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Butson-Hadamard matrices in association schemes of class 6 on Galois - - PowerPoint PPT Presentation

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Butson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4 Akihiro Munemasa Tohoku University (joint work with Takuya Ikuta) November 17, 2017 Combinatorics Seminar Shanghai Jiao Tong University A.

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Butson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4

Akihiro Munemasa Tohoku University (joint work with Takuya Ikuta) November 17, 2017 Combinatorics Seminar Shanghai Jiao Tong University

  • A. Munemasa

Galois rings November 17, 2017 1 / 15

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Hadamard matrices and association schemes

Goethals-Seidel (1970), regular symmetric Hadamard matrices with constant diagonal are equivalent to certain strongly regular graphs (symmetric association schemes of class 2).

  • A. Munemasa

Galois rings November 17, 2017 2 / 15

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Hadamard matrices and association schemes

Goethals-Seidel (1970), regular symmetric Hadamard matrices with constant diagonal are equivalent to certain strongly regular graphs (symmetric association schemes of class 2). From real (HH⊤ = nI) to complex (HH∗ = nI): real Hadamard (±1) ⊂ Butson-Hadamard (roots of unity) ⊂ Complex Hadamard (absolute value 1) ⊂ Inverse-orthogonal = type II

  • A. Munemasa

Galois rings November 17, 2017 2 / 15

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Hadamard matrices and association schemes

Goethals-Seidel (1970), regular symmetric Hadamard matrices with constant diagonal are equivalent to certain strongly regular graphs (symmetric association schemes of class 2). From real (HH⊤ = nI) to complex (HH∗ = nI): real Hadamard (±1) ⊂ Butson-Hadamard (roots of unity) ⊂ Complex Hadamard (absolute value 1) ⊂ Inverse-orthogonal = type II Jaeger-Matsumoto-Nomura (1998): type II matrices Chan-Godsil (2010): complex Hadamard Ikuta-Munemasa (2015): complex Hadamard

  • A. Munemasa

Galois rings November 17, 2017 2 / 15

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Complex Hadamard matrices

An n × n matrix H = (hij) is called a complex Hadamard matrix if HH∗ = nI and |hij| = 1 (∀i, j).

  • A. Munemasa

Galois rings November 17, 2017 3 / 15

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Complex Hadamard matrices

An n × n matrix H = (hij) is called a complex Hadamard matrix if HH∗ = nI and |hij| = 1 (∀i, j). It is called a Butson-Hadamard matrix if all hij are roots of unity. It is called a (real) Hadamard matrix if all hij are ±1.

  • A. Munemasa

Galois rings November 17, 2017 3 / 15

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SLIDE 7

Complex Hadamard matrices

An n × n matrix H = (hij) is called a complex Hadamard matrix if HH∗ = nI and |hij| = 1 (∀i, j). It is called a Butson-Hadamard matrix if all hij are roots of unity. It is called a (real) Hadamard matrix if all hij are ±1. The 5th workshop on Real and Complex Hadamard Matrices and Applications, July, 2017, Budapest, aimed at

  • A. Munemasa

Galois rings November 17, 2017 3 / 15

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SLIDE 8

Complex Hadamard matrices

An n × n matrix H = (hij) is called a complex Hadamard matrix if HH∗ = nI and |hij| = 1 (∀i, j). It is called a Butson-Hadamard matrix if all hij are roots of unity. It is called a (real) Hadamard matrix if all hij are ±1. The 5th workshop on Real and Complex Hadamard Matrices and Applications, July, 2017, Budapest, aimed at

1

The Hadamard conjecture: a (real) Hadamard matrix exists for every order which is a multiple of 4 (yes for order ≤ 664).

  • A. Munemasa

Galois rings November 17, 2017 3 / 15

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Complex Hadamard matrices

An n × n matrix H = (hij) is called a complex Hadamard matrix if HH∗ = nI and |hij| = 1 (∀i, j). It is called a Butson-Hadamard matrix if all hij are roots of unity. It is called a (real) Hadamard matrix if all hij are ±1. The 5th workshop on Real and Complex Hadamard Matrices and Applications, July, 2017, Budapest, aimed at

1

The Hadamard conjecture: a (real) Hadamard matrix exists for every order which is a multiple of 4 (yes for order ≤ 664).

2

Complete set of mutually unbiased bases (MUB) exists for non-prime power dimension? For example, 6.

