Difgerence set in association scheme Hiroki Kajiura Hiroshima - PowerPoint PPT Presentation

Difgerence set in association scheme Hiroki Kajiura Hiroshima University Joint work with Makoto Matsumoto, Takayuki Okuda (Hiroshima Univ.) August 14, 2019 arXv:1903.00697 is written a this talk.

Table of theorems in this talk. . 3 relationships between difgerence sets in commutative association 2 some examples of difgerence set in association scheme, 1 (classical) difgerence set and difgerence set in association scheme, We will introduce TODO schemes and a kind of QMC(quasi-Monte Carlo) [if we have time]. Defjnition 1 (Difgerence set in association scheme). Let X := ( X, { R i } d i =0 ) be an association scheme and Y ⊂ X . Y is a difgerence set in X ⇒ λ 1 ( Y ) = λ 2 ( Y ) = · · · = λ d ( Y ) : ⇐ k 1 k 2 k d Where, we put k i is a i -th valency on X and λ i ( Y ) := # ( R i ∩ ( Y × Y )) .

Introduction: Defjnition of (classical) difgerence sets. Defjnition 3 ((classical) difgerence set). Let G be a fjnite group. Defjnition 2 ((classical) λ –number). Y ⊂ G ， a ∈ G ， (classical) λ –number in Y of a λ a := # { ( x, y ) ∈ Y 2 | a = x − 1 y } . Y ⊂ G is a (classical) difgerence set : ⇐ ⇒ ∃ λ ∈ N s . t . ∀ a ∈ G \{ 0 } , λ = λ a .

Introduction: An example of (classical) difg. set. 0 5 4 3 2 1 6 Theorem 1 ((classical) difgerence set on Z /7 Z ). This following Y ⊂ Z /7 Z is a (classical) difgerence set: Y := { 0 , 1 , 3 } 0 − 0 = 0 , 0 − 1 = 6 , 0 − 3 = 4 , { 0 , 1 , 3 } 1 − 0 = 1 , 1 − 1 = 0 , 1 − 3 = 5 , 3 − 0 = 3 , 3 − 1 = 2 , 3 − 3 = 0 . a visualization of Z /7 Z . Then λ 1 = λ 2 = λ 3 = λ 4 = λ 5 = λ 6 = 1 , therefore, we proved that Y is a (classical) difgerence set.

Difgerence set in association scheme. , Defjnition 1 (Difgerence set in association scheme). 1 An association scheme from a fjnite group, We introduce the following examples: In this talk Let X := ( X, { R i } d i =0 ) be an association scheme, Y ⊂ X , k i be a i -th valency on X . ⇒ λ 1 ( Y ) = λ 2 ( Y ) = · · · = λ d ( Y ) Y is a difgerence set in X : ⇐ k 1 k 2 k d Where λ i ( Y ) := # ( R i ∩ ( Y × Y )) = # { ( x, y ) ∈ Y × Y | ( x, y ) ∈ R i } is called “ λ –number in Y of a ” . 2 A group association scheme, especially, the dihedral group of order 16 , 3 J (5 , 2) : the Johnson scheme on 5 points set with class 2 .

Example: Association scheme from fjnite groups. Proposition 1. Let G be a fjnite group. Defjnition 4 (Association scheme from G ). G := ( G, { R a } a ∈ G ) is an association scheme from G ⇒ ∀ a ∈ G, R a := { ( x, y ) ∈ G × G | x − 1 y = a } . : ⇐ Remark) k a = 1 . Let G be an association scheme from G , Y ⊂ G , Y is a difgerence set in G ⇐ ⇒ Y is (classical) difgerence set. Remark) { 0 , 1 , 3 } ⊂ Z /7 Z is a difgerence set in association scheme from Z /7 Z .

Example: Group association scheme Part 1. Defjnition 4 (Group association scheme). Proposition 2. Let G be a fjnite group, C ( G ) be the set of conjugacy classes of G . G C := ( G, { R [ a ] } [ a ] ∈ C ( G ) ) is an group association scheme ⇒ ∀ [ a ] ∈ C ( G ) , R [ a ] := { ( x, y ) ∈ G × G | x − 1 y ∈ [ a ] } . : ⇐ Remark) G C is a commutative association scheme and k [ a ] = # [ a ] for [ a ] ∈ C ( G ) . Defjnition 5 (conjugative λ –number). Y ⊂ G , a ∈ G , An conjugative λ –number in Y of [ a ] λ [ a ] := # { ( x, y ) ∈ Y × Y | x − 1 y ∈ [ a ] } . G C : group association scheme, Y ⊂ G , Y is difgerence set in G C ⇐ ⇒ λ [ a ] / # [ a ] is constant on [ a ] ∈ C ( G ) .

