Non-commutative association schemes of rank 6 M. Muzychuk (joint - - PowerPoint PPT Presentation

Non-commutative association schemes of rank 6

• M. Muzychuk (joint work with A. Herman and B. Xu),

Netanya Academic College, Israel

, October, 2016, Pilsen, Czech Republick

Non-commutative association schemes of rank 6

• 1. Y. Asaba and A. Hanaki, A construction of integral standard

generalized table algebras from parameters of projective geometries, Israel

• J. Math., 194, (2013), 395-408.
• 2. A. Hanaki and P.-H. Zieschang, on imprimitive noncommutative

association schemes of order 6, Comm. Algebra, 42 (3), (2014), 1151-1199.

• 3. M. Yoshikawa, On noncommutative integral standard table algebras in

dimension 6, Comm. Algebra, 42 (2014), 2046-2060.

• 4. B. Drabkin and C. French, On a class of noncommutative imprimitive

association schemes of rank 6, Comm. Algebra, 43 (9), (2015), 4008-4041.

• 5. C. French and P.-H. Zieschang, On the normal structure of

noncommutative association schemes of rank 6, Comm. Algebra, 44 (3), 2016, 1143-1170.

Notation

Notation

If R, S ⊆ X 2 are binary relations on a finite set X, then

1 R(x) := {y ∈ X | (x, y) ∈ R}; 2 Rt := {(x, y) ∈ X 2 | (y, x) ∈ R} 3 RS is the relational product of R and S

Notation

If R, S ⊆ X 2 are binary relations on a finite set X, then

1 R(x) := {y ∈ X | (x, y) ∈ R}; 2 Rt := {(x, y) ∈ X 2 | (y, x) ∈ R} 3 RS is the relational product of R and S

If F is a field, then

1 MX(F) is the matrix algebra; 2 IX is the identity matrix; 3 JX is all one matrix; 4 ⊤ is is matrix transposition; 5 Y is the characteristic vector of Y ⊆ X.

Association schemes

Association schemes

Definition A pair X = (X, R = {R0, ..., Rd}) is called an association scheme iff

Association schemes

Definition A pair X = (X, R = {R0, ..., Rd}) is called an association scheme iff

1 R is a partition of X 2 and R0 = {(x, x) | x ∈ X};

Association schemes

Definition A pair X = (X, R = {R0, ..., Rd}) is called an association scheme iff

1 R is a partition of X 2 and R0 = {(x, x) | x ∈ X}; 2 ∀i∈{0,...,d} ∃i′∈{0,...,d} s.t. Rt i = Ri′;

Association schemes

Definition A pair X = (X, R = {R0, ..., Rd}) is called an association scheme iff

1 R is a partition of X 2 and R0 = {(x, x) | x ∈ X}; 2 ∀i∈{0,...,d} ∃i′∈{0,...,d} s.t. Rt i = Ri′; 3 for any triple i, j, k ∈ {0, ..., d} and any pair (x, y) ∈ Rk the

intersection number pk

ij := |Ri(x) ∩ Rj′(y)| depends only on

i, j, k.

Association schemes

Definition A pair X = (X, R = {R0, ..., Rd}) is called an association scheme iff

1 R is a partition of X 2 and R0 = {(x, x) | x ∈ X}; 2 ∀i∈{0,...,d} ∃i′∈{0,...,d} s.t. Rt i = Ri′; 3 for any triple i, j, k ∈ {0, ..., d} and any pair (x, y) ∈ Rk the

intersection number pk

ij := |Ri(x) ∩ Rj′(y)| depends only on

i, j, k.

1 (X, Ri) - basic (di)graphs of X; 2 |X| - the order of X; 3 |R| - the rank of X; 4 vi := p0 ii′ - the valency of (X, Ri).

Adjacency (BM-) algebra of a scheme

Theorem Let Ai be the adjacency matrix of the basic graph (X, Ri). Then the linear span AF := A0, ..., Ad is a subalgebra of the matrix algebra MX(F). Moreover IX, JX ∈ AF, A⊤

F = AF and

AiAj = d

k=0 pk ijAk.

