Hemisystem-like structures in finite geometries · 2017-10-25 · Motivation The Higman-Sims group HS Discovered by Donald G. Higman and Charles Sims (1968); ...a lecture by Marshall

Hemisystem-like structures in finite geometries

John Bamberg

Motivation

The Higman-Sims group HS

• Discovered by Donald G. Higman and Charles Sims (1968);

• . . . a lecture by Marshall Hall Jnr on J2, which has a rank 3 action on 100elements.

• Higman and Sims looked for more simple groups amongst rank 3 groupsacting on 100 elements and found HS .

• |HS | = 44352000

• associated rank 3 graph on 100 elements is the Higman-Sims graph.

• PSU(3, 5) : 2 < HS

• PSU(3, 5) : 2 also has a rank 3 action, on 50 elements.

• We can decompose HS-graph into two Hoffman-Singleton graphs.

• So we have a rank 3 graph composed of two rank 3 graphs.

Motivation





• |HS | = 44352000


• PSU(3, 5) : 2 < HS




Motivation





• |HS | = 44352000


• PSU(3, 5) : 2 < HS




Motivation





• |HS | = 44352000


• PSU(3, 5) : 2 < HS




Generalising rank 3 graphs

Strongly regular graphs

Regular graph such that there are two constants λ and µ such that

• any pair of adjacent vertices have λ common neighbours;

• and any pair of non-adjacent vertices have µ commonneighbours.

LemmaA connected graph is strongly regular if and only if it has 3 distincteigenvalues.

Generalising rank 3 graphs

Strongly regular graphs

Regular graph such that there are two constants λ and µ such that

• any pair of adjacent vertices have λ common neighbours;

• and any pair of non-adjacent vertices have µ commonneighbours.

LemmaA connected graph is strongly regular if and only if it has 3 distincteigenvalues.

Strongly regular decompositions and subconstituents

• M. S. Smith (1975)1

• Cameron – Goethals – Seidel (1978)2

• Cameron – Delsarte – Goethals (1979)3

• Cameron – MacPherson (1985)4

• D. G. Higman (1988) 5

• Haemers and D. G. Higman (1989)6

• Kasikova (1997)7

1‘On rank 3 permutation groups’, J. Algebra (1975)

2‘Strongly regular graphs having strongly regular subconstituents’, J. Algebra 55 (1978)

3‘Hemisystems, orthogonal configurations and dissipative conference matrices’, Philips I. Res. 34 (1979)

4‘Rank three permutation groups with rank three subconstituents’, J. Combin. Theory Ser. B 39 (1985)

5‘Strongly regular designs and coherent configurations of type

[3 2

3

]’, European J. Combin. 9 (1988)

6‘Strongly regular graphs with strongly regular decomposition’, Linear Algebra Appl. 114/115 (1989)

7‘Distance-regular graphs with strongly regular subconstituents’, J. Algebraic Combin. 6 (1997)

Geometries yielding strongly regular graphs

• Polar spaces & generalised quadrangles

• Partial geometries

• Partial quadrangles

Polar spaces

Definition of Polar Space

• Point/line geometry

• 1. Every two points lie on at most one line.

partial linear space

2. “All or one” axiom:

3. Non-degeneracy: no point is collinear with all points.

• ‘Just one’: Generalised quadrangle.

GQ = rank 2 polar space

• Subspace: any two points are collinear.

〈X 〉 := points on all lines spanned by pairs of points in X

Polar spaces



• 1. Every two points lie on at most one line. partial linear space







Polar spaces











Polar spaces







• ‘Just one’: Generalised quadrangle. GQ = rank 2 polar space



Polar spaces











• The intersection of two subspaces is again a subspace.

• The set of elements contained in a subspace forms aprojective space.

• Finite: there is a rank.

• The maximal subspaces have a common rank.

Theorem (Buekenhout-Shult-Tits-Veldkamp)

A finite polar space of rank > 3 arises from a vector spaceequipped with a bilinear, Hermitian, or quadratic form.

