On Pseudo-cyclic Coherent Configurations

Definitions and Preliminaries Pseudo-cyclic Schemes

On Pseudo-cyclic Coherent Configurations Reza Sharafdini Mathematics Department Faculty of Sience Persian Gulf University, Bushehr, IRAN.

First Algebraic Graph Theory Conference, Shahid Rajaee University Teacher Tracing, 20-22 October 2011, Tehran, IRAN.

Reza Sharafdini

On Pseudo-cyclic Coherent Configurations


Representation theory of schemes

Coherent Configurations (Schemes) Let V be a finite set and R a set of non-empty binary relations on V . A pair C = (V, R) is called a coherent configuration or a a scheme on V if the following conditions are satisfied: R forms a partition of the set V × V . ∆V = (v, v) | v ∈ V is a union of elements of R. for every R ∈ R, Rt := {(v, u) | (u, v) ∈ R} ∈ R.

R

for every R, S, T ∈ R, the number | w ∈ V | (u, w) ∈ R, (w, v) ∈ S |

u

>w

T

does not depend on the choice of (u, v) ∈ T and is denoted by cTRS . Reza Sharafdini


S

/v



Coherent Configurations (Schemes) Let V be a finite set and R a set of non-empty binary relations on V . A pair C = (V, R) is called a coherent configuration or a a scheme on V if the following conditions are satisfied: R forms a partition of the set V × V . ∆V = (v, v) | v ∈ V is a union of elements of R. for every R ∈ R, Rt := {(v, u) | (u, v) ∈ R} ∈ R.

R

for every R, S, T ∈ R, the number | w ∈ V | (u, w) ∈ R, (w, v) ∈ S |

u

>w

T

does not depend on the choice of (u, v) ∈ T and is denoted by cTRS . Reza Sharafdini


S

/v



Strongly Regular Graphs and Schemes

Example (Strongly Regular Graphs) Let (V, R) be a strongly regular graph with the parameters (v, k, λ, µ). Then it is known that the pair (V, {∆V , R, Rc }), where Rc := (V × V ) \ (∆V ∪ R) is a scheme for which R Rc X Rt = R, |V | = k, , c∆ RR = k, , cRR = λ, , cRR = µ.

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Fibers and basis relations Let C = (V, R) be a scheme. Since ∆V is a union of elements of R, we may define the following set: Fib(C) = {X ⊆ V | ∆X ∈ R};

∆X = {(x, x) | x ∈ X}.

Each element of the set Fib(C) is called a fiber of C. For each X, Y ∈ Fib(C) it is easy to see that X × Y is a union of elements of R. Therefore, R=

[

RX,Y

(disjoint union),

X,Y ∈Fib(C)

where for X, Y ∈ Fib(C) we set RX,Y = {R ∈ R | R ⊆ X × Y } .

Reza Sharafdini





∆X = {(x, x) | x ∈ X}.


[

RX,Y

(disjoint union),

X,Y ∈Fib(C)


Reza Sharafdini





∆X = {(x, x) | x ∈ X}.


[

RX,Y

(disjoint union),

X,Y ∈Fib(C)


Reza Sharafdini




Homogeneous Components Definition A scheme C = (V, R) is called commutative if cTRS = cTSR for all R, S, T ∈ R. Definition A scheme C is called homogeneous or (association scheme) if |Fib(C)| = 1 or equivalently, if ∆V ∈ R. It is known that commutative schemes are homogeneous ones. Lemma Given X ∈ Fib(C), CX := (X, RX,X ) is a homogeneous scheme called the homogeneous component of C corresponding to X.

Reza Sharafdini





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Reza Sharafdini




Algebraic approach to schemes (Adjacency Algebras)

Given a scheme C = (V, R), for each R ∈ R we define AR ∈ MatV (C) as follows: (AR )u,v =

1 , (u, v) ∈ R 0 , O.W.

The matrix AR is called the adjacency X matrix of R. For every R, S ∈ R, AR AS = cTRS AT .

T ∈R Therefore, A(C) := AR | R ∈ R C ≤ MatV (C) become an algebra called the adjacency algebra of C over C.

Reza Sharafdini






1 , (u, v) ∈ R 0 , O.W.



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1 , (u, v) ∈ R 0 , O.W.