  • A. Munemasa

Galois rings November 17, 2017 3 / 15

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SLIDE 10

Complex Hadamard matrices

An n × n matrix H = (hij) is called a complex Hadamard matrix if HH∗ = nI and |hij| = 1 (∀i, j). It is called a Butson-Hadamard matrix if all hij are roots of unity. It is called a (real) Hadamard matrix if all hij are ±1. The 5th workshop on Real and Complex Hadamard Matrices and Applications, July, 2017, Budapest, aimed at

1

The Hadamard conjecture: a (real) Hadamard matrix exists for every order which is a multiple of 4 (yes for order ≤ 664).

2

Complete set of mutually unbiased bases (MUB) exists for non-prime power dimension? For example, 6.

3

Understand the space of complex Hadamard matrices of order 6.

  • A. Munemasa

Galois rings November 17, 2017 3 / 15

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Coherent Algebras and Coherent Configuration

Let G be a finite permutation group acting on a finite set X. From the set of orbits of X × X, one defines adjacency matrices A0, A1, . . . , Ad with

d

  • i=0

Ai = J (all-one matrix).

  • A. Munemasa

Galois rings November 17, 2017 4 / 15

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SLIDE 12

Coherent Algebras and Coherent Configuration

Let G be a finite permutation group acting on a finite set X. From the set of orbits of X × X, one defines adjacency matrices A0, A1, . . . , Ad with

d

  • i=0

Ai = J (all-one matrix). Then the linear span A0, A1, . . . , Ad is closed under multiplication and transposition (→ coherent algebra, coherent configuration).

  • A. Munemasa

Galois rings November 17, 2017 4 / 15

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SLIDE 13

Coherent Algebras and Coherent Configuration

Let G be a finite permutation group acting on a finite set X. From the set of orbits of X × X, one defines adjacency matrices A0, A1, . . . , Ad with

d

  • i=0

Ai = J (all-one matrix). Then the linear span A0, A1, . . . , Ad is closed under multiplication and transposition (→ coherent algebra, coherent configuration). If G acts transitively, we may assume A0 = I (→ Bose-Mesner algebra of an association scheme).

  • A. Munemasa

Galois rings November 17, 2017 4 / 15

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SLIDE 14

Coherent Algebras and Coherent Configuration

Let G be a finite permutation group acting on a finite set X. From the set of orbits of X × X, one defines adjacency matrices A0, A1, . . . , Ad with

d

  • i=0

Ai = J (all-one matrix). Then the linear span A0, A1, . . . , Ad is closed under multiplication and transposition (→ coherent algebra, coherent configuration). If G acts transitively, we may assume A0 = I (→ Bose-Mesner algebra of an association scheme). If G contains a regular subgroup N, we may identify X with N, Ai ↔ Ti ⊆ N, and N =

d

  • i=0

Ti, T0 = {1N}, C[N] ⊇

  • g∈Ti

g | 0 ≤ i ≤ d.

  • A. Munemasa

Galois rings November 17, 2017 4 / 15

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Schur rings

N =

d

  • i=0

Ti, T0 = {1N}, C[N] ⊇ A =

  • g∈Ti

g | 0 ≤ i ≤ d (subalgebra).

  • A. Munemasa

Galois rings November 17, 2017 5 / 15

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Schur rings

N =

d

  • i=0

Ti, T0 = {1N}, C[N] ⊇ A =

  • g∈Ti

g | 0 ≤ i ≤ d (subalgebra). A is called a Schur ring if, in addition {T −1

i

| 0 ≤ i ≤ d} = {Ti | 0 ≤ i ≤ d}, where T −1 = {t−1 | t ∈ T } for T ⊆ N.

  • A. Munemasa

Galois rings November 17, 2017 5 / 15

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Schur rings

N =

d

  • i=0

Ti, T0 = {1N}, C[N] ⊇ A =

  • g∈Ti

g | 0 ≤ i ≤ d (subalgebra). A is called a Schur ring if, in addition {T −1

i

| 0 ≤ i ≤ d} = {Ti | 0 ≤ i ≤ d}, where T −1 = {t−1 | t ∈ T } for T ⊆ N. Examples: AGL(1, q) > G > N = GF (q) (cyclotomic).

  • A. Munemasa

Galois rings November 17, 2017 5 / 15

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AGL(1, q) > G > N = GF (q) (cyclotomic)

More generally, R : R× > G > N = R : a ring.

  • A. Munemasa

Galois rings November 17, 2017 6 / 15

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AGL(1, q) > G > N = GF (q) (cyclotomic)

More generally, R : R× > G > N = R : a ring. In Ito-Munemasa-Yamada (1991), we wanted to construct an association scheme with eigenvalue a multiple of i = √−1.

