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Reversible Menon-Hadamard Difference Sets in Abelian 2-groups Jordan D. Webster Mid Michigan Community College October 10, 2015 Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

Introduction A ( v , k , λ ) difference set D in a group G of order v is a k -subset of G such that each group element other than the identity appears exactly λ times in the multiset { d 1 d − 1 : d 1 , d 2 ∈ D } . 2 A Menon-Hadamard difference set has parameters (4 m 2 , 2 m 2 − m , m 2 − m ) for some m ∈ N . Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

Example Example: { x , x 3 , y , y 3 , xy , x 3 y 3 } is a (16 , 6 , 2) Menon-Hadamard difference set in C 4 × C 4 = < x , y : x 4 = y 4 = [ x , y ] = 1 > . x 3 y 3 x 3 y 3 x y xy x 3 x 2 x 3 y x 3 y 3 x 2 y 3 1 y x 2 xy 3 x 2 y y 3 1 x xy y 3 xy 3 x 3 y 3 y 2 x 3 y 2 1 x x 3 y y 2 xy 2 x 3 1 y xy x 3 y 3 y 3 x 2 y 3 x 3 x 3 y 2 x 2 y 2 1 x 2 y xy 2 x 2 y 2 1 xy y x Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

Example Example: { x , x 3 , y , y 3 , xy , x 3 y 3 } is a (16 , 6 , 2) Menon-Hadamard difference set in C 4 × C 4 = < x , y : x 4 = y 4 = [ x , y ] = 1 > . x 3 y 3 x 3 y 3 x y xy x 3 x 2 x 3 y x 3 y 3 x 2 y 3 1 y x 2 xy 3 x 2 y y 3 1 x xy y 3 xy 3 x 3 y 3 y 2 x 3 y 2 1 x x 3 y y 2 xy 2 x 3 1 y xy x 3 y 3 y 3 x 2 y 3 x 3 x 3 y 2 x 2 y 2 1 x 2 y xy 2 x 2 y 2 1 xy y x Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

Reversibility Our Example D = { x , x 3 , y , y 3 , xy , x 3 y 3 } in C 4 × C 4 Notice for each element in D , the inverse element is also in D . A difference set D is Reversible if for each d ∈ D , d − 1 ∈ D . Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

What we know Abelian non-2 groups do not have reversible difference sets Some Abelian 2 groups with reversible difference sets: C 4 , ( C 2 r ) (2) , ( C 2 2 r ) (3) Direct products of groups that contain reversible difference sets have reversible difference sets. Some Abelian 2 groups don’t have reversible difference sets. C 8 × C 2 , C 64 × ( C 16 ) (2) . Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

What we don’t/didn’t know Odd powers on groups with C 2 : Example ( C 32 ) (3) × C 2 Odd powers on groups without C 2 : Example ( C 128 ) (3) × ( C 32 ) (3) Single cyclic group in direct product with no match Example ( C 32 ) (2) × C 16 Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

What we don’t/didn’t know Odd powers on groups with C 2 : Example ( C 32 ) (3) × C 2 Odd powers on groups without C 2 : Example ( C 128 ) (3) × ( C 32 ) (3) Single cyclic group in direct product with no match Example ( C 32 ) (2) × C 16 Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

This Talk Build a difference set in ( C 8 ) (3) × C 2 Convince that the pattern extends to ( C 2 2 r +1 ) (3) × C 2 Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

The Group Ring The group ring C [ G ]. � This is the ring of all formal sums of the form a g g where g ∈ G a g ∈ C . Addition in this ring is defined pointwise. (4 g 1 + 5 g 2 ) + ( − 2 g 1 + 8 g 2 ) = 2 g 1 + 13 g 2 Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

The Group Ring Multiplication is done with consideration of both the multiplication in C and by the “multiplication” of the group G . (4 g 1 + 5 g 2 )(2 g 3 ) = 8 g 1 g 3 + 10 g 2 g 3 Multiplication distributes over addition. Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

