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.”. – P. Delsarte’s thesis, p. 33 quote “If it isn’t in Delsarte’s thesis, it’s in Haemers’ thesis”. – A. B.

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 { 1 } { 2 } { 3 } { 4 } 1 1 0 0 { 1 , 2 }   1 0 1 0 � { 1 , 3 } 1 y ⊆ b   1 0 0 1 { 1 , 4 }   I i ( b , y ) =   0 1 1 0 0 otherwise { 2 , 3 }     0 1 0 1 { 2 , 4 }   0 0 1 1 { 3 , 4 } designs revisited I � 1 b ∈ B χ B I i = |B| χ B ( b ) = v jI i , ∀ 0 � i � t 0 otherwise

1 B = { 123 , 145 , 167 , 246 , 257 , 347 , 356 } 2 4 7 3 5 6 All 3-subsets: { 123 , 124 , 125 , 126 , 127 , 134 , . . . , 567 } χ B = (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) χ B I 0 = (7) χ B I 1 = (3 , 3 , 3 , 3 , 3 , 3 , 3) χ B I 2 = (1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1) χ B I 3 = χ B

the Johnson scheme J (7 , 3) two 3-subsets can interact in four different ways: equal A 0 = I differ in 1 element A 1 differ in 2 elements A 2 differ in 3 elements A 3 adjacency matrices A 0 A 1 A 2 A 3 (simultaneous) eigenspaces E 0 E 1 E 2 E 3 dimensions 1 6 14 14 inclusion matrices I 0 I 1 I 2 I 3 ranks 1 7 21 35 j ⊤ � n.b., I 0 = � and I 3 = I .

eigenspaces: {E 0 , E 1 , . . . , E d } . There is an ordering so that i � RowSpace ( I i ) = E j j =0 projections The projection E i to E i is a minimal idempotent for the Johnson scheme. χ B I i = |B| jI i ∀ 0 � i � t v χ B E i = |B| jE i ∀ 0 � i � t v = 0 ∀ 0 < i � t n.b., E 0 = 1 v J

designs revisited II χ B E i = 0 , ∀ 0 < i � t

what if you change the scheme? Hamming scheme: orthogonal arrays Grassmann scheme: q -designs S n -scheme: λ -transitive sets of permutations T -design, T ⊆ { 1 , . . . , d } χ B E i = 0 , ∀ i ∈ T

Remark Since E 2 i = E i and each E i is positive semidefinite, we have χ B E i = 0 ⇐ ⇒ � χ B E i � = 0 ⇒ ( χ B E i )( χ B E i ) ⊤ = 0 ⇐ ⇒ χ B E i E ⊤ i χ ⊤ ⇐ B = 0 ⇒ χ B E i χ ⊤ ⇐ B = 0 . Analogue (coclique): χ B A i χ ⊤ B = 0 .

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 µ λ A 2 = ( k − µ ) I + ( λ − µ ) A + µ J three distinct eigenvalues k θ τ E k = 1 three minimal idempotents n J E θ E τ

Take a minimal idempotent E � = 1 n J and suppose χ D E = 0. A = β I + γ J + α E χ D A = βχ D + γ |D| j = h 1 · χ D + h 2 · ( j − χ D ) Theorem χ D E = 0 = ⇒ there exist constants h 1 , h 2 , � h 2 d h 1 if v ∈ D | Γ( v ) ∩ D| = h 2 if v / ∈ D h 1 That is, D is intriguing.

Theorem (Haemers 1995) Let Γ be a connected k-regular graph, and Y ⊆ V Γ . Let N be the number of ordered pairs of adjacent vertices of Y . Then θ min | Y | + k − θ min | Y | 2 � N � θ max | Y | + k − θ max | Y | 2 n n Moreover, if equality holds in one of the inequalities above, then Y is intriguing. org Eisfeld to association schemes 1 generalised by J¨ explored further by De Bruyn and Suzuki 2 1 Theorem 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).

