hemisystem-like structures in finite geometries · 2017-10-25 · motivation the higman-sims group...
TRANSCRIPT
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Hemisystem-like structures in finite geometries
John Bamberg
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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.
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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.
![Page 4: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/4.jpg)
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.
![Page 5: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/5.jpg)
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.
![Page 6: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/6.jpg)
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.
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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.
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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)
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Geometries yielding strongly regular graphs
• Polar spaces & generalised quadrangles
• Partial geometries
• Partial quadrangles
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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
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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
![Page 12: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/12.jpg)
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
![Page 13: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/13.jpg)
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
![Page 14: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/14.jpg)
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
![Page 15: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/15.jpg)
• 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.
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Theorem (Buekenhout-Shult-Tits-Veldkamp)
A finite polar space of rank > 3 arises from a vector spaceequipped with a bilinear, Hermitian, or quadratic form.
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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
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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
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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)
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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
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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).
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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).
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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)
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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)
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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)
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• 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
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• 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
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• 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
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• 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
![Page 30: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/30.jpg)
• 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
![Page 31: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/31.jpg)
• 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.
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• 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.
![Page 33: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/33.jpg)
• 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.
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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
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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
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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
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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)
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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)
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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)
![Page 40: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/40.jpg)
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)
![Page 41: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/41.jpg)
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.
![Page 42: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/42.jpg)
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.
![Page 43: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/43.jpg)
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.
![Page 44: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/44.jpg)
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).
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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.
![Page 46: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/46.jpg)
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.
![Page 47: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/47.jpg)
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.
![Page 48: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/48.jpg)
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.
![Page 49: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/49.jpg)
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
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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
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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!
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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!
![Page 53: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/53.jpg)
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!
![Page 54: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/54.jpg)
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!
![Page 55: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/55.jpg)
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)
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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)
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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)
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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.
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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.
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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
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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
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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)
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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)
![Page 64: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/64.jpg)
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)
![Page 65: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/65.jpg)
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)
![Page 66: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/66.jpg)
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)
![Page 67: 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](https://reader033.vdocument.in/reader033/viewer/2022041810/5e5764dc126e261e63571c92/html5/thumbnails/67.jpg)
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)