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1 , (u, v) ∈ R 0 , O.W.



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Central Primitive Idempotents It is well-know that A := A(C) is semisimple algebra since it is self-adjoint, i.e., it is closed under complex conjugate transpose map. Therefore, by the Wedderburn Theorem, A is isomorphic to the direct sum of full matrix algebras. A=

M

AP ∼ =

P ∈P(C)

M

MatnP (C),

(1)

P ∈P(C)

where P(C) denotes the set of all central primitive idempotents of A. It follows that |R| =

X

n2P ,

P ∈P(C)

where mP :=

|V | =

X

mP nP .

(2)

P ∈P(C)

rank(P ) . nP Reza Sharafdini





M

AP ∼ =

P ∈P(C)

M

MatnP (C),

(1)

P ∈P(C)


X

n2P ,

P ∈P(C)

where mP :=

|V | =

X

mP nP .

(2)

P ∈P(C)






M

AP ∼ =

P ∈P(C)

M

MatnP (C),

(1)

P ∈P(C)


X

n2P ,

P ∈P(C)

where mP :=

|V | =

X

mP nP .

(2)

P ∈P(C)





Set P0 =

X

JX /|X|,

where JX =

X

AR .

R∈RX,X

X∈Fib(C)

Then P0 ∈ P(C), which is called principal. It is known that (mP0 , nP0 ) = (1, | Fib(C)|).

(3)

By P ] (C) := P(C) \ {P0 }.

Reza Sharafdini




Set P0 =

X

JX /|X|,

where JX =

X

AR .

R∈RX,X

X∈Fib(C)

Then P0 ∈ P(C), which is called principal. It is known that (mP0 , nP0 ) = (1, | Fib(C)|).

(3)

By P ] (C) := P(C) \ {P0 }.

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Embedding Theorem Below for X ∈ Fib(C) and P ∈ P(C) put PX := P IX (where IX = A∆X ) and set n o n o PX (C) = P ∈ P(C) | PX 6= 0 , Supp(P ) = X ∈ Fib(C) | PX 6= 0 . Proposition (S. Evdokimov, I. Ponomarenko 1997) Let C = (V, R) be a scheme. Then the following hold: (i) For each X ∈ Fib(C) the mapping PX (C) −→ P(CX ) (P 7→ PX ) is bijective. (ii) For all P ∈ P(C) and X ∈ Supp(P ), X nP = nPX , mP = mPX . X∈Supp(P ) Reza Sharafdini




Embedding Theorem Below for X ∈ Fib(C) and P ∈ P(C) put PX := P IX (where IX = A∆X ) and set n o n o PX (C) = P ∈ P(C) | PX 6= 0 , Supp(P ) = X ∈ Fib(C) | PX 6= 0 . Proposition (S. Evdokimov, I. Ponomarenko 1997) Let C = (V, R) be a scheme. Then the following hold: (i) For each X ∈ Fib(C) the mapping PX (C) −→ P(CX ) (P 7→ PX ) is bijective. (ii) For all P ∈ P(C) and X ∈ Supp(P ), X nP = nPX , mP = mPX . X∈Supp(P ) Reza Sharafdini



Pseudo-cyclic Schemes

Definition (Pseudo-cyclic Schemes) A scheme C (not necessarily homogeneous) is called P pseudo-cyclic if sPnm is constant for each P ∈ P ] (C) where P sP := | Supp(P )|. P = k for P ∈ P ] (C). A scheme C is called k-pseudo-cyclic if sPnm P Remark Note that, sP = 1 if C is homogenous. Thus the preceding definition generalizes the notion of pseudo-cyclic homogenous schemes.

Reza Sharafdini




Definition (Pseudo-cyclic Schemes) A scheme C (not necessarily homogeneous) is called P pseudo-cyclic if sPnm is constant for each P ∈ P ] (C) where P sP := | Supp(P )|. P A scheme C is called k-pseudo-cyclic if sPnm = k for P ∈ P ] (C). P Remark Note that, sP = 1 if C is homogenous. Thus the preceding definition generalizes the notion of pseudo-cyclic homogenous schemes.

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Definition (Pseudo-cyclic Schemes) A scheme C (not necessarily homogeneous) is called P pseudo-cyclic if sPnm is constant for each P ∈ P ] (C) where P sP := | Supp(P )|. P A scheme C is called k-pseudo-cyclic if sPnm = k for P ∈ P ] (C). P Remark Note that, sP = 1 if C is homogenous. Thus the preceding definition generalizes the notion of pseudo-cyclic homogenous schemes.

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Non-commutative pseudo-cyclic homogeneous schemes

Let G ≤ Sym(V ) be permutation group. Then it is known that (V, Orb(G, V × V ) is a scheme called the 2-orbit scheme of G on V. Theorem (M. Muzychok, I. Ponomarenko 2009) Let G ≤ Sym(V ) be a Frobenius group with a point stabilizer K. Then the 2-orbit scheme C of G is |K|-pseudo-cyclic. Moreover, C is commutative if and only if the kernel of G is Abelian.

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Non-commutative pseudo-cyclic homogeneous schemes

Let G ≤ Sym(V ) be permutation group. Then it is known that (V, Orb(G, V × V ) is a scheme called the 2-orbit scheme of G on V. Theorem (M. Muzychok, I. Ponomarenko 2009) Let G ≤ Sym(V ) be a Frobenius group with a point stabilizer K. Then the 2-orbit scheme C of G is |K|-pseudo-cyclic. Moreover, C is commutative if and only if the kernel of G is Abelian.

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1-pseudo-cyclic schemes

Theorem (S. Evdokimov, I. Ponomarenko 1997) Let C = (V, R) be a scheme. Then the following (i) and (ii) hold: (i) nP ≤ sP mP for all P ∈ P(C), (ii) nP = sP mP for all P ∈ P(C) if and only if C is quasi-regular, i.e., |X| = |RX,X | for each X ∈ Fib(C). Corollary A scheme C is 1-pseudo-cyclic if and only if C is quasi-regular.

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1-pseudo-cyclic schemes

Theorem (S. Evdokimov, I. Ponomarenko 1997) Let C = (V, R) be a scheme. Then the following (i) and (ii) hold: (i) nP ≤ sP mP for all P ∈ P(C), (ii) nP = sP mP for all P ∈ P(C) if and only if C is quasi-regular, i.e., |X| = |RX,X | for each X ∈ Fib(C). Corollary A scheme C is 1-pseudo-cyclic if and only if C is quasi-regular.

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Tensor Products

Definition (Tensor Products) Let C1 = (V1 , R1 ) and C2 = (V2 , R2 ) be schemes.

R1 ⊗ R2 = {R1 ⊗ R2 | R1 ∈ R1 , R2 ∈ R2 ,

(4)

where n o R1 ⊗ R2 = (u1 , u2 ), (v1 , v2 ) | (u1 , v1 ) ∈ R1 , (u2 , v2 ) ∈ R2 . Then C = V1 × V2 , R1 ⊗ R2 is a scheme called the tensor N product of the schemes C1 and C2 and is denoted by C1 C2 .

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Tensor Products

Example Let C be a scheme and Tt a trivial scheme on t points (i.e, A(Tt ) ∼ = Matt (C)). Then

P(C

O

n o Tt ) = P ⊗ It | P ∈ P(C) ;

nP ⊗It = tnP , mP ⊗It = mP , sP ⊗It = t. This implies that C

N

Tt is k-pseudo-cyclic if and only if C is so.

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Direct Sums Definition Let C1 = (V1 , R1 ) and C2 = (V2 , R2 ) be schemes. Put n o R := R1 ∪R2 ∪ X ×Y | X ∈ Fib(Ci ), Y ∈ Fib(Cj ), 1 ≤ i, j ≤ 2 , then C1 C2 := (V1 ∪ V2 , R) is a scheme which is called the direct sum of C1 and C2 . Example Let C be a scheme. Then C is k-pseudo-cyclic if and only if so is C C, since n o n o P ] (C C) = P ⊕ 0 | P ∈ P ] (C) ∪ 0 ⊕ P | P ∈ P ] (C) . Reza Sharafdini









Combinatorial analogue of pseudo-cyclic schemes Theorem (M. Muzychok, I. Ponomarenko 2009) Let C be a homogeneous scheme. Then the following are equivalent: 1

C = (V, R) is k-pseudo-cyclic.

2

V dR = k (dR := c∆ RRt ) for each R ∈ R \ {∆V }, and

C(R) =

X

cR SS t = k − 1.