D in the Group Ring � Difference set in the ring is D = d . d ∈ D � Also Notation of D ( − 1) = d − 1 d ∈ D DD ( − 1) = ( k − λ ) + λ ( G ) Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

D in the Group Ring � Difference set in the ring is D = d . d ∈ D If D is reversible then D = D ( − 1) DD = ( k − λ ) + λ ( G ) Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

Transformation We wish to substitute the difference set for something else. The Hadamard transform: � D = G − 2 D . If a (4 m 2 , 2 m 2 − m , m 2 − m ) difference set D exists, then D ( − 1) = 4 m 2 D � � For remainder of talk, we say that a (4 m 2 , 2 m 2 − m , m 2 − m ) difference set is an element of the group ring � D with coefficients of D ( − 1) = 4 m 2 . ± 1 and has the property that � D � Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

Character Values For abelian groups, a character of the group is a homomophism into the set of complex numbers. If G is abelian, then the complete set of characters forms the dual group G ∗ . Extend each character to C -algebra homomorphism. � � χ ( a g g ) = a g χ ( g ) Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

Character Values Theorem (Turyn 1965) � D is a Menon-Hadamard difference set in an abelian group of order 4 m 2 if and only if it is an element of the group ring with ± 1 coefficients and for each χ ∈ G ∗ , we have χ ( � D ) χ ( � D ( − 1) ) = 4 m 2 . Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

Character Example The group C 4 = < x : x 4 = 1 > . Each character is homomorphism so each is defined by where it sends generator(s) of the group. If χ is a character, then ( χ ( x )) 4 = 1 2 π i 4 ) j for 1 ≤ j ≤ 4. χ j ( x ) = ( e The dual group C ∗ 4 = { χ j : 1 ≤ j ≤ 4 } Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

Character Values for Reversible We know χ ( � D ) χ ( � D ( − 1) ) = 4 m 2 D ( − 1) = � When D is reversible, � D and D )) 2 = 4 m 2 ( χ ( � χ ( � D ) = ± 2 m Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

Building a Better Basis The ring C [ G ] has the standard basis { g : g ∈ G } . Create elements for each character value. � 1 χ ( g ) g − 1 e χ = | G | g ∈ G These idempotents form basis for C [ G ]. Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

The Better Part In the basis { e χ ; χ ∈ G ∗ } we take advantage of character values. χ ( e χ ′ ) = δ χ,χ ′ � Let Y = c χ e χ χ ∈ G ∗ So χ ( Y ) = χ ( c χ ) Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

The Better Part � We write � D = c χ e χ . χ ∈ G ∗ So χ ( � D ) = χ ( c χ ) Each χ ( c χ ) must be a complex number of modulus 2 m . If � D is reversible then χ ( c χ ) = ± 2 m Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

The Better Part We have a Menon-Hadamard Difference set ( � D ) if ◮ ± 1 are coefficients on each group element g ◮ Each coefficient, c χ , on e χ has the property that χ ( c χ ) is a complex number of modulus 2 m . Reversible if χ ( c χ ) = ± 2 m Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

The Better Part (2) � 1 χ ( g ) g − 1 Recall that each idempotent is | G | g ∈ G Difference set � D has coefficients in {− 1 , 1 } . Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

The Better Part (2) Idempotents with the same kernel may be added together and we may use the same coefficient for all of them. Sums of idempotents with the same kernel are called rational idempotents. [ e χ ] is be the rational idempotent containing e χ and all other e χ ′ such that ker ( χ ) = ker ( χ ′ ). The [ e χ ] exist in the group ring Q [ G ] Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

The Better Part (2) We have a Hadamard Difference set if ◮ ± 1 are coefficients on each group element g ◮ Each coefficient, c χ , on [ e χ ] has the property that χ ( c χ ) is a complex number of modulus 2 m . Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

Tiles A tile T is an element of the group ring with the following properties: ◮ Coefficients of g are all in the set { 1 , 0 , − 1 } ◮ For every character χ either χ ( T ) = 0 or χ ( T ) is a complex number of modulus 2 m Sums of rational idempotents with appropriate aliases create tiles. Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups