For any incident point-plane pair (P, H) we have: Star(P ) L + line(h) L = x + (q + 1) pen(p, H) L.
7 Cameron-Liebler classes in P G(3, q) Let χ L be the characteristic function of a set of lines L in P G(3, q). A set of lines L in P G(3, q) is a Cameron-Liebler line class with parameter x in P G(3, q) if one of the following holds: (i) For every line l, {m L \ {l} : m meets l} = (q + 1)x + (q 2 1)χ L (l). (ii) For any incident point-plane pair (P, H) we have: Star(P ) L + line(h) L = x + (q + 1) pen(p, H) L. (iii) For a pair of skew lines l, l from P G(3, q) the number of lines meeting l and l from L is x + q(χ L (l) + χ L (l )).
8 Cameron-Liebler classes in P G(3, q) Let χ L be the characteristic function of a set of lines L in P G(3, q). A set of lines L in P G(3, q) is a Cameron-Liebler line class with parameter x in P G(3, q) if one of the following holds: (i) For every line l, {m L \ {l} : m meets l} = (q + 1)x + (q 2 1)χ L (l). (ii) For any incident point-plane pair (P, H) we have: Star(P ) L + line(h) L = x + (q + 1) pen(p, H) L. (iii) For a pair of skew lines l, l from P G(3, q) the number of lines meeting l and l from L is x + q(χ L (l) + χ L (l )). (iv) For any line-spread S we have S L = x.
9 Cameron-Liebler classes in P G(3, q) A set of lines L in P G(n, q) is a Cameron-Liebler line class with parameter x in PG(3,q) if one of the following holds: (i) For every line l, {m L \ {l} : m meets l} = (q + 1)x + (q 2 1)χ L (l). (ii) For any incident incident point-plane pair (P, H) we have: Star(P ) L + line(h) L = x + (q + 1) pen(p, H) L. (iii) For a pair of skew lines l, l from P G(3, q) the number of common neighbors from L is x + q(χ L (l) + χ L (l )) (iv) For any line-spread S we have S L = x. (v) L is a completely regular code in G q (4, 2) of valency (q + 1)x and strength 0 as a q-design.
10 Cameron-Liebler classes in P G(3, q): constructions Cameron and Liebler 82:, (x=0), line(h), Star(P ) (x=1), Star(P ) line(h) for nonincident pair (P,H) (x=2). Drudge 99: classification of Cameron-Liebler classes in P G(3, 3) (P G(n, 3)), a new example for x = 5. Bruen and Drudge 98: an example in P G(3, q), x = (q 2 + 1)/2 Govaerts and Pentilla 05: an example in P G(3, 4), x = 7 Rodgers 11: examples in P G(3, q) for some odd q 200, x = (q 2 1)/2.
11 Cameron-Liebler classes in P G(3, q): nonexistence results Pentilla 91: x 3, 4 if q 5.
12 Cameron-Liebler classes in P G(3, q): nonexistence results Pentilla 91: x 3, 4 if q 5. Bruen, Drudge 98: x / {3,..., q}. Drudge 99: approach via blocking sets in P G(2, q), x / {3,..., e(q)}, where 1 + q + e(q) is the size of the smallest nontrivial blocking set in P G(2, q). De Beule, Hallez, Storme 08: x / {3,..., q/2}. Metsch 10: x / {3,..., q}.
13 Cameron-Liebler classes in P G(3, q): nonexistence results Govaerts, Pentilla 05: x 4, 5 in P G(3, 4).
14 Pattern Let L be a Cameron-Liebler line class in P G(3, q).
15 Pattern Let L be a Cameron-Liebler line class in P G(3, q). Consider line l P G(3, q) that belongs to Star(P 1 ),..., Star(P q+1 ), line(h 1 ),..., line(h q+1 ).
16 Pattern Let L be a Cameron-Liebler line class in P G(3, q). Consider line l P G(3, q) that belongs to Star(P 1 ),..., Star(P q+1 ), line(h 1 ),..., line(h q+1 ). The pattern w.r.t. l is the matrix T of order q + 1: T ij = (pen(p i, H j ) \ l) L for any i, j {1,..., q + 1}.
17 The properties of pattern Lemma Let L be a Cameron-Liebler line class in P G(3, q), T := (t ij ) be a pattern w.r.t. a line l. Then the following hold: 0 t ij q for all i, j {1,..., q + 1}; q+1 i,j=1 t ij = x(q + 1) + χ L (l)(q 2 1); q+1 q+1 t kj + t il = x + (q + 1)(t kl + χ L (l)), k, l; j=1 q+1 i,j=1 i=1 t 2 ij = (x χ L ) 2 + q(x χ L ) + χ L (l)q 2 (q + 1).
