Automorphisms of association schemes whose relations are of valency - - PDF document

Automorphisms of association schemes whose relations are of valency one or three

Mitsugu Hirasaka Department of Mathematics Pusan National University Joint Work with Jeong Rye Park June 2-5, 2014 Modern Trends in Algebraic Graph Theory at Villanova University, Villanova, Pennsilvania

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Notations for association schemes X: a finite set (X, {Ri}d

i=0) : an association scheme

{Ai}d

i=0 : the set of adjacency matrices

A0 = I, AT

i = Ai′,

AiAj = d

i=0 pk ijAk,

ki = p0

ii′: the valency of Ri

Ri(x) = {y ∈ X | (x, y) ∈ Ri}. X : a finite set (X, S) where S is a partition of X × X {σs | s ∈ S} : the set of adjacency matrices σ1 = I, σ∗

s = σs∗,

σsσt =

u∈S astuσu,

ns = ass∗1 : the valency of s, xs = {y ∈ X | (x, y) ∈ s}.

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In this talk we assume that S# = S \ {1}, N(S) = {ns | s ∈ S#}, and we shall write s · t =

u∈S astuu

instead of σsσt =

u∈S astuσu,

and st = {u ∈ S | astu > 0}. r : X × X → S, (x, y) → r(x, y) where (x, y) ∈ r(x, y). Aut(X, S) = {ρ ∈ Sym(X) | ∀x, y, r(x, y) = r(xρ, yρ)}. An orbital means an orbit of the induced action

• f G on X × X where G ≤ Sym(X).

We say that (X, S) is regular if S is the set of

• rbitals of a regular group.

We say that (X, S) is Frobenius if S is the set of

• rbitals of a Frobenius group.

We say that (X, S) is schurian if S is the set of

• rbitals of a transitive group.

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Known Results N(S) = {k} (called k-equivalenced).

• 1. N(S) = {1}. For example,

       

1 1 1

   ,   

1 1 1

   ,   

1 1 1

       

• Theorem. ([Zieschang, 1996])

N(S) = {1} ⇐ ⇒ (X, S) is regular.

• 2. N(S) = {2}. For example,

                      

1 1 1 1 1

       

,

       

1 1 1 1 1 1 1 1 1 1

       

,

       

1 1 1 1 1 1 1 1 1 1

                      

• Theorem. ([Muzychuk, Zieschang, 2008])

N(S) = {2} ⇒ (X, S) is Frobenius.

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• 3. N(S) = {3}. For example,

{Cay(F7, {0}), Cay(F7, {1, 2, 4}), Cay(F7, {2, 5, 6})} Theorem. (see [H, Kyoung-Tark Kim, Jeong Rye Park, 2014]) N(S) = {3} ⇒ (X, S) is Frobenius.

• 4. N(S) = {4}. For example,

H(2, 3) : the Hamming scheme

• Theorem. ([Jeong Rye Park, 2014])

N(S) = {4} ⇒ Aut(X, S) is transitive.

• 5. N(S) = {k}. For example,

Cyclotomic schemes of valency k

• Theorem. (see [Muzychuk, Ponomarenko, 2011])

Suppose N(S) = {k}, ∀s ∈ S#,

t∈S as tt∗ = k − 1.

If |S| ≥ 4(k − 1)k3 then (X, S) is Frobenius.

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• 6. N(S) = {1, 2} (called quasi-thin).

For example, The set of orbitals of the action of G

• n G/H where |H| = 2.
• Lemma. ([H, Muzychuk, 2002])

Suppose N(S) = {1, 2}. For x ∈ X, φx : X → X, y → y′ where yr(x, y) = {y, y′} is an element of Aut(X, S). Remark.(see [Hanaki, Miyamoto], and [M.Klin, M.Muzychuk, C.Pech, A.Woldar, P.-H.Zieschang] ) (1) as16-173 is a non-schurian quasi-thin scheme. (2) as28-176 is a non-schurian quasi-thin scheme.

• Theorem. ([Muzychuk, Ponomarenko, 2012])

Suppose N(S) = {1, 2}. If (X, S) is not schurian then (X, S) is a Kleinian scheme of index 4 or 7.

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Our Main Focus N(S) = {1, 3}. For example, The set of orbitals of the action of G

• n G/H where |H| = 3.

