Construction of Some New Projective Planes G. Eric Moorhouse, - - PowerPoint PPT Presentation

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Construction of Some New Projective Planes G. Eric Moorhouse, - - PowerPoint PPT Presentation

Dec 28, 2023 135 likes •412 views

Construction of Some New Projective Planes G. Eric Moorhouse, University of Wyoming http://math.uwyo.edu/~moorhous/pub/planes/ n 2 3 4 5 7 8 9 11 number of planes 1 1 1 1 1 1 4 1 of order n known n 13 16 17 19 23 25 27

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Construction of Some New Projective Planes

  • G. Eric Moorhouse, University of Wyoming

http://math.uwyo.edu/~moorhous/pub/planes/

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n 2 3 4 5 7 8 9 11 number of planes

  • f order n known

1 1 1 1 1 1 4 1 n 13 16 17 19 23 25 27 29 number of planes

  • f order n known

1 22 1 1 1 193 13 1

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Known Planes of Order 25

Translation planes a1,…,a8; b1,…,b8; s1,…,s5 classified by Czerwinski & Oakden (1992)

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The Wyo m ing Plains

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|Aut(w1)| = 19200 |Aut(w2)| = 3200

The Wyo m ing Planes

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Thanks to my coauthor…

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Where do the new planes come from?

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1 1 2 2 3 3 4 4 1 3 4 2 quotient by t, an automorphism of

  • rder 2
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1 1 2 2 3 3 4 4 1 3 4 2 1 1 2 2 3 3 4 4

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Given a projective plane P with involution t 2 Aut(Π), let Π/t be the incidence structure induced on point and line orbits of size 2. Π/t yields a cell complex Δ having

  • vertices (0-cochains): points, blocks of Π/t
  • edges (1-cochains): flags of Π/t
  • square faces (2-cochains): “digons” of Π/t

C i = C i(Δ,F2) = F2-space of i-cochains H 1 = H 1(Δ,F2) = ker d1 / im d0 C 0 ! C 1 ! C2 d1 d0 coboundary map

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C i = C i(Δ,F2) = F2-space of i-cochains H 1 = H 1(Δ,F2) = ker d1 / im d0 C 0 ! C 1 ! C2 d1 d0 coboundary map If H 1 = 0 then Π/t lifts uniquely back to Π. Theorem. Equivalence classes of pairs (Π,t) covering the same Π/t

  • rbits of Aut(Π/t)
  • n H 1(Δ,F2)
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C i = C i(Δ,F2) = F2-space of i-cochains H 1 = H 1(Δ,F2) = ker d1 / im d0 C 0 ! C 1 ! C2 d1 d0 coboundary map If H 1 = 0 then Π/t lifts uniquely back to Π. Theorem. Equivalence classes of pairs (Π,t) covering the same Π/t

  • rbits of Aut(Π/t)
  • n H 1(Δ,F2)

actually, a particular coset of H1=Z1/B1 in C1/B1…

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In all cases I have examined, dim H 1(Δ,F2) ∙ 4. If H 1 = 0 then Π/t lifts uniquely back to Π. Theorem. Equivalence classes of pairs (Π,t) covering the same Π/t

  • rbits of Aut(Π/t)
  • n H 1(Δ,F2)
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For any given plane Π,

  • compute G=Aut(Π);
  • find a representative t for each conjugacy class
  • f involutions in G;
  • compute H 1;
  • if H 1 ≠ 0, find orbits of Aut(Π/t) on H 1.
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Known Planes of Order 25

Translation planes a1,…,a8; b1,…,b8; s1,…,s5 classified by Czerwinski & Oakden (1992)

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Other instances of non-unique lifting (among planes of order 16)

Johnson-Walker plane Dempwolff plane quotient

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Other instances of non-unique lifting (among planes of order 16)

Lorimer-Rahilly plane derived semifield plane quotient

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Other instances of non-unique lifting (among planes of order 16)

semifield plane

  • ver F4

quotient semifield plane

  • ver F2
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Other instances of non-unique lifting (among planes of order 16)

Mathon plane quotient dual Mathon plane

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Other instances of non-unique lifting (among planes of order 9)

Desarguesian plane quotient Hughes plane

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Why only consider involutions t 2 Aut(Π)?

In this case the problem of lifting Π/t to Π amounts to solving a linear system. In any double cover, the fibres are necessarily

t-orbits for some involution t.

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Also tested:

  • the 18 smallest known generalised quadrangles;
  • the 4 smallest known generalised hexagons;
  • the smallest known generalised octagon.

In only one case is H 1 nontrivial: dim H 1 = 1 for the GQ with s=3, t=5. We are having to solve linear systems over F2 with thousands of unknowns. Space constraints (computer memory) is the chief limitation in testing larger generalised polygons.