18 Cameron-Liebler line classes in P G(3, 4) The parameter x {0, 1, 2, 3, 4, 5, 6, 7, 8}
19 Cameron-Liebler line classes in P G(3, 4) The parameter x {0, 1, 2, 3, 4, 5, 6, 7, 8} Govaerts, Pentilla 05: x {0!, 1!, 2!, 3, 4, 5, 6?, 7!?, 8?}
20 Cameron-Liebler line classes in P G(3, 4) The parameter x {0, 1, 2, 3, 4, 5, 6, 7, 8} Govaerts, Pentilla 05: x {0!, 1!, 2!, 3, 4, 5, 6?, 7!?, 8?} The approach via patterns The cases x = 4, 8 are eliminated by the approach solely, the cases x = 5, 6, 7 are solved by the approach and additional considerations.
21 Cameron-Liebler line class with x = 7 in P G(3, 4) Govaerts, Pentilla 05: Let (P, H) be nonincident point-hyperplane pair in P G(3, 4), O be a hyperoval in H. Let C be the set of lines incident to the points of O and P. Then C, all 2-secants of C and all lines in H external to O form a Cameron Liebler line class with parameter x = 7.
22 Cameron-Liebler line class with x = 7 in P G(3, 4) Govaerts, Pentilla 05: Let (P, H) be nonincident point-hyperplane pair in P G(3, 4), O be a hyperoval in H. Let C be the set of lines incident to the points of O and P. Then C, all 2-secants of C and all lines in H external to O form a Cameron Liebler line class with parameter x = 7.
23 Cameron-Liebler line class with x = 7 in P G(3, 4) Let (P, H) be nonincident point-hyperplane pair in P G(3, 4), O be a hyperoval in H. Let C be the set of lines incident to the points of O and P. Then C, all 2-secants of C and all lines in H external to O form a Cameron Liebler line class with parameter x = 7. w.r.t. l L we have the following patterns: w.r.t. l / L: , ,,
24 Cameron-Liebler line class with x = 7 in P G(3, 4) Let (P, H) be nonincident point-hyperplane pair in P G(3, 4), O be a hyperoval in H. Let C be the set of lines incident to the points of O and P. Then C, all 2-secants of C and all lines in H external to O form a Cameron Liebler line class with parameter x = 7. w.r.t. l L we have the following patterns: w.r.t. l / L: , ,,
25 Cameron-Liebler line classes in P G(3, 4) Theorem There only Cameron-Liebler classes in P G(3, 4) are the classes with x = 0, 1, 2 and the class with x = 7 constructed by Govaerts and Pentilla unique up to collineation.
26 The pattern approach for small q Pairs (q, x) excluded by the approach. q x total 4 3,4,8 3 of 8 5 3,4,7,11 4 of ,4,5,6,7,11,12,14,15,19,20,22,23 13 of ,4,5,6,8,12,14,15,17,21,23,24,26,30,32 15 of ,4,5,7,8,9,11,13,14,15,18,19,23,24,27, 28,29,31,33,34,35, 38,39 23 of ,...,9,11,12,14,15,19,20,22,23,27 28,30,31,35,36,38,39,43,44,46,47,51,52,54,55,59,60 35 of 61
27 Cameron-Liebler line classes in P G(n, q) A set of lines L in P G(n, q) is a Cameron-Liebler line class with parameter x if one of the following holds: for every line l, {m L\{l} : m meets l} = (q+1)x+(q n q 2 1)χ L (l).
28 Cameron-Liebler line classes in P G(n, 4) Drudge s Lemma Let L be a Cameron Liebler line class in P G(n, q), n > 3, consider a 3-dimensional projective subspace P G(3, q) of P G(n, q). Then line(x) L is a Cameron Liebler line class in P G(3, q). Theorem The only Cameron-Liebler classes to exist in P G(n, 4) are: empty set (x = 0), line(h), Star(P ) (x=1), Star(P ) line(h) for nonincident pair (P, H) (x=2) and the class with x = 7 in P G(3, 4) constructed by Govaerts and Pentilla, which is unique up to a collineation.
29 Conclusion The patterns approach for studying Cameron-Liebler line classes. Classification of Cameron-Liebler line classes in P G(n, 4). Promising results for other values of q.
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