Ranking patterns of unfolding models of codimension one Hidehiko - PowerPoint PPT Presentation

K = K v of unfolding model • Getting back to the unfolding model, we can write K = u + col W as K = K v with v := u − proj col W ( u ) { I m − W ( W T W ) − 1 W T } = u . 37

V : x 1 + ... + x m = 0 u v K = u + col W = K v 0 c ol W 38

RP UF ( µ 1 , . . . , µ m ) = RP(v( µ 1 , . . . , µ m )) • v is a function of µ 1 , . . . , µ m : v = v( µ 1 , . . . , µ m ) . • We can write RP UF ( µ 1 , . . . , µ m ) as RP UF ( µ 1 , . . . , µ m ) = { ( i 1 · · · i m ) : K v( µ 1 ,..., µ m ) ∩ C i 1 ··· i m ̸ = ∅} = RP(v( µ 1 , . . . , µ m )) . 39

Stability of RP(v( µ 1 , . . . , µ m )) • m -tuple ( µ 1 , . . . , µ m ) changes just a little, → v = v( µ 1 , . . . , µ m ) and thus K v change just a little, → the braid slice by K v remains exactly the same. • ( µ 1 , . . . , µ m ) changes beyond a certain extent, → the braid slice changes. 40

K v ’ � K v C 213 C 213 C 123 D ’ v ’ C 231 C 231 v D C 321 C 312 C 321 41

v ̸ = 0 Stability of RP(˜ v ) , ˜ • Let’s investigate the stability of the braid v ̸ = 0 , in general. slices RP(˜ v ) , ˜ 42

Table of Contents 1. Introduction 2. Unfolding as a braid slice 3. All braid slices 4. Unrealizable braid slices 5. Number of ranking patterns 6. Inequivalent ranking patterns 7. Concluding remarks 43

3 All braid slices • We will investigate the stability of the braid slices RP(˜ v ) , ˜ v ̸ = 0 , and find all (different) braid slices. 44

All-subset arrangement A m • First, define the all-subset arrangement A m in V by A m := { H I : I ⊂ { 1 , . . . , m } , 1 ≤ | I | ≤ m − 1 } , ∑ H I := { ( x 1 , . . . , x m ) T ∈ V : x i = 0 } ⊂ V. i ∈ I 45

m =3: V : x 1 + x 2 + x 3 =0 3 x 3 = 0 ( x 1 + x 2 = 0) H { 3 } = H { 1,2 } x 1 = 0 ( x 2 + x 3 = 0) x 2 = 0 ( x 1 + x 3 = 0) H { 2 } = H { 1,3 } H { 1 } = H { 2,3 } 46

Chambers of A m • Let Ch ( A m ) := { chambers D of A m } . D 47

Generic braid slices • We only consider RP(˜ v ) for “regular” ˜ v : ∪ ⊔ v ∈ V \ ˜ H = D. H ∈A m D ∈ Ch ( A m ) • We will say the braid slices are “generic” in this case. 48

v ) , ∀ ˜ v ∈ D RP D := RP(˜ • For ˜ v ’s in a chamber D ∈ Ch ( A m ) , RP(˜ v ) are the same: v ∈ D. RP(˜ v ) : the same for all ˜ We can define RP D := RP(˜ v ) , ˜ v ∈ D . • For different chambers D and D ′ , RP(˜ v ) changes: RP D ̸ = RP D ′ for D ̸ = D ′ . 49

~ ~ ~ ~ K v ’ � K v C 213 C 213 C 123 D ’ v ’ C 231 C 231 v D C 321 C 312 C 321 50

{ chambers of A m } ↔ { generic braid slices } • Therefore, ✓ ✏ Proposition: { D } ↔ { RP D } We have a bijection Ch ( A m ) → { RP D : D ∈ Ch ( A m ) } , D �→ RP D = RP(˜ v ∈ D. v ) , ˜ ✒ ✑ 51

v( µ 1 , . . . , µ m ) ∈ V \ ∪ H ∈A m H • Let’s get back to the unfolding model. • We can show ∪ ⊔ v( µ 1 , . . . , µ m ) ∈ V \ H = D. H ∈A m D ∈ Ch ( A m ) D v ( µ 1 ,..., µ m ) 52

RP(v( µ 1 , . . . , µ m )) : generic • Thus, RP UF ( µ 1 , . . . , µ m ) = RP(v( µ 