AF is called the adjacency / Bose-Mesner algebra of X. The basis A0, ..., Ad is called the standard basis of AF.

Adjacency (BM-) algebra of a scheme

Theorem Let Ai be the adjacency matrix of the basic graph (X, Ri). Then the linear span AF := A0, ..., Ad is a subalgebra of the matrix algebra MX(F). Moreover IX, JX ∈ AF, A⊤

F = AF and

AiAj = d

k=0 pk ijAk.

AF is called the adjacency / Bose-Mesner algebra of X. The basis A0, ..., Ad is called the standard basis of AF. A scheme is called commutative if it’s BM-algebra is commutative.

Adjacency (BM-) algebra of a scheme

Theorem Let Ai be the adjacency matrix of the basic graph (X, Ri). Then the linear span AF := A0, ..., Ad is a subalgebra of the matrix algebra MX(F). Moreover IX, JX ∈ AF, A⊤

F = AF and

AiAj = d

k=0 pk ijAk.

AF is called the adjacency / Bose-Mesner algebra of X. The basis A0, ..., Ad is called the standard basis of AF. A scheme is called commutative if it’s BM-algebra is commutative. A scheme is called symmetric (antisymmetric) if all it’s non-reflexive relations are symmetric (antisymmetric. resp.).

Adjacency (BM-) algebra of a scheme

Theorem Let Ai be the adjacency matrix of the basic graph (X, Ri). Then the linear span AF := A0, ..., Ad is a subalgebra of the matrix algebra MX(F). Moreover IX, JX ∈ AF, A⊤

F = AF and

AiAj = d

k=0 pk ijAk.

AF is called the adjacency / Bose-Mesner algebra of X. The basis A0, ..., Ad is called the standard basis of AF. A scheme is called commutative if it’s BM-algebra is commutative. A scheme is called symmetric (antisymmetric) if all it’s non-reflexive relations are symmetric (antisymmetric. resp.). Proposition A symmetric scheme is commutative.

Adjacency (BM-) algebra of a scheme

Theorem Let Ai be the adjacency matrix of the basic graph (X, Ri). Then the linear span AF := A0, ..., Ad is a subalgebra of the matrix algebra MX(F). Moreover IX, JX ∈ AF, A⊤

F = AF and

AiAj = d

k=0 pk ijAk.

AF is called the adjacency / Bose-Mesner algebra of X. The basis A0, ..., Ad is called the standard basis of AF. A scheme is called commutative if it’s BM-algebra is commutative. A scheme is called symmetric (antisymmetric) if all it’s non-reflexive relations are symmetric (antisymmetric. resp.). Proposition A symmetric scheme is commutative. Conjecture (Evdokimov - Ponomarenko) Primitive antisymmetric scheme is commutative.

Imprimitive association schemes

Definition The scheme is imprimitive if there exists a non-reflexive basic graph which is not strongly connected.

Imprimitive association schemes

Definition The scheme is imprimitive if there exists a non-reflexive basic graph which is not strongly connected. Proposition Let X = (X, R = {Ri}d

i=0) be an association scheme and

AF = A0, ..., Ad its BM-algebra, char(F) = 0. The following conditions are equivalent (a) X is imprimitive; (b) ∃ I ⊂ {0, ..., d} s.t. |I| > 1 and

i∈I Ri is an equivalence

relation on X; (c) ∃ I ⊂ {0, ..., d} s.t. I ′ = I and Aii∈I is a subalgebra of AF, char(F) = 0. The subset {Ri}i∈I is called a closed subset of R.