Remark.([Hanaki, Miyamoto]) (1) as27-473 is a non-schurian scheme; (2) as27-475 is a non-schurian scheme; (3) as27-477 is a non-schurian scheme. We wish to find a non-trivial automorphism with a fixed point. For the remainder of this talk we assume that (X, S) is an association scheme with N(S) = {1, 3}. We define τ : S → Z≥0 such that τ(s) = max{ass∗u | u ∈ S#} for s ∈ S. (1) For s, t ∈ S, nsnt =

u∈S astunu ∈ {1, 3, 9}

and lcm(ns, nt) | astunu; (2) τ(s) = 0 ⇐ ⇒ ns = 1 and S = τ−1(0) ∪ τ−1(1) ∪ τ−1(2) ∪ τ−1(3); (3) For s ∈ τ−1(1) ∪ τ−1(2) and t ∈ S there exists u ∈ S such that atu∗s = 1; (4) For s ∈ S, τ(s) = τ(s∗).

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Please watch a movie on the white board. Fix x ∈ X. We define Γx to be {ρ ∈ Sym(X) | ∀t ∈ S, xt is an orbit of ρ }. For ρ ∈ Γx we define a binary relation ∼ρ on S such that s ∼ρ t if and only if r(y, z) = r(yρ, zρ) for each (y, z) ∈ xs × xt. Lemma 1. For ρ ∈ Γx, ∼ρ is the trivial equiva- lence relation if and only if ρ ∈ Aut(X, S). Lemma 2. For s ∈ τ−1(1) ∪ τ−1(2) there exists ρ ∈ Γx such that s ∼ρ t for each t ∈ S and u ∼ρ v for all u, v ∈ S with nunv ≤ 3. Theorem. Suppose u ∼ρ v, u ∼ρ w, v ≁ρ w and u ∈ τ−1(1) ∪ τ−1(2) for v, w ∈ S \ τ−1(0). Then we have the following: (1) (X, S) is not commutative; (2) For s ∈ uu∗ \ {1}, τ(s) = 3.

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Sketch of the Proof Define

s∈S αss, s∈S βss := s∈S αs¯

βsns. Key Lemma. For all a, b, c ∈ S \ τ−1(0) with b = c and ab ∩ ac = ∅ we have the following: (1) If τ(a) = τ(b) = 3 then cb∗ ⊆ τ−1(0), (2) If τ(a) = 1 and b∗b = c∗c ⊆ τ−1(0) then c · b∗ = 3d for some d ∈ a∗a with τ(d) = 3. (Proof) Note that 0 < a · b, a · c = a∗ · a, c · b∗ = a∗ · a − 3 · 1, c · b∗. (1) τ(a∗) = τ(a) = 3 implies that c = t·b for some t ∈ a∗a ∩ τ−1(0). Since τ(b) = 3, cb∗ = (tb)b∗ ⊆ τ−1(0). (2) By the assumption, c · b∗, c · b∗ = c∗ · c, b∗ · b = 27. This implies that c · b∗ = 3d for some d ∈ a∗a with d ∈ τ−1(3).

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Please watch a movie on the white board. Suppose that |u∗v| = |u∗w| = |v∗w| = 3, (u∗v ∪ u∗w ∪ v∗w) ∩ τ−1(0) = ∅ u∗v = {a0, a1, a2}, u∗w = {b0, b1, b2}, v∗w = {c0, c1, c2}. By the assumption on ∼ρ, we have a∗

i · bj = c0 + c1 + c2 for all i, j ∈ {0, 1, 2};

ai · cj = b0 + b1 + b2 for all i, j ∈ {0, 1, 2}; bi · c∗

j = a0 + a1 + a2 for all i, j ∈ {0, 1, 2}.

Recall

s∈S αss, s∈S βss := s∈S αs¯

βsns. Lemma 4. We have bi · b∗

i , aj · a∗ k = a∗ j · bi, a∗ k · bi = 9, τ(ai) = 2 and

b∗

i · bi, a∗ k · aj = b∗ i · bi − 3 · 1, a∗ k · aj if j = k.

Lemma 5. For s, t ∈ S \ τ−1(0), s · s∗ = t · t∗ ⇐ ⇒ t∗ · s, t∗ · s > 9.

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Lemma 6. We have a1 · a∗

1 = a2 · a∗ 2 = a3 · a∗ 3,

b1 · b∗

1 = b2 · b∗ 2 = b3 · b∗ 3,

c1 · c∗

1 = c2 · c∗ 2 = c3 · c∗ 3,

a∗

1 · a1 = a∗ 2 · a2 = a∗ 3 · a3,

b∗

1 · b1 = b∗ 2 · b2 = b∗ 3 · b3,

c∗

1 · c1 = c∗ 2 · c2 = c∗ 3 · c3,

(Proof for the first case) Recall b∗

i · a∗ j = c0 + c1 + c2 for all i, j ∈ {0, 1, 2};

τ(b∗

i ) = 2 since b∗ i · bi − 3 · 1, a∗ k · aj = 9.