Generalised quadrangles

Generalised quadrangle

Given a point P and ` which are not incident, there is a unique linem on P concurrent with `.

`

P

Order (s, t)

s + 1 points on a line, t + 1 lines through a point

Generalised quadrangles

Generalised quadrangle

Given a point P and ` which are not incident, there is a unique linem on P concurrent with `.

`

P

Order (s, t)

s + 1 points on a line, t + 1 lines through a point

Classical generalised quadrangles (those arising fromsesquilinear and quadratic forms)

GQ order GQ order

W(3, q) (q, q) Q(4, q) (q, q)H(3, q2) (q2, q) Q−(5, q) (q, q2)H(4, q2) (q2, q3)

m-coversA set of lines M of a GQ(s, t) is an m-cover 8 if every point lieson m elements of M.

Figure: A 2-cover of W(3, 2).

80 < m < t + 1

A geometry arising...

Suppose we have an m-cover.

• New points: elements of the m-cover.

• New lines: the points of the GQ.

Figure: The Petersen graph from a 2-cover of W(3, 2).

A geometry arising...

Suppose we have an m-cover.

• New points: elements of the m-cover.

• New lines: the points of the GQ.

Figure: The Petersen graph from a 2-cover of W(3, 2).

m-covers of classical generalised quadrangles

• State of play:

W(3, q) Not too manyQ(4, q), q odd Many, m evenQ−(5, q) Many!H(4, q2) Many found9, m > 1H(3, q2) Hemisystems, q odd

• Segre (1965):An m-cover of H(3, q2), q odd, has m = q+1

2 .

• A hemisystem is an m-cover where m = t+12 .

9JB, Devillers, Schillewaert, ‘Weighted intriguing sets of finite generalised quadrangles’, JAC (2012)


• State of play:



2 .




• State of play:



2 .



• Segre (1965):There exists a unique hemisystem of H(3, 32).

• Bruen & Hirschfeld (1978):No m-cover exists of H(3, q2), q even.

• J. A. Thas (1981):

Hemisystem of H(3, q2) −→ partial quadrangle andstrongly regular graph

Strongly regular decomposition

Complement of a hemisystem is a hemisystem ⇒ concurrencygraph of H(3, q2) is the sum of two strongly regular graphs.

Segre hemisystem of H(3, 32) −→ Sims-Gewirtz graph



• J. A. Thas (1981):







• J. A. Thas (1981):







• J. A. Thas (1981):







• J. A. Thas (1981):





• Cameron, Delsarte & Goethals (1979):

Hemisystem of GQ(q2, q) −→ partial quadrangle andstrongly regular graph

• Martin, Muzychuk, van Dam (2013):

Hemisystem of GQ(q2, q) −→4-class imprimitive cometric Q-antipodal association schemethat is not metric

• H. Suzuki (1998), Cerzo & Suzuki (2009),H. & R. Tanaka (2011):

Imprimitive cometricassociation scheme withfirst multiplicity> 3

−→ Q-bipartite or Q-antipodal.















Partial quadrangle (P. J. Cameron 1975)

• Given a point P and ` which are not incident, there isat most one line m on P concurrent with `.

• There exists µ such that any two non-collinear points X andY are collinear to µ common points.

• New points: elements of m-cover

• New lines: points of GQ











Known PQs

• Triangle-free strongly regular graphs:pentagon, Petersen graph, Clebsch graph, Sims-Gewirtz graph,Hoffman-Singleton graph, M22-graph, Higman-Sims graph

• Three exceptional examples:Coxeter cap, Hill 56-cap, Hill 78-cap, 430-cap of PG(6, 4)?

• GQ(q, q2) minus a point

• arise from a hemisystem of a GQ(q2, q)

Known PQs





Known PQs





Known PQs





Recent times

• Thas (1995):Conjectured that no hemisystem of H(3, q2) exists for q > 3.

• Cossidente – Penttila (2005):For each odd prime power q, there exists a hemisystem ofH(3, q2).

• JB – Giudici – Royle (2010):Every flock quadrangle of order (q2, q), q odd, has ahemisystem.

Recent times




Recent times




Theorem (JB, Giudici, Royle)

The hemisystems of the flock quadrangles of order (52, 5) areknown:

Group Size Construction/Author(s)

PΣL(2, 25) 15600 Cossidente–Penttila(3 · A7).2 15120 Cossidente–Penttila

Table: The hemisystems of H(3, 52).

Group Size Construction/Author(s)

C 25 : (C4 × S3) 600 BGR

AGL(1, 5)× S3 120 Bamberg–De Clerck–DuranteS3 6 New

Table: The hemisystems of FTWKB(5) (up to complements).

Infinite families for H(3, q2)?

Invariant under a Singer element

• Cyclic semiregular element10 K of order q2 − q + 1.

• K -invariant hemisystems exist forq ∈ 3, 5, 7, 9, 11, 17, 19, 23, 27, 29.

Open problem

Do examples like this exist for all odd prime powers q 6≡ 1(mod 12)?

10In the dual, Q−(5, q): take GF(q6) equipped with B(x, y) := Tr

q6→q(xyq

3). Take ω = ζ(q3−1)(q+1)

where 〈ζ〉 = GF(q6)∗. Then K := 〈ω〉 is irreducible and acts semiregularly on points.





Open problem



q6→q(xyq

3). Take ω = ζ(q3−1)(q+1)






Open problem



q6→q(xyq

3). Take ω = ζ(q3−1)(q+1)






Open problem



q6→q(xyq

3). Take ω = ζ(q3−1)(q+1)


Theorem (JB, Lee, Momihara, Xiang11)

There is a hemisystem of H(3, q2) for every prime power q ≡ 3(mod 4), each admitting C(q3+1)/4 : C3.

Still open

q ≡ 1, 5, 9 (mod 12)

11Combinatorica, to appear

Theorem (JB, Lee, Momihara, Xiang11)

There is a hemisystem of H(3, q2) for every prime power q ≡ 3(mod 4), each admitting C(q3+1)/4 : C3.

Still open

q ≡ 1, 5, 9 (mod 12)

11Combinatorica, to appear

Dualising

• The dual of a generalised quadrangle of order (s, t) is a GQ of order (t, s).

GQ order GQ order

W(3, q) (q, q) Q(4, q) (q, q)H(3, q2) (q2, q) Q−(5, q) (q, q2)H(4, q2) (q2, q3) DH(4, q2) (q3, q2)

• Dual polar space:

Points Maximals of a given polar space PLines Second-to-maximals of P

• m-cover Set of maximals of a polar space such that every point iscovered m times.

m-ovoid Set of points of a polar space such that every maximal iscovered m times.

Confused yet!

Dualising


GQ order GQ order






Confused yet!

Dualising


GQ order GQ order






Confused yet!

Dualising


GQ order GQ order






Confused yet!

Hemisystems of regular near polygons

Regular near polygon (Shult & Yanushka12)

Incidence geometry of points and lines such that the collinearitygraph is distance regular and

For every point P and every line `, there exists a unique point on `nearest to P.

• Diameter 2⇒ Generalised quadrangle

• Dual polar spaces

• Geometries for J2, M24.

12‘Near n-gons and line systems’, Geom. Dedicata 9, (1980)

Theorem (F. Vanhove13)

If Γ is a regular near 2d-gon of order (s, t) with s > 1 and

cj = (s2j − 1)/(s2 − 1),

for some j ∈ 2, . . . , d, then m-ovoids can only exist form = (s + 1)/2.

Theorem (F. Vanhove)

Let S be a ((q + 1)/2)-ovoid in the dual polar graph Γ fromH(2d − 1, q2) with q odd. The induced subgraph on S isdistance-regular with classical parameters:

(d , b, α, β) =

(d ,−q,−

(q + 1

2

),−

((−q)d + 1

2

))

13‘A Higman inequality for regular near polygons’, JAC (2011)

Theorem (F. Vanhove13)

If Γ is a regular near 2d-gon of order (s, t) with s > 1 and

cj = (s2j − 1)/(s2 − 1),

for some j ∈ 2, . . . , d, then m-ovoids can only exist form = (s + 1)/2.