A concrete example

A(X) =                 1 2 2 1 3 4 5 5 4 1 1 2 2 4 3 4 5 5 2 1 1 2 5 4 3 4 5 2 2 1 1 5 5 4 3 4 1 2 2 1 4 5 5 4 3 3 5 4 4 5 2 1 1 2 5 3 5 4 4 2 2 1 1 4 5 3 5 4 1 2 2 1 4 4 5 3 5 1 1 2 2 5 4 4 5 3 2 1 1 2                

Main sources of AS

Main sources of AS

1 group theory;

Main sources of AS

1 group theory; 2 merging of classes;

Main sources of AS

1 group theory; 2 merging of classes; 3 finite geometry and design theory;

Main sources of AS

1 group theory; 2 merging of classes; 3 finite geometry and design theory; 4 the others.

Schemes coming from groups

Let G = Hg0H ∪ ... ∪ HgdH be a double coset decomposition of a finite group G (g0 = e) w.r. to a proper subgroup H. On the set of left cosets G/H = {gH | g ∈ G} define relations Ri, i = 0, ..., d via (g1H, g2H) ∈ Ri ⇐ ⇒ Hg −1

1 g2H = HgiH.

Schemes coming from groups

Let G = Hg0H ∪ ... ∪ HgdH be a double coset decomposition of a finite group G (g0 = e) w.r. to a proper subgroup H. On the set of left cosets G/H = {gH | g ∈ G} define relations Ri, i = 0, ..., d via (g1H, g2H) ∈ Ri ⇐ ⇒ Hg −1

1 g2H = HgiH.

Proposition The set of relations Ri, i = 0, ..., d form an association scheme on the set G/H. Association schemes of this type are called Schurian. The BM-algebra of this scheme is known as the Hecke algebra of the pair (G, H).

Schemes coming from groups

Let G = Hg0H ∪ ... ∪ HgdH be a double coset decomposition of a finite group G (g0 = e) w.r. to a proper subgroup H. On the set of left cosets G/H = {gH | g ∈ G} define relations Ri, i = 0, ..., d via (g1H, g2H) ∈ Ri ⇐ ⇒ Hg −1

1 g2H = HgiH.

Proposition The set of relations Ri, i = 0, ..., d form an association scheme on the set G/H. Association schemes of this type are called Schurian. The BM-algebra of this scheme is known as the Hecke algebra of the pair (G, H). Example If H = {e}, then the relations Ri are permutations of G which form a regular permutation group on G isomorphic to G. All basic relations of this scheme are thin (have valency one). The BM-algebra of this scheme is isomorphic to F[G].

Class merging (fusion and fission schemes)

Definition Let X = (X, R = {Ri}d

i=0) and X′ = (X, R′ = {R′ i }d′ i=0) be two

association schemes with the same point set X. We say that X′ is a fusion of X (or X is a fission of X′) iff each R′

i is a union of some

Rj.

Class merging (fusion and fission schemes)

Definition Let X = (X, R = {Ri}d

i=0) and X′ = (X, R′ = {R′ i }d′ i=0) be two

association schemes with the same point set X. We say that X′ is a fusion of X (or X is a fission of X′) iff each R′

i is a union of some

Rj. Proposition X′ = (X, R′ = {R′

i }d′ i=0) is a fusion of X = (X, R = {Ri}d i=0) iff

there exists a partition T0, ..., Td′ of {0, 1, ..., d} such that

1 T0 = {0}; 2 ∀i ∃ j T ′ i = Tj; 3 ∀i R′ i = j∈Ti Ri.

Flag scheme of a projective plane

Let Π = (P, L) be a projective plane of order n. Denote by F the set of flags (p, ℓ) of the plane Π. Define two relations on F as following S := {((p1, ℓ1), (p2, ℓ2)) | p1 = p2, ℓ1 = ℓ2}, T := {((p1, ℓ1), (p2, ℓ2)) | P1 = p2, ℓ1 = ℓ2}. Then the relations 1F, S, T, ST, TS, TST form an association scheme of rank 6 on F called the flag scheme of a projective plane.