Example: Group association scheme Part 2. Proposition 2. We have Let G be a fjnite group, G := ( G, { R a } a ∈ G ) be an association scheme from G , G C := ( G, { R [ a ] } [ a ] ∈ C ( G ) ) be a group association scheme and Y ⊂ G be a difgerence set in G C with λ = λ [ a ] / # [ a ] for any [ a ] ∈ C ( G ) . 1 { (classical) difg. set } = { difg. set in G } ⊂ { difg. set in G C } , 2 The complement of Y is a difgerence set in G C ,

Example: Group association scheme Part 3 difgerence set: 2 order of product of distinct odd primes[Shiu, 2007], 1 order of prime power[Fan, Shiu and Ma, 1985][Deng, 2004], set[Fan, Shiu and Ma, 1985]. We already known following cases: We conjecture that dihedral group has only “trivial” (classical) difgerence Remark. integer[Deng, 2004]. Theorem 2. s, r | s 2 = r 8 = srsr � � Let G := D 8 = be the dihedral group of order 16 , D C 8 be the group association scheme on D 8 . The following Y ⊂ D 8 is a difgerence set on D C 8 and non-(classical) Y := { e, r, s, sr 3 , sr 5 , sr 7 } . 3 order of 2 p t , where p is a prime number and t is a positive

Example: J (5 , 2) Part 1. V := { x ⊂ { 1 , 2 , . . . , 5 } | # x = 2 } ， Defjnition 4 (Johnson scheme on 5 points set with class 2 ). J (5 , 2) := ( V, { R i } 2 i =0 ) is called the Johnson scheme on 5 points set with class 2 , if R i := { ( x, y ) ⊂ V × V | # ( x ∩ y ) = 2 − i } .

or their complements. Example: J (5 , 2) Part 2. Let J (5 , 2) := ( V, { R i } 2 i =0 ) be the Johnson scheme on 5 points set with class 2 . Theorem 2 (difgerence set in J (5 , 2) ). 1 The following subsets Y 1 , Y 2 and Y 3 in J (5 , 2) are all difgerence sets: 1 Y 1 := {{ 0 , 1 } , { 2 , 3 } , { 0 , 2 }} , 2 Y 2 := {{ 0 , 1 } , { 2 , 3 } , { 0 , 2 } , { 1 , 3 }} , 3 Y 3 := {{ 0 , 1 } , { 2 , 3 } , { 3 , 4 } , { 0 , 3 }} . 2 Any difgerence set in J (5 , 2) is conjugate to Y 1 , Y 2 , Y 3

Remark Before now Association scheme: non-commutative OK! After now Association scheme: non-commutative NG! (only commutative association scheme)

In a sense that, Relationships between difg. sets in comm. a.s. and QMC. The point of this talk. Let X := ( X, { R i } d i =0 ) be a commutative association scheme, Y ⊂ X . 1 1 � � I X ( f ) := f ( x ) , I Y ( f ) := f ( x ) . # X # Y x ∈ X x ∈ Y Aim of a kind of QMC(quasi-Monte Carlo) on X . We want to fjnd(or construction) a following Y ⊂ X : “ I Y ( f ) good approximates I X ( f ) for many f ”, in this talk, Y is called QMC point set on X . “best” QMC point set on X ⇐ ⇒ difgerence set in X .

quasi-Monte Carlo (QMC) Fact. 1 (Error evaluation of Monte Carlo Integration). we want to fjnd point set that choose “deterministically” converge quickly! Idea of QMC with to reduce the error by an order of magnitude(very ineffjcient). i.e., we need 100 times the points to use Monte Carlo integration In this talk, its point set is called “QMC point set”. f : [0 , 1) s − → R , “ integrable ” , Y ⊂ [0 , 1) s : “Random” N -points subset. Defjnition 5 (Monte Carlo Integration on [0 , 1) s ). 1 � Monte Carlo Integration I Y ( f ) := f ( x ) . # Y x ∈ Y � � � = O ( N − 1/2 ) . � Err ( f ; Y ) := [0 , 1) s f ( x ) dx − I Y ( f ) � � � For many f ∈ C X ,

Koksma–Hlawka inequality Theorem (Koksma–Hlawka inequality[E. Hlawka, 1964]). How to fjnd QMC point set? f : a function of bounded variation, Y ⊂ [0 , 1) s : fjnite subset, Err ( f ; Y ) ≤ V ( f ) D ⋆ ( Y ) , Where, V ( f ) is the (Hardy and Krause) variation of f , D ⋆ ( X ) is the “star-Discrepancy” of X . We search fjnite subset Y on [0 , 1) s that has small D ⋆ ( Y ) .

A kind of QMC on commutative association scheme. We show the fmlowing theorem[K., M.Matsumoto and T.Okuda, 2019]: Fact. (Koksma-Hlawka type inequalities on a comm. a.s.) Where X := ( X, { R i } d i =0 ) :comm. a.s., f ∈ C X , Y ⊂ G , | I X ( f ) − I Y ( f ) | ≤ � f � J D ( Y ) d � � f � J := � E j f � dim V j , j =1 � ∂ j ( Y ) 1 � D ( X ) := max , ∂ j ( Y ) := E j ( x, y ) . # Y 2 dim V j 1 ≤ j ≤ d x,y ∈ Y { E j } d j =0 is primitive idempotents for matrix product in Bose-Mesner alg. on X , V j := E j C X .

relationships between difg. sets in comm. a.s. and QMC. Theorem [K., M. Matsumoto and O.Takayuki, 2019] X := ( X, { R i } d i =0 ) : comm. a.s., Y ⊂ X , � 1/ # Y − 1/ # X D ( Y ) ≥ , # X − 1 Especially, equality holds ifg Y is a difgerence set in X .

Summary of this talk. Summary of this talk. 1 (classical) difgerence set and difgerence set in association scheme, 2 some examples of difgerence set in association scheme, 3 relationships between difgerence sets in commutative association schemes and a kind of QMC(quasi-Monte Carlo). Future issues Find some applications (engineering/physics/mathematics) of Theorem 1. arXv:1903.00697 is written a this talk.