Theorem (F. Vanhove)

Let S be a ((q + 1)/2)-ovoid in the dual polar graph Γ fromH(2d − 1, q2) with q odd. The induced subgraph on S isdistance-regular with classical parameters:

(d , b, α, β) =

(d ,−q,−

(q + 1

2

),−

((−q)d + 1

2

))

13‘A Higman inequality for regular near polygons’, JAC (2011)

Theorem (M. Lee)

If DH(5, q2), q odd, possesses an m-ovoid, then so too doesDW(5, q). There are no m-ovoids of DW(5, 3) or DW(5, 5).

Conjecture

DW(5, q), q odd, has no m-ovoids for 0 < m < q + 1.

Theorem (M. Lee)

If DH(5, q2), q odd, possesses an m-ovoid, then so too doesDW(5, q). There are no m-ovoids of DW(5, 3) or DW(5, 5).

Conjecture

DW(5, q), q odd, has no m-ovoids for 0 < m < q + 1.

More hemisystem-like stuff

Hemisystems of Q(6, q) (Cossidente & Pavese, 2016)

• constructed hemisystems of Q(6, q), q odd

• each admitting the group PSL(2, q2)

• other sporadic examples (e.g., Q(6, 3), A5)

Relative hemisystem (Penttila and Williford, JCTA, 2011);analogue for q even

• Take a subquadrangle Q′ of order (q, q) away from a generalisedquadrangle Q of order (q2, q).

• Relative hemisystem: Half the external lines of Q\Q′ such that eachpoint of Q\Q′ has its lines halved.

Classical caseQ = H(3, q2), Q′ = W(3, q), q even

More hemisystem-like stuff

Hemisystems of Q(6, q) (Cossidente & Pavese, 2016)

• constructed hemisystems of Q(6, q), q odd

• each admitting the group PSL(2, q2)

• other sporadic examples (e.g., Q(6, 3), A5)

Relative hemisystem (Penttila and Williford, JCTA, 2011);analogue for q even

• Take a subquadrangle Q′ of order (q, q) away from a generalisedquadrangle Q of order (q2, q).

• Relative hemisystem: Half the external lines of Q\Q′ such that eachpoint of Q\Q′ has its lines halved.

Classical caseQ = H(3, q2), Q′ = W(3, q), q even

Q = H(3, q2), Q′ = W(3, q), q even

• Relative hemisystem −→ Q-bipartite 4-class association scheme, notP-polynomial (nor dual thereof) and not Q-antipodal.

• All known infinite families of Q-polynomial schemes which are neitherP-polynomial nor duals thereof are imprimitive and Q-antipodal.

• Penttila & Williford (1995): construction for each q even, admittingPΩ−(4, q).

• Cossidente (2013): construction for each q even, admitting PSL(2, q).

• . . . also an another family14 admitting groups of order q2(q + 1).

• Cossidente & Pavese (2014): construction arising from a Suzuki-Titsovoid for q = 8.

• JB, Lee, Swartz15: unified construction.

14‘A new family of relative hemisystems on the hermitian surface’, DCC (2014)

15‘A note on relative hemisystems of Hermitian generalised quadrangles’, DCC (to appear)

Q = H(3, q2), Q′ = W(3, q), q even










Q = H(3, q2), Q′ = W(3, q), q even










Q = H(3, q2), Q′ = W(3, q), q even










Q = H(3, q2), Q′ = W(3, q), q even










Q = H(3, q2), Q′ = W(3, q), q even










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Hemisystem-like structures in finite geometries · 2017-10-25 · Motivation The Higman-Sims group HS Discovered by Donald G. Higman and Charles Sims (1968); ...a lecture by Marshall