Flag scheme of a projective plane

Let Π = (P, L) be a projective plane of order n. Denote by F the set of flags (p, ℓ) of the plane Π. Define two relations on F as following S := {((p1, ℓ1), (p2, ℓ2)) | p1 = p2, ℓ1 = ℓ2}, T := {((p1, ℓ1), (p2, ℓ2)) | P1 = p2, ℓ1 = ℓ2}. Then the relations 1F, S, T, ST, TS, TST form an association scheme of rank 6 on F called the flag scheme of a projective plane. The flag scheme is non-commutative and imprimitive.

Flag scheme of a projective plane

Let Π = (P, L) be a projective plane of order n. Denote by F the set of flags (p, ℓ) of the plane Π. Define two relations on F as following S := {((p1, ℓ1), (p2, ℓ2)) | p1 = p2, ℓ1 = ℓ2}, T := {((p1, ℓ1), (p2, ℓ2)) | P1 = p2, ℓ1 = ℓ2}. Then the relations 1F, S, T, ST, TS, TST form an association scheme of rank 6 on F called the flag scheme of a projective plane. The flag scheme is non-commutative and imprimitive. The flag scheme is Schurian iff the plane is Desarguesian.

Flag scheme of a projective plane

Let Π = (P, L) be a projective plane of order n. Denote by F the set of flags (p, ℓ) of the plane Π. Define two relations on F as following S := {((p1, ℓ1), (p2, ℓ2)) | p1 = p2, ℓ1 = ℓ2}, T := {((p1, ℓ1), (p2, ℓ2)) | P1 = p2, ℓ1 = ℓ2}. Then the relations 1F, S, T, ST, TS, TST form an association scheme of rank 6 on F called the flag scheme of a projective plane. The flag scheme is non-commutative and imprimitive. The flag scheme is Schurian iff the plane is Desarguesian. The flag scheme has a rank 4 fusion: {1F, S ∪ T, ST ∪ TS, STS}.

AS of small rank

AS of small rank

Proposition A rank two scheme on a point set X is trivial: IX, X 2 \ IX.

AS of small rank

Proposition A rank two scheme on a point set X is trivial: IX, X 2 \ IX. Schemes of rank three. X = (X, {R0, R1, R2}) with 1′ = 2, 2′ = 1 (antisymmetric case) or 1′ = 1, 2′ = 2 (symmetric case).

AS of small rank

Proposition A rank two scheme on a point set X is trivial: IX, X 2 \ IX. Schemes of rank three. X = (X, {R0, R1, R2}) with 1′ = 2, 2′ = 1 (antisymmetric case) or 1′ = 1, 2′ = 2 (symmetric case). In the first case the parameters are completely determined by the degree |X|. The basic graphs form a pair of doubly regular tournaments.

AS of small rank

Proposition A rank two scheme on a point set X is trivial: IX, X 2 \ IX. Schemes of rank three. X = (X, {R0, R1, R2}) with 1′ = 2, 2′ = 1 (antisymmetric case) or 1′ = 1, 2′ = 2 (symmetric case). In the first case the parameters are completely determined by the degree |X|. The basic graphs form a pair of doubly regular tournaments. In the second case the basic graphs form a complementary pair of strongly regular graphs. The parameters are completely determined by p0

11, p1 11, p2 11.

BM-algebra of an association scheme.

Theorem (Weisfeiler-Leman, Higman) Let X = (X, R) be a scheme. It’s BM-algebra F[R] is semisimple if char(F) = 0. If, in addition, F is algebraically closed, then F[R] ∼ = ⊕k

i=0Mmi(F), with m0 = 1.

In particular, |R| = k

i=1 m2 i .

BM-algebra of an association scheme.

Theorem (Weisfeiler-Leman, Higman) Let X = (X, R) be a scheme. It’s BM-algebra F[R] is semisimple if char(F) = 0. If, in addition, F is algebraically closed, then F[R] ∼ = ⊕k

i=0Mmi(F), with m0 = 1.

In particular, |R| = k

i=1 m2 i .

Theorem (W-L, H) A scheme of rank less than 6 is commutative.