  • A. Munemasa

Galois rings November 17, 2017 6 / 15

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AGL(1, q) > G > N = GF (q) (cyclotomic)

More generally, R : R× > G > N = R : a ring. In Ito-Munemasa-Yamada (1991), we wanted to construct an association scheme with eigenvalue a multiple of i = √−1. Not possible with R = GF (q).

  • A. Munemasa

Galois rings November 17, 2017 6 / 15

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SLIDE 21

AGL(1, q) > G > N = GF (q) (cyclotomic)

More generally, R : R× > G > N = R : a ring. In Ito-Munemasa-Yamada (1991), we wanted to construct an association scheme with eigenvalue a multiple of i = √−1. Not possible with R = GF (q). GF (p) ֒ → GF (pe) Zpn ֒ → GR(pn, e)

  • A. Munemasa

Galois rings November 17, 2017 6 / 15

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AGL(1, q) > G > N = GF (q) (cyclotomic)

More generally, R : R× > G > N = R : a ring. In Ito-Munemasa-Yamada (1991), we wanted to construct an association scheme with eigenvalue a multiple of i = √−1. Not possible with R = GF (q). GF (p) ֒ → GF (pe) Zpn ֒ → GR(pn, e) A Galois ring R = GR(pn, e) is a commutative local ring with characteristic pn, whose quotient by the maximal ideal pR is GF (pe).

  • A. Munemasa

Galois rings November 17, 2017 6 / 15

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Structure of GR(pn, e)

Let R = GR(pn, e) be a Galois ring. Then |R| = pne, pR is the unique maximal ideal, |R×| = |R \ pR| = pne − p(n−1)e = (pe − 1)p(n−1)e, R× = T × U, T ∼ = Zpe−1, |U| = p(n−1)e. Now specialize pn = 4, consider GR(4, e).

  • A. Munemasa

Galois rings November 17, 2017 7 / 15

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Structure of GR(4, e)

Let R = GR(4, e) be a Galois ring of characteristic 4. Then |R| = 4e, 2R is the unique maximal ideal, |R×| = |R \ 2R| = 4e − 2e = (2e − 1)2e, R× = T × U, T ∼ = Z2e−1, U = 1 + 2R ∼ = Ze

2.

  • A. Munemasa

Galois rings November 17, 2017 8 / 15

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Structure of GR(4, e)

Let R = GR(4, e) be a Galois ring of characteristic 4. Then |R| = 4e, 2R is the unique maximal ideal, |R×| = |R \ 2R| = 4e − 2e = (2e − 1)2e, R× = T × U, T ∼ = Z2e−1, U = 1 + 2R ∼ = Ze

2.

To construct a Schur ring, we need to partition R = R× ∪ 2R (into even smaller parts). In Ito-Munemasa-Yamada (1991), the

  • rbits of a subgroup of the form T × U0 < R× were used.

Ma (2007) also considered orbits of a subgroup containing T .

  • A. Munemasa

Galois rings November 17, 2017 8 / 15

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U0 as a subgroup of U of index 2

R = GR(4, e), 2R is the unique maximal ideal, R× = T × U, T ∼ = Z2e−1, U = 1 + 2R ∼ = Ze

2

the principal unit group.

  • A. Munemasa

Galois rings November 17, 2017 9 / 15

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U0 as a subgroup of U of index 2

R = GR(4, e), 2R is the unique maximal ideal, R× = T × U, T ∼ = Z2e−1, U = 1 + 2R ∼ = Ze

2

the principal unit group. There is a bijection GF (2e) = R/2R ← T ∪ {0} → 2R → U, a + 2R ← a → 2a → 1 + 2a.

  • A. Munemasa

Galois rings November 17, 2017 9 / 15

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U0 as a subgroup of U of index 2

R = GR(4, e), 2R is the unique maximal ideal, R× = T × U, T ∼ = Z2e−1, U = 1 + 2R ∼ = Ze

2

the principal unit group. There is a bijection GF (2e) = R/2R ← T ∪ {0} → 2R → U, a + 2R ← a → 2a → 1 + 2a. So the “trace-0” additive subgroup of GF (2e) is mapped to P0 and U0 with |2R : P0| = |U : U0| = 2.

  • A. Munemasa

Galois rings November 17, 2017 9 / 15

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U0 as a subgroup of U of index 2

R = GR(4, e), 2R is the unique maximal ideal, R× = T × U, T ∼ = Z2e−1, U = 1 + 2R ∼ = Ze

2

the principal unit group. There is a bijection GF (2e) = R/2R ← T ∪ {0} → 2R → U, a + 2R ← a → 2a → 1 + 2a. So the “trace-0” additive subgroup of GF (2e) is mapped to P0 and U0 with |2R : P0| = |U : U0| = 2. Assume e is odd. Then 1 / ∈ “trace-0” subgroup, so 2 / ∈ P0 and −1 = 3 / ∈ U0.

  • A. Munemasa

Galois rings November 17, 2017 9 / 15

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Partition of R = GR(4, e)

Assume e is odd. Then 2 / ∈ P0, −1 / ∈ U0. R× = T × U, T ∼ = Z2e−1, 2R = P0 ∪ (2 + P0), U = U0 ∪ (−U0).

  • A. Munemasa

Galois rings November 17, 2017 10 / 15

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SLIDE 31

Partition of R = GR(4, e)

Assume e is odd. Then 2 / ∈ P0, −1 / ∈ U0. R× = T × U, T ∼ = Z2e−1, 2R = P0 ∪ (2 + P0), U = U0 ∪ (−U0). Then U0 acts on R, and the orbit decomposition is R =

t∈T

tU0 ∪ (−tU0)

  • ∪

a∈2R

{a}

  • =
  • A. Munemasa

Galois rings November 17, 2017 10 / 15

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SLIDE 32

Partition of R = GR(4, e)

Assume e is odd. Then 2 / ∈ P0, −1 / ∈ U0. R× = T × U, T ∼ = Z2e−1, 2R = P0 ∪ (2 + P0), U = U0 ∪ (−U0). Then U0 acts on R, and the orbit decomposition is R =

t∈T

tU0 ∪ (−tU0)

  • ∪

a∈2R

{a}

  • = U0 ∪ (−U0) ∪

 

  • t∈T \{1}

tU0   ∪  

  • t∈T \{1}

(−tU0)   ∪ {0} ∪ (P0 \ {0}) ∪ (2 + P0).

  • A. Munemasa

Galois rings November 17, 2017 10 / 15

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SLIDE 33

R \ {0} is partitioned into 6 parts

T0 = {0}, T1 =

t∈T \{1} tU0,

T2 =

t∈T \{1}(−tU0),

T3 = U0, T4 = −U0, T5 = P0 \ {0}, T6 = 2 + P0.

  • A. Munemasa

Galois rings November 17, 2017 11 / 15

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SLIDE 34

R \ {0} is partitioned into 6 parts

T0 = {0}, T1 =

t∈T \{1} tU0,

T2 =

t∈T \{1}(−tU0),

T3 = U0, T4 = −U0, T5 = P0 \ {0}, T6 = 2 + P0.

Theorem (Ikuta-M., 2017+)

1

{T0, T1, . . . , T6} defines a Schur ring on GR(4, e),

2

The matrices A0 + ǫ1i(A1 − A2) + ǫ2i(A3 − A4) + A5 + A6, A0 + ǫ1i(A1 − A2) + ǫ2(A3 + A4) + A5 − A6 are the only hermitian complex Hadamard matrices in its Bose-Mesner algebra, where ǫ1, ǫ2 ∈ {±1}.

  • A. Munemasa

Galois rings November 17, 2017 11 / 15

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SLIDE 35

Proof

T0 = {0}, T1 =

t∈T \{1} tU0,

T2 =

t∈T \{1}(−tU0),

T3 = U0, T4 = −U0, T5 = P0 \ {0}, T6 = 2 + P0.

  • A. Munemasa

Galois rings November 17, 2017 12 / 15

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SLIDE 36

Proof

T0 = {0}, T1 =

t∈T \{1} tU0,

T2 =

t∈T \{1}(−tU0),

T3 = U0, T4 = −U0, T5 = P0 \ {0}, T6 = 2 + P0.

Theorem (Ikuta-M., 2017+)

1

{T0, T1, . . . , T6} defines a Schur ring on GR(4, e).

  • A. Munemasa

Galois rings November 17, 2017 12 / 15

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SLIDE 37

Proof

T0 = {0}, T1 =

t∈T \{1} tU0,

T2 =

t∈T \{1}(−tU0),

T3 = U0, T4 = −U0, T5 = P0 \ {0}, T6 = 2 + P0.

Theorem (Ikuta-M., 2017+)

1

{T0, T1, . . . , T6} defines a Schur ring on GR(4, e).

Proof.

Compute the character sums (χ = χb: additive character of R)

  • α∈Tj

χ(a) =

  • α∈Tj

√ −1

tr(ab)

(b ∈ Ti), show that this is independent of b ∈ Ti, depends only on i.

  • A. Munemasa

Galois rings November 17, 2017 12 / 15

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SLIDE 38

Proof

Theorem (Ikuta-M., 2017+)

2

The matrices A0 + ǫ1i(A1 − A2) + ǫ2i(A3 − A4) + A5 + A6, A0 + ǫ1i(A1 − A2) + ǫ2(A3 + A4) + A5 − A6 are the only hermitian complex Hadamard matrices in its Bose-Mesner algebra, where ǫ1, ǫ2 ∈ {±1}.

  • A. Munemasa

Galois rings November 17, 2017 13 / 15

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SLIDE 39

Proof

Theorem (Ikuta-M., 2017+)

2

The matrices A0 + ǫ1i(A1 − A2) + ǫ2i(A3 − A4) + A5 + A6, A0 + ǫ1i(A1 − A2) + ǫ2(A3 + A4) + A5 − A6 are the only hermitian complex Hadamard matrices in its Bose-Mesner algebra, where ǫ1, ǫ2 ∈ {±1}.

Proof.

Suppose H = 6

i=0 wiAi is a hermitian complex Hadamard matrix.

Since the Bose-Mesner algebra is isomorphic to a subalgebra of the group ring of R, the relation HH∗ = nI can be translated in terms

  • f additive characters of R. Then one obtains a system of quadratic

equations in wi’s.

  • A. Munemasa

Galois rings November 17, 2017 13 / 15

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SLIDE 40

Example

H = A0 + i(A1 + A3) − i(A2 + A4) + (A5 + A6)

  • A. Munemasa

Galois rings November 17, 2017 14 / 15

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SLIDE 41

Example

H = A0 + i(A1 + A3) − i(A2 + A4) + (A5 + A6) ∈ A0, A1 + A3, A2 + A4, A5 + A6.

  • A. Munemasa

Galois rings November 17, 2017 14 / 15

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SLIDE 42

Example

H = A0 + i(A1 + A3) − i(A2 + A4) + (A5 + A6) ∈ A0, A1 + A3, A2 + A4, A5 + A6. Smaller Schur ring defined by T0 = {0}, T1 ∪ T3 =

  • t∈T

tU0, T2 ∪ T4 =

  • t∈T

(−tU0), T5 ∪ T6 = 2R \ {0}.

  • A. Munemasa

Galois rings November 17, 2017 14 / 15

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SLIDE 43

Example

H = A0 + i(A1 + A3) − i(A2 + A4) + (A5 + A6) ∈ A0, A1 + A3, A2 + A4, A5 + A6. Smaller Schur ring defined by T0 = {0}, T1 ∪ T3 =

  • t∈T

tU0, T2 ∪ T4 =

  • t∈T

(−tU0), T5 ∪ T6 = 2R \ {0}. This defines a nonsymmetric amorphous association scheme of Latin square type L2e,1(2e) in the sense of Ito-Munemasa-Yamada (1991).

  • A. Munemasa

Galois rings November 17, 2017 14 / 15

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SLIDE 44

Theorem (Ikuta-M. (2017+))

Let A0 + w1A1 + w1A⊤

1 + w3A3

be a hermitian complex Hadamard matrix contained in the Bose-Mesner algebra A = A0, A1, A2 = A⊤

1 , A3 of a 3-class

nonsymmetric association scheme. Then A is amorphous of Latin square type La,1(a), and w1 = ±i, w3 = 1.

  • A. Munemasa

Galois rings November 17, 2017 15 / 15

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SLIDE 45

Theorem (Ikuta-M. (2017+))

Let A0 + w1A1 + w1A⊤

1 + w3A3

be a hermitian complex Hadamard matrix contained in the Bose-Mesner algebra A = A0, A1, A2 = A⊤

1 , A3 of a 3-class

nonsymmetric association scheme. Then A is amorphous of Latin square type La,1(a), and w1 = ±i, w3 = 1. This can be regarded as a nonsymmetric analogue of

Theorem (Goethals-Seidel (1970))

Let H = A0 + A1 − A2 be a (real) Hadamard matrix contained in the Bose-Mesner algebra A = A0, A1, A2 of a 2-class symmetric association scheme. Then A is (amorphous) of Latin or negative Latin square type.

  • A. Munemasa

Galois rings November 17, 2017 15 / 15