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Delsarte designs . . . from finite polar spaces John Bamberg The - - PowerPoint PPT Presentation
Delsarte designs . . . from finite polar spaces John Bamberg The - - PowerPoint PPT Presentation
Delsarte designs . . . from finite polar spaces John Bamberg The University of Western Australia quote This motivates the present general definition, the conjecture being that T-designs will often have interesting properties..
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t − (v, k, λ) design
k-subsets B (blocks) of a v-set such that every t-subset is contained in exactly λ blocks. inclusion matrices: k versus i Ii(b, y) =
- 1
y ⊆ b
- therwise
{1} {2} {3} {4} {1,2}
1 1
{1,3}
1 1
{1,4}
1 1
{2,3}
1 1
{2,4}
1 1
{3,4}
1 1
designs revisited I
χB(b) =
- 1
b ∈ B
- therwise
χBIi = |B| v jIi, ∀0 i t
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1 3 5 2 4 6 7 B = {123, 145, 167, 246, 257, 347, 356} All 3-subsets: {123, 124, 125, 126, 127, 134, . . . , 567} χB = (1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0) χBI0 = (7) χBI1 = (3, 3, 3, 3, 3, 3, 3) χBI2 = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) χBI3 = χB
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the Johnson scheme J(7, 3)
two 3-subsets can interact in four different ways: equal A0 = I differ in 1 element A1 differ in 2 elements A2 differ in 3 elements A3 adjacency matrices A0 A1 A2 A3 (simultaneous) eigenspaces E0 E1 E2 E3 dimensions 1 6 14 14 inclusion matrices I0 I1 I2 I3 ranks 1 7 21 35 n.b., I0 =
- j⊤
and I3 = I.
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eigenspaces: {E0, E1, . . . , Ed}. There is an ordering so that RowSpace(Ii) =
i
- j=0
Ej
projections
The projection Ei to Ei is a minimal idempotent for the Johnson scheme. χBIi = |B| v jIi ∀0 i t χBEi = |B| v jEi ∀0 i t = 0 ∀0 < i t n.b., E0 = 1
v J
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designs revisited II
χBEi = 0, ∀0 < i t
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what if you change the scheme?
Hamming scheme: orthogonal arrays Grassmann scheme: q-designs Sn-scheme: λ-transitive sets of permutations
T-design, T ⊆ {1, . . . , d}
χBEi = 0, ∀i ∈ T
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Remark
Since E 2
i = Ei and each Ei is positive semidefinite, we have
χBEi = 0 ⇐ ⇒ χBEi = 0 ⇐ ⇒ (χBEi)(χBEi)⊤ = 0 ⇐ ⇒ χB EiE ⊤
i χ⊤ B = 0
⇐ ⇒ χB Ei χ⊤
B = 0.
Analogue (coclique): χB Ai χ⊤
B = 0.
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strongly regular graphs
k-regular graph such that
◮ every pair of adjacent vertices has λ common neighbours ◮ every pair of non-adjacent vertices has µ common neighbours
λ µ
A2 = (k − µ)I + (λ − µ)A + µJ three distinct eigenvalues k θ τ three minimal idempotents Ek = 1
nJ
Eθ Eτ
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Take a minimal idempotent E = 1
nJ and suppose χDE = 0.
A = βI + γJ + αE χD A = βχD + γ|D| j = h1 · χD + h2 · (j − χD)
Theorem
χDE = 0 = ⇒ there exist constants h1, h2, |Γ(v) ∩ D| =
- h1
if v ∈ D h2 if v / ∈ D That is, D is intriguing. h2 h1
d
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Theorem (Haemers 1995)
Let Γ be a connected k-regular graph, and Y ⊆ V Γ. Let N be the number
- f ordered pairs of adjacent vertices of Y . Then
θmin|Y | + k − θmin n |Y |2 N θmax|Y | + k − θmax n |Y |2 Moreover, if equality holds in one of the inequalities above, then Y is intriguing. generalised by J¨
- rg Eisfeld to association schemes1
explored further by De Bruyn and Suzuki2
1Theorem 2 of ‘Subsets of association schemes corresponding to eigenvectors of the
Bose-Mesner algebra’. Bull. Belg. Math. Soc. Simon Stevin 5 (1998).
2‘Intriguing sets of vertices of regular graphs’. Graphs & Combinatorics 26 (2010).
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polar spaces
geometry of totally singular subspaces of a sesquilinear or quadratic form introduced by Freudenthal, Tits, Veldkamp (1950’s) collinearity graph is strongly regular
classical
symplectic
- rthogonal (elliptic, parabolic, hyperbolic)
Hermitian W(d, q) Q−(d, q), Q(d, q), Q+(d, q) H(d, q)
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eigenvalues: degree positive one negative one eigenspaces: j E+ E− interesting cases: – {1}-designs {2}-designs
a {2}-design: points in a generator π (dimension d)
χπA =
- qd − 1
- χπ + qd − 1
q − 1 j The χπ span all of j ⊕ E+.
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anti-designs
Theorem (Roos 1982)
Let D be a T-design and let A be a U-design with T ∩ U = ∅. Then |D ∩ A| = |D||A| n .