BM-algebra of an association scheme.

Theorem (Weisfeiler-Leman, Higman) Let X = (X, R) be a scheme. It’s BM-algebra F[R] is semisimple if char(F) = 0. If, in addition, F is algebraically closed, then F[R] ∼ = ⊕k

i=0Mmi(F), with m0 = 1.

In particular, |R| = k

i=1 m2 i .

Theorem (W-L, H) A scheme of rank less than 6 is commutative. Corollary A BM-algebra of a non-commutative rank six scheme over algebraically closed field F of characteristic zero is isomorphic to F ⊕ F ⊕ M2(F).

Reality based algebras (H. Blau)

An associative complex algebra A with a distinguished basis B = {b0, ..., bd} is called a reality-based algebra (RBA) if it satisfies the following conditions

Reality based algebras (H. Blau)

An associative complex algebra A with a distinguished basis B = {b0, ..., bd} is called a reality-based algebra (RBA) if it satisfies the following conditions

1 There exists an involutive anti-automorphism ∗ of A such that

B∗ = B;

Reality based algebras (H. Blau)

An associative complex algebra A with a distinguished basis B = {b0, ..., bd} is called a reality-based algebra (RBA) if it satisfies the following conditions

1 There exists an involutive anti-automorphism ∗ of A such that

B∗ = B;

2 B2 ⊆ RB, i.e. bibj = d k=0 pk ijbk with pk ij ∈ R;

Reality based algebras (H. Blau)

An associative complex algebra A with a distinguished basis B = {b0, ..., bd} is called a reality-based algebra (RBA) if it satisfies the following conditions

1 There exists an involutive anti-automorphism ∗ of A such that

B∗ = B;

2 B2 ⊆ RB, i.e. bibj = d k=0 pk ijbk with pk ij ∈ R; 3 b0 is the identity of A;

Reality based algebras (H. Blau)

An associative complex algebra A with a distinguished basis B = {b0, ..., bd} is called a reality-based algebra (RBA) if it satisfies the following conditions

1 There exists an involutive anti-automorphism ∗ of A such that

B∗ = B;

2 B2 ⊆ RB, i.e. bibj = d k=0 pk ijbk with pk ij ∈ R; 3 b0 is the identity of A; 4 p0 ij = 0 ⇐

⇒ j = i∗ and p0

ii∗ > 0.

Reality based algebras (H. Blau)

An associative complex algebra A with a distinguished basis B = {b0, ..., bd} is called a reality-based algebra (RBA) if it satisfies the following conditions

1 There exists an involutive anti-automorphism ∗ of A such that

B∗ = B;

2 B2 ⊆ RB, i.e. bibj = d k=0 pk ijbk with pk ij ∈ R; 3 b0 is the identity of A; 4 p0 ij = 0 ⇐

⇒ j = i∗ and p0

ii∗ > 0.

A degree map δ : A → C is an algebra homomorphism which satisfy δ(bi) = δ(bi∗) > 0. Proposition Each RBA algebra has at most one degree map.

Table algebras

A RBA (A, B) is called a (generalized) table algebra if pk

ij ≥ 0 for

all i, j, k.

Table algebras

A RBA (A, B) is called a (generalized) table algebra if pk

ij ≥ 0 for

all i, j, k. Theorem Every table algebra has a degree map δ : A → R>0.

Table algebras

A RBA (A, B) is called a (generalized) table algebra if pk

ij ≥ 0 for

all i, j, k. Theorem Every table algebra has a degree map δ : A → R>0. Definition A table algebra is integral iff its structure constants and degrees are rational integers.

Table algebras

A RBA (A, B) is called a (generalized) table algebra if pk

ij ≥ 0 for

all i, j, k. Theorem Every table algebra has a degree map δ : A → R>0. Definition A table algebra is integral iff its structure constants and degrees are rational integers. Definition A table algebra is called standard if δ(bi) = p0

ii∗.

Each complex BM-algebra provides an example of a standard integral table algebra.

