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.

Let X := (X, {Ri}d

i=0) be an association scheme and Y ⊂ X.

Defjnition 1 (Difgerence set in association scheme). Y is a difgerence set in X :⇐ ⇒ λ1(Y ) k1 = λ2(Y ) k2 = · · · = λd(Y ) kd . Where, we put ki is a i-th valency on X and λi(Y ) := #(Ri ∩ (Y × Y )). TODO We will introduce

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) [if we have time].

Introduction: Defjnition of (classical) difgerence sets.

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

Introduction: An example of (classical) difg. set.

Theorem 1 ((classical) difgerence set on Z/7Z). This following Y ⊂ Z/7Z is a (classical) difgerence set:

Y := {0, 1, 3}

0 − 0 = 0, 0 − 1 = 6, 0 − 3 = 4, 1 − 0 = 1, 1 − 1 = 0, 1 − 3 = 5, 3 − 0 = 3, 3 − 1 = 2, 3 − 3 = 0.

1 2 3 4 5 6

{0, 1, 3} a visualization of Z/7Z.

Then λ1 = λ2 = λ3 = λ4 = λ5 = λ6 = 1, therefore, we proved that Y is a (classical) difgerence set.

Difgerence set in association scheme.

Let X := (X, {Ri}d

i=0) be an association scheme, Y ⊂ X,

ki be a i-th valency on X. Defjnition 1 (Difgerence set in association scheme). Y is a difgerence set in X :⇐ ⇒ λ1(Y ) k1 = λ2(Y ) k2 = · · · = λd(Y ) kd , Where λi(Y ) := #(Ri ∩ (Y × Y )) = #{(x, y) ∈ Y × Y | (x, y) ∈ Ri} is called “λ–number in Y of a”. In this talk We introduce the following examples:

1 An association scheme from a fjnite group, 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.

Let G be a fjnite group. Defjnition 4 (Association scheme from G). G := (G, {Ra}a∈G) is an association scheme from G :⇐ ⇒ ∀a ∈ G, Ra := {(x, y) ∈ G × G | x−1y = a}. Remark) ka = 1. Proposition 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/7Z is a difgerence set in association scheme from Z/7Z.

Example: Group association scheme Part 1.

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

Example: Group association scheme Part 2.

Let G be a fjnite group, G := (G, {Ra}a∈G) be an association scheme from G, GC := (G, {R[a]}[a]∈C(G)) be a group association scheme and Y ⊂ G be a difgerence set in GC with λ = λ[a]/#[a] for any [a] ∈ C(G). Proposition 2. We have

1 {(classical) difg. set} = {difg. set in G} ⊂ {difg. set in GC}, 2 The complement of Y is a difgerence set in GC,

Example: Group association scheme Part 3

Let G := D8 =

• s, r | s2 = r8 = srsr
• be the dihedral group of order 16,

DC

8 be the group association scheme on D8.

Theorem 2. The following Y ⊂ D8 is a difgerence set on DC

8 and non-(classical)

difgerence set:

Y := {e, r, s, sr3, sr5, sr7}.

Remark. We conjecture that dihedral group has only “trivial” (classical) difgerence set[Fan, Shiu and Ma, 1985]. We already known following cases:

1 order of prime power[Fan, Shiu and Ma, 1985][Deng, 2004], 2 order of product of distinct odd primes[Shiu, 2007], 3 order of 2pt, where p is a prime number and t is a positive

integer[Deng, 2004].

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, {Ri}2

i=0) is called the Johnson scheme on 5 points set with

class 2, if Ri := {(x, y) ⊂ V × V | #(x ∩ y) = 2 − i}.

Example: J(5, 2) Part 2.

Let J(5, 2) := (V, {Ri}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 Y1, Y2 and Y3 in J(5, 2) are all difgerence sets: 1 Y1 := {{0, 1}, {2, 3}, {0, 2}}, 2 Y2 := {{0, 1}, {2, 3}, {0, 2}, {1, 3}}, 3 Y3 := {{0, 1}, {2, 3}, {3, 4}, {0, 3}}. 2 Any difgerence set in J(5, 2) is conjugate to Y1, Y2, Y3

• r their complements.

Remark

Before now

Association scheme: non-commutative OK!

After now

Association scheme: non-commutative NG! (only commutative association scheme)

Relationships between difg. sets in comm. a.s. and QMC.

Let X := (X, {Ri}d

i=0) be a commutative association scheme, Y ⊂ X.

IX(f) := 1 #X

• x∈X

f(x), IY (f) := 1 #Y

• x∈Y

f(x). Aim of a kind of QMC(quasi-Monte Carlo) on X. We want to fjnd(or construction) a following Y ⊂ X:

“IY (f) good approximates IX(f) for many f”,

in this talk, Y is called QMC point set on X. The point of this talk. In a sense that,

“best” QMC point set on X ⇐ ⇒ difgerence set in X.

quasi-Monte Carlo (QMC)

f : [0, 1)s − → R, “integrable” , Y ⊂ [0, 1)s :“Random” N-points subset. Defjnition 5 (Monte Carlo Integration on [0, 1)s). Monte Carlo Integration IY (f) := 1 #Y

• x∈Y

f(x).

• Fact. 1 (Error evaluation of Monte Carlo Integration).

Err(f; Y ) :=

• [0,1)s f(x)dx − IY (f)
• = O(N−1/2).

i.e., we need 100 times the points to use Monte Carlo integration with to reduce the error by an order of magnitude(very ineffjcient). Idea of QMC For many f ∈ CX, we want to fjnd point set that choose “deterministically” converge quickly! In this talk, its point set is called “QMC point set”.

Koksma–Hlawka inequality

Theorem (Koksma–Hlawka inequality[E. Hlawka, 1964]). 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. How to fjnd QMC point set?

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.)

X := (X, {Ri}d

i=0):comm. a.s., f ∈ CX, Y ⊂ G,

|IX(f) − IY (f)| ≤ fJ D(Y )

Where fJ :=

d

• j=1

Ejf dim Vj, D(X) := max

1≤j≤d

∂j(Y ) dim Vj , ∂j(Y ) :=

• 1

#Y 2

• x,y∈Y

Ej(x, y). {Ej}d

j=0 is primitive idempotents for matrix product in Bose-Mesner alg.

• n X, Vj := EjCX.

relationships between difg. sets in comm. a.s. and QMC.

X := (X, {Ri}d

i=0): comm. a.s., Y ⊂ X,

Theorem [K., M. Matsumoto and O.Takayuki, 2019]

D(Y ) ≥

• 1/#Y − 1/#X

#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.