S∈R

Problem Is it possible to have a combinatorial description of pseudo-cyclic schemes? I do not know yet. Reza Sharafdini





2


C(R) =

X

cR SS t = k − 1.

S∈R






2


C(R) =

X

cR SS t = k − 1.

S∈R




Homogenous Components Lemma Let C be a scheme. If for each X ∈ Fib(C), CX is k-pseudo-cyclic, then C is k-pseudo-cyclic. Lemma Let for each P ∈ P ] (C), and for all X, Y ∈ Supp(P ), nPX = nPY . Then, C is k-pseudo-cyclic if and only if so is CX , for each X ∈ Fib(C). Question Does it make sense to consider the assumption of the previous Lemma? Reza Sharafdini









Balanced Schemes Definition (Balanced Schemes) A scheme C = (V, R) is said to be balanced if |RX,Y | is constant for all X, Y ∈ Fib(C). Theorem Let C = (V, R) be a scheme. Then the following are equivalent: 1

C is a balanced scheme.

2

For each X ∈ Fib(C) the mapping P(C) −→ P(CX ) (P 7→ PX ) is bijective with nP = | Fib(C)|nPX .

Corollary Let C be a balanced scheme. Then, C is k-pseudo-cyclic if and only if so is CX , for each X ∈ Fib(C). Reza Sharafdini





2







2





Lemma Let C be a scheme and k a positive integer. Then the following are equivalent: 1

For each P ∈ P ] (C) and X ∈ Supp(P ), mP = knPX .

2

For each X ∈ Fib(C), CX is k-pseudo-cyclic.

Definition (Quasi-commutative schemes) A scheme C is called quasi-commutative if each homogeneous component of C is commutative.

Reza Sharafdini



Lemma Let C be a scheme and k a positive integer. Then the following are equivalent: 1

For each P ∈ P ] (C) and X ∈ Supp(P ), mP = knPX .

2

For each X ∈ Fib(C), CX is k-pseudo-cyclic.

Definition (Quasi-commutative schemes) A scheme C is called quasi-commutative if each homogeneous component of C is commutative.

Reza Sharafdini



If C is is a quasi commutative scheme, then for all X ∈ Fib(C), P P ∈ PX (C), nPX = 1. Therefore, nP = X Supp(P ) nPX = sP and we have the following: Lemma Let C be a scheme and k a positive integer. Then the following are equivalent: 1

mP = k for each P ∈ P ] (C).

2

C is quasi-commutative, and CX is k-pseudo-cyclic for each X ∈ Fib(C).

3

C is quasi-commutative and k-pseudo-cyclic.

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If C is is a quasi commutative scheme, then for all X ∈ Fib(C), P P ∈ PX (C), nPX = 1. Therefore, nP = X Supp(P ) nPX = sP and we have the following: Lemma Let C be a scheme and k a positive integer. Then the following are equivalent: 1

mP = k for each P ∈ P ] (C).

2

C is quasi-commutative, and CX is k-pseudo-cyclic for each X ∈ Fib(C).

3

C is quasi-commutative and k-pseudo-cyclic.

Reza Sharafdini



How about Schurity? A scheme C = (V, R) is said to be Schurian if R = Orb(G, V × V ) for some G ≤ Sym(V ). Proposition (M. Hirasaka 2001) Let C = (V, R) be a homogeneous scheme. If dR = 2 for each R ∈ R \ {∆V }, then C is Schurian. Specifically, C is symmetrization of an Abelian group of odd order. Corollary Each 2-pseudo-cyclic homogeneous scheme is Schurian. Proof. It is known that a homogeneous scheme C = (V, R) is V k-pseudocyclic if and only if dR = k (dR := c∆ RRt ) for each P R R ∈ R \ {∆V }, and C(R) = S∈R cSS t = k − 1. Thus, by the preceding proposition we are done. Reza Sharafdini









Problem Let C be scheme such that CX is 2-pseudocyclic, for each X ∈ Fib(C). Then C is Schurian. Thank you for your kind attention!!

Reza Sharafdini



Problem Let C be scheme such that CX is 2-pseudocyclic, for each X ∈ Fib(C). Then C is Schurian. Thank you for your kind attention!!

Reza Sharafdini