Representation Theory

X = (X, R = {R0, ..., R5}) is a non-commutative rank 6 scheme; A = A0, ..., A5 is a BM-algebra of X; δi is the valency of Ri; n =

i δi is the degree of X;

τ(

i xiAi) = x0n is the standard character of A.

Representation Theory

X = (X, R = {R0, ..., R5}) is a non-commutative rank 6 scheme; A = A0, ..., A5 is a BM-algebra of X; δi is the valency of Ri; n =

i δi is the degree of X;

τ(

i xiAi) = x0n is the standard character of A.

Theorem

1 RB ∼

= R ⊕ R ⊕ M2(R);

2 (x, y, Z)t = (x, y, Z ⊤); 3 the relations R1, R2, R3 are symmetric while Rt 4 = R5, Rt 5 = R4

An isomorphism has a form a → (δ(a), φ(a), B(a)) where δ, φ and B are irreducible real representations of A. Their dimensions are 1, 1, 2.

Irreducible characters

Irr(A) = {δ, φ, χ} where χ(a) := tr(B(a)), a ∈ A. Decomposition of the standard character τ: τ = δ + mφφ + mχχ, where mφ and mχ are the multiplicities. Abbreviation δi := δ(Ai), φi := φ(Ai), χi = χ(Ai), Bi := B(Ai).

The character table A0 A1 A2 A3 A4 A5 δ 1 δ1 δ2 δ3 δ4 δ5 = δ4 φ 1 φ1 φ2 φ3 φ4 φ5 = φ4 χ 2 χ1 χ2 χ3 χ4 χ5 = χ4

Orthogonality relations

Orthogonality relations

Rows orthogonality 1 + δ1 + δ2 + δ3 + 2δ4 = n 1 + φ2

1

δ1 + φ2

2

δ2 + φ2

3

δ3 + 2 φ2

4

δ4

=

n mφ

1 + χ2

1

δ1 + χ2

2

δ2 + χ2

3

δ3 + 2 χ2

4

δ4

=

n mχ

1 + φ1 + φ2 + φ3 + 2φ4 = 1 + χ1 + χ2 + χ3 + 2χ4 = 1 + φ1χ1

δ1

+ φ2χ2

δ2

+ φ3χ3

δ3

+ 2 φ4χ4

δ4

=

Orthogonality relations

Rows orthogonality 1 + δ1 + δ2 + δ3 + 2δ4 = n 1 + φ2

1

δ1 + φ2

2

δ2 + φ2

3

δ3 + 2 φ2

4

δ4

=

n mφ

1 + χ2

1

δ1 + χ2

2

δ2 + χ2

3

δ3 + 2 χ2

4

δ4

=

n mχ

1 + φ1 + φ2 + φ3 + 2φ4 = 1 + χ1 + χ2 + χ3 + 2χ4 = 1 + φ1χ1

δ1

+ φ2χ2

δ2

+ φ3χ3

δ3

+ 2 φ4χ4

δ4

= Columns orthogonality 1 + mφ + 2mχ = n; ∀i=1,...,5 1 + mφφi + mχχi =

Necessary conditions

Proposition The numbers δi, φi, χi, i = 1, ...4 are integers and

1 ∀i δi > 0; 2 |φi| ≤ δi, |χi| ≤ 2δi.

Proposition The set {φi/δi}4

i=1 contains at least two numbers. If it contains

exactly two numbers, then the scheme has a symmetric rank three fusion scheme whose BM-algebra is the center of A.

Main Results

Main Results

Theorem A Let δ1, ..., δ4 and φ1, ..., φ4 be arbitrary 4-tuples of real numbers satisfying δi > 0, i = 1, ..., 4, 1 + φ1 + φ2 + φ3 + 2φ4 = 0 and φ3/δ3 = φ4/δ4. Then there exist matrices B0 = I2, B1, ..., B4 ∈ M2(R) s.t. the algebra A := R ⊕ R ⊕ M2(R) becomes a RBA w.r.t. a basis bi = (δi, φi, Bi), i = 0, ..., 4; b5 = (δ4, φ4, B⊤

4 ). The basis is unique

up to an orthogonal conjugation. The mapping bi → δi is a degree map of a RBA (A, B).

Main Results

Theorem A Let δ1, ..., δ4 and φ1, ..., φ4 be arbitrary 4-tuples of real numbers satisfying δi > 0, i = 1, ..., 4, 1 + φ1 + φ2 + φ3 + 2φ4 = 0 and φ3/δ3 = φ4/δ4. Then there exist matrices B0 = I2, B1, ..., B4 ∈ M2(R) s.t. the algebra A := R ⊕ R ⊕ M2(R) becomes a RBA w.r.t. a basis bi = (δi, φi, Bi), i = 0, ..., 4; b5 = (δ4, φ4, B⊤

4 ). The basis is unique

up to an orthogonal conjugation. The mapping bi → δi is a degree map of a RBA (A, B). Theorem B The structure constants are uniquely determined by the numbers δ1, ..., δ4, φ1, ..., φ4 and they belong either to the field F := Q[δ1, ..., δ4, φ1, ..., φ3] or to a quadratic extension of F.

Outline of the proof

Input: positive reals δ1, ..., δ4 and reals φ1, φ2, φ3.

Outline of the proof

Input: positive reals δ1, ..., δ4 and reals φ1, φ2, φ3. Output: Distinguished basis B = {bi}5

i=0 of A = R ⊕ R ⊕ M2(R).

Outline of the proof

Input: positive reals δ1, ..., δ4 and reals φ1, φ2, φ3. Output: Distinguished basis B = {bi}5

i=0 of A = R ⊕ R ⊕ M2(R).

Step 1. Computation of character table. φ4 := −(1 + φ1 + φ2 + φ3)/2; n := 1 + δ1 + δ2 + δ3 + 2δ4; mφ =

n 1+

φ2 1 δ1 + φ2 2 δ2 + φ2 2 δ3 +2 φ2 4 δ4

; mχ := (n − 1 − mφ)/2; χi = − δi+mφφi

mχ

, i = 1, ..., 4

Outline of the proof

Step 2. Computation of matrices B0, B1, ..., B5

Outline of the proof

Step 2. Computation of matrices B0, B1, ..., B5 B0 = I2, B1, B2, B3 are symmetric and B5 = B⊤

4 ;

• i Bi

= O2;

• i

φi δi Bi

= O2; Bi, Bj = δinδij − δiδj − mφφiφj Using the above expressions one can express B3 and B4 + B5 as linear combinations of B0, B1, B2.

Outline of the proof

Step 2. Computation of matrices B0, B1, ..., B5 B0 = I2, B1, B2, B3 are symmetric and B5 = B⊤

4 ;

• i Bi

= O2;

• i

φi δi Bi

= O2; Bi, Bj = δinδij − δiδj − mφφiφj Using the above expressions one can express B3 and B4 + B5 as linear combinations of B0, B1, B2. The matrix B4 − B5 is skew-symmetric and satisfies B4 − B5, B4 − B5 = 2nδ4/mχ. Hence B4 − B5 =  

• nδ4

mχ

−

• nδ4

mχ

  It is enough to find B1 and B2

Outline of the proof

B1 and B2 are symmetric = ⇒ B1 = r1 s1

• ,

B2 = r2 u2 u2 s2

• The matrices satisfy the following equations

δ1 + mφφ1 + mχ(r1 + s1) = δ2

1 + mφφ2 1 + mχ(r2 1 + s2 1)

= nδ1 δ2 + mφφ2 + mχ(r2 + s2) = δ1δ2 + mφφ1φ2 + mχ(r1r2 + s1s2) = δ2

2 + mφφ2 2 + mχ(r2 2 + s2 2 + 2u2 2)

= nδ2 The structure constants are computed by the formula pk

ij =

1 nδk

• δiδjδk + mφφiφjφk + mχtr(BiBjB⊤

k )

Enumeration results

We have enumerated all feasible parameters of standard integral table algebras up to order 150. Among them 5 primitive table algebras were found N n δ, φ (mφ, mχ) TA1 81 [10, 10, 20, 20], [1, 1, −7, 2] (20, 30) TA2 96 [19, 19, 19, 19], [−5, −5, 3, 3] (19, 38) TA3 96 [19, 19, 19, 19], [−1, −1, −1, 1] (76, 19/2) TA4 96 [19, 19, 19, 19], [3, 3, 3, −5] (19, 38) TA5 120 [17, 17, 51, 17], [−3, −3, 3, 1] (51, 34)

Primitive rank six schemes

Theorem There is no primitive non-commutative rank six scheme of order ≤ 150.

• Proof. The third parameter set has non-integral mutlplicity. The

algebra TA1 has a rank four fusion with degrees 1, 20, 20, 40 and multiplicities 1, 20, 30, 30. According to van Dam classification it does not exist. The algebras TA3 and TA4 have rank three fusion with degrees 1, 38, 57. According to Brouwer’s table an SRG with such parameters doesn’t exists. TA5 violates the condition pi

ijδi ≡ 0(mod 2) whenever

i∗ = i, j∗ = j.

Open problems

Open problems

Problem Find an example of a primitive non-commutative association scheme of rank 6 (if it exists).

Open problems

Problem Find an example of a primitive non-commutative association scheme of rank 6 (if it exists). Problem Find an example of a schurian primitive non-commutative association scheme of rank 6 (if it exists).

Open problems

Problem Find an example of a primitive non-commutative association scheme of rank 6 (if it exists). Problem Find an example of a schurian primitive non-commutative association scheme of rank 6 (if it exists). Lemma (Munemasa) There is no schurian primitive association schemes of rank 6 with less than 1600 points.

Bipartite association schemes

Definition An association scheme X = (X, R = {R0, ..., Rd}) is called bipartite if there exists a partition X = X1 ∪ X2 such that (X1 × X1) ∪ (X2 × X2) =

i∈I Ri for some I ⊂ {0, ..., d}.

The vector e := X 1 − X 2 is a common eigenvector for all of the matrices Ai. More precisely, Aie = δie if Ri ⊆ X 2

1 ∪ X 2 2 and

Aie = −δie otherwise. Thus a bipartition determines a

• ne-dminesional real representation of the BM-algebra φ : A → C

via Aie = φ(Ai)e. It’s multiplicty is one. Proposition A scheme X is bipartite iff its BM-algebra has a one dimensional real non-principal representation of multiplicity one.

Semidirect product

Let X = (Y , R = {R0, R1, R2}) be a rank three scheme s.t. the linear map A0 → A0, A1 → A2, A2 → A1 is an automorphism of its BM-algebra A0, A1, A2. Theorem The matrices

• A0

O O A0

• ,
• A1

O O A2

• ,

A2 O O A1

• ,

O A0 A0 O

• ,

O A1 A2 O

• ,

O A2 A1 O

• form a standard basis of a rank 6 scheme on the set X × {1, 2}.

This scheme is bipartite and non-commutative. It is called semidirect product of X and S2, notation X ⋊ S2

Bipartite non-commutative schemes of rank six

Proposition A non-commutative rank six scheme is bipartite iff mφ = 1. Let X = (X, R = {R0, R1, R2, R3, R4, R5 = Rt

4}) be a scheme in

the title and X = X1 ∪ X2 be its bipartition. Theorem The scheme X is either a semidirect product or the following conditions hold

1 (X1 × X1) ∪ (X2 × X2) = R0 ∪ R4 ∪ R5; 2 (X1, X2, Ri ∩ X1 × X2) is a symmetric 2-design for i = 1, 2, 3; 3 the numbers δ1, δ2, δ3 determines the rest of numercial

parameters of the scheme.

Thank you!