If τ(b∗

i ) = 1, then b∗ i ·bi = 3·1+d+d∗ for some d ∈

S \τ−1(0). Since a∗

k ·aj ∈ {d+2d∗, 2d+d∗, 3d, 3d∗}

for all distinct j, k, we have a∗

k · aj, a∗ k · ai > 9

for each case. Therefore, Lemma 6 follows from Lemma 5. If τ(b∗

j) = 3, then b∗ i · bi = 3 · 1 + 3t + 3t∗ for some

t ∈ τ−1(0), and hence aj ∈ {aj · t, ak · t∗}, implying aj · a∗

j = (ak · t) · (t∗ · ak) = ak · a∗ k.

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Lemma 7. We have (τ(ai), τ(bi)) ∈ {(1, 3), (3, 1)}, (Proof) By Lemma 4, 2 / ∈ {τ(ai), τ(bi), τ(bi)}. Thus, it suffices to exclude the cases (τ(ai), τ(bi)) ∈ {(1, 1), (3, 3)}. Suppose (τ(ai), τ(bi)) = (1, 1), Then aj · a∗

k ∈ {e + 2e∗, 2e + e∗}

where bi · b∗

i = 3 · 1 + e + e∗.

This implies that ai · a∗

j, ai · a∗ k = a∗ i · ai, a∗ k · aj ≥ 12,

which contradicts τ(a∗

i ) = τ(ai) = 1.

Suppose (τ(ai), τ(bi)) = (3, 3), Then t ∈ aja∗

k

where bi · b∗

i = 3 · 1 + 3t + 3t∗.

Since τ(aj) = 3 and aj = t · ak, aja∗

k ⊆ τ−1(0).

Since u∗u ∩ aja∗

k = ∅, it contradicts τ(u) ∈ {1, 2}.

Without loss of generality we may assume that (τ(ai), τ(bi)) = (1, 3). By Key Lemma, τ(d) = 3 where d ∈ u∗u ∩ b1b∗

2.

Thus, the first second statement of Theorem is proved. Suppose that (X, S) is commutative. Then d ∈ u∗u ∩ b1b∗

2 = b∗ 2b1 ⊆ τ−1(0)

since τ(bi) = τ(b∗

i ) = 3, a contradiction.

This completes the theorem.

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Main Theorem. Let (X, S) be an association scheme with N(S) = {1, 3} having u ∈ S with τ(u) ∈ {1, 2}. Then, for each x ∈ X, Aut(X, S)x is non-trivial if one of the following holds: (1) τ(s) ∈ {1, 2} for s ∈ u∗u \ {1}; (2) (X, S) is commutative. Remark. (1) Each of as27-473, as27-475 and as27-477 is commutative; (2) Now I don’t have any idea to remove the as- sumptions in Main Theorem. (3) I am looking for other type of non-schurian schemes with N(S) = {1, 3}.

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[Hanaki, Miyamoto], Classification of association schemes with small

• rders, http://kissme.shinshu-u.ac.jp/as/.

[Hirasaka, Muzychuk, 2002] Association schemes generated by a non- symmetric relation of valency 2. Algebraic and topological methods in graph theory (Lake Bled, 1999). Discrete Math. 244 (2002), no. 1-3, 109-135. [M.Hirasaka, K.Kim, J.Park, 2014] Every 3-equivalenced association scheme is Frobenous to appear in J. Alg. Comb.. [M.Klin, M.Muzychuk, C.Pech, A.Woldar, P.-H.Zieschang] Association schemes on 28 points as mergings of a half-homogeneous coherent configuration, European J. Combin. 28 (2007), no. 7, 1994-2025. [M.Muzychuk, I.Ponomarenko, 2012] On pseudocyclic association schemes, Ars Math, Cotem., 5 (2012) 125 [M.Muzychuk, I.Ponomarenko, 2012] On quasi-thin association schemes, Journal of Algebra 351 (2012) 467489. [M.Muzychuk, P.-H.Zieschan, 2008g] On association schemes all ele- ments of which have valency 1 or 2, Discrete Math. 308 (2008), no. 14, 3097-3103. [P.-H.Zieschang, 1996], An algebraic approach to association schemes, Lecture Notes in Mathematics, vol. 1628, Springer-Verlag, Berlin, 1996. [P.-H.Zieschang] Theory of association schemes, Springer Monograph in Mathematics, Springer-Verlag, Berlin, 2005. 14

Thank you for your attention.

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