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 orthogonal (elliptic, parabolic, hyperbolic) Hermitian Q − ( d , q ), Q( d , q ), Q + ( d , q ) W( d , q ) H( d , q )

eigenvalues: degree positive one negative one E + E − eigenspaces: � j � interesting cases: – { 1 } -designs { 2 } -designs a { 2 } -design: points in a generator π (dimension d ) χ π + q d − 1 � q d − 1 � χ π A = q − 1 j The χ π span all of � j � ⊕ E + .

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 = ( χ D E i )( χ A E i ) ⊤ = χ D E i χ ⊤ A = χ D ( I − 1 n J ) χ ⊤ A = χ D χ ⊤ n χ D J χ ⊤ A − 1 A = | D ∩ A | − 1 n | D || A |

{ 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 χ S A = h 1 · χ S + h 2 · ( j − χ S ) Theorem An m-ovoid and an i-tight set meet in m · i elements.

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 χ D with x ∈ R n (or C n ). T -design xE i = 0, ∀ i ∈ T ⇒ x · y = ( x · j )( y · j ) Roos’ Theorem T ∩ U = 0 = n

Some applications Vanhove (2009) A partial spread of H(2 d − 1 , q 2 ), d odd, has size at most q d + 1. De Bruyn & Vanhove (2013) No generalised hexagon of order ( s , s 3 ), s � 2, can have 1-ovoids. B., De Beule, Ihringer (2017) No ovoids of H(2 r − 1 , q 2 ) can exist for r > q 3 − q 2 + 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?

other interesting designs and antidesigns, and connections m -systems of polar spaces distance- j -ovoids of generalised hexagons SPG-reguli pseudo-ovoids of PG (4 m + 3 , q ) Cameron-Liebler line classes of PG (3 , q ) Erd˝ os-Ko-Rado sets partial quadrangles cometric association schemes

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

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 m � − 3+ √ 9+4 q r W(2 r − 1 , q ), r > 2, q odd , (B., Kelly, Law, Penttila) 2 q − 2 Problem Improve the lower bound on m , when q is odd and r > 2. Conjecture? m � q r − 2

hermitian spaces H(3 , q 2 ) many H(4 , q 2 ) none known! H(5 , q 2 ) various H(6 , q 2 ) none known! conjecture For all 1 � m � q 3 , there are no m -ovoids of H(4 , q 2 ). known Thas 1981: no examples for m = 1 B., Kelly, Law, Penttila 2007: m < √ q . B., Devillers, Schillewaert 2012: m < − 3 q − 3+ √ 4 q 5 − 4 q 4 +5 q 2 − 2 q +1 2( q 2 − q − 2)

orthogonal 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?

non-embeddable GQ DH(4 , q 2 ): 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) T 3 ( O ), q even: don’t exist T 2 ( O ), q even: ?

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 → H(2 n + 1 , q 2 )) (e.g., W(2 n + 1 , q ) ֒

B., Kelly, Law, Penttila 2007: no irreducible i -tight sets in W(2 r − 1 , q ), H(2 r − 1 , q 2 ), or Q + (2 r − 1 , q ), for i small compared to q De Beule-Govaerts-Hallez-Storme 2009: H(2 n + 1 , q ) √ i < q 5 / 8 / 2 + 1 = ⇒ reducible or Baer. Beukemann & Metsch 2013: Q + (2 n + 1 , q ) 1 � n � 3 and i � q , or n � 4 and i < q and q � 71 = ⇒ reducible.

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(2 n + 1 , q ) i < ( q 2 / 3 − 1) / 2 = ⇒ reducible or Baer.

reducibility sometimes an irreducible i -tight set is the complement of a reducible i ′ -tight set. S 7 acting on H(3 , 3 2 ) orbit 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.

other things m -ovoids of regular near polygons (Jesse Lansdown’s talk) relative m -ovoids of polar spaces (Francesco Pavese’s talk) i -tight sets of hyperbolic quadrics (Sasha Gavrilyuk’s talk) research directions interplay of tight sets with m -ovoids to prove non-existence results constructions of examples re-contextualisation of old results