Proof.
0 = (χDEi)(χAEi)⊤ = χDEiχ⊤
A
= χD(I − 1
nJ)χ⊤ A
= χDχ⊤
A − 1 nχDJχ⊤ A
= |D ∩ A| − 1 n |D||A|
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{1}-designs are m-ovoids
must meet {2}-designs in (product of sizes divided by n) = ⇒ must meet sets of points in a generator in a constant = ⇒ m-ovoid {1}-designs {2}-designs i-tight sets m-ovoids
intriguing sets
χSA = h1 · χS + h2 · (j − χS)
Theorem
An m-ovoid and an i-tight set meet in m · i elements.
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Quote
“The definition of T-designs in a symmetric association scheme (X, R) can be extended so as to admit the possibility of ‘repeated points’.” – P. Delsarte’s thesis, p. 34
Generalisation
Replace χD with x ∈ Rn (or Cn). T-design xEi = 0, ∀i ∈ T Roos’ Theorem T ∩ U = 0 = ⇒ x · y = (x·j)(y·j)
n
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Some applications
Vanhove (2009)
A partial spread of H(2d − 1, q2), d odd, has size at most qd + 1.
De Bruyn & Vanhove (2013)
No generalised hexagon of order (s, s3), s 2, can have 1-ovoids.
B., De Beule, Ihringer (2017)
No ovoids of H(2r − 1, q2) can exist for r > q3 − q2 + 2. Excellent resources Ferdinand Ihringer’s Masters Thesis: Faszinierende Mengen in Polarr¨ aumen Fr´ ed´ eric Vanhove’s PhD Thesis: Incidence geometry from an algebraic graph theory point of view Jan De Beule’s talk (2014): Do i-tight sets and m-ovoids hate each other?
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- ther interesting designs and antidesigns, and connections
m-systems of polar spaces distance-j-ovoids of generalised hexagons SPG-reguli pseudo-ovoids of PG(4m + 3, q) Cameron-Liebler line classes of PG(3, q) Erd˝
- s-Ko-Rado sets
partial quadrangles cometric association schemes
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what do we know about m-ovoids?
underlying points of an m-system is an m′-ovoid (Shult + Thas 1994) hemisystems sometimes give rise to two-character sets
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m-ovoids of symplectic spaces
W(3, q), q even Many W(3, q), q odd Many, m even W(5, q), q even m = q + 1, (Cossidente, Pavese) W(5, q), q odd Various W(2r − 1, q), r > 2, q odd m −3+√9+4qr
2q−2
, (B., Kelly, Law, Penttila)
Problem
Improve the lower bound on m, when q is odd and r > 2. Conjecture? m qr−2
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hermitian spaces
H(3, q2) many H(4, q2) none known! H(5, q2) various H(6, q2) none known!
conjecture
For all 1 m q3, there are no m-ovoids of H(4, q2).
known
Thas 1981: no examples for m = 1 B., Kelly, Law, Penttila 2007: m < √q. B., Devillers, Schillewaert 2012: m < −3q−3+√
4q5−4q4+5q2−2q+1 2(q2−q−2)
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- rthogonal spaces
Q(4, q) interesting Q(6, q) many? Q−(5, q) hemisystems Q−(7, q) from 1-systems; m = q + 1 Q+(5, q) Many Q+(7, q) ?
question
Do there exist 2-ovoids of Q(4, q) for q > 5?
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non-embeddable GQ
DH(4, q2): constructions for many m, lower bound m f (q)? Flock GQ: hemisystems (Cameron, Goethals, Seidel 1979) Payne derived GQ: some examples (B., Devillers, Schillewaert 2012) T3(O), q even: don’t exist T2(O), q even: ?
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what do we know about i-tight sets?
sometimes gives rise to two-character sets i-tight sets of orthogonal (hyperbolic) spaces ≡ Cameron-Liebler line classes of PG(3, q) reducible examples: union of generators Baer examples: union of Baer-subgeometries (e.g., W(2n + 1, q) ֒ → H(2n + 1, q2))
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B., Kelly, Law, Penttila 2007: no irreducible i-tight sets in W(2r − 1, q), H(2r − 1, q2), or Q+(2r − 1, q), for i small compared to q De Beule-Govaerts-Hallez-Storme 2009:
H(2n + 1, q)
i < q5/8/ √ 2 + 1 = ⇒ reducible or Baer. Beukemann & Metsch 2013:
Q+(2n + 1, q)
1 n 3 and i q, or n 4 and i < q and q 71 = ⇒ reducible.
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Metsch 2016: reducibility in the ‘other’ polar spaces De Beule & Metsch 2017: i-tight set of H(4, q) and H(6, q) is reducible if i is small compared to q. Naki´ c and Storme 2017
H(2n + 1, q)
i < (q2/3 − 1)/2 = ⇒ reducible or Baer.
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reducibility
sometimes an irreducible i-tight set is the complement of a reducible i′-tight set. S7 acting on H(3, 32)
- rbit lengths on points: 70, 210
the set of size 210 is an irreducible 21-tight set but it is the complement of a partial spread of 7 lines.
proposal
study minimal partitions of the points into i-tight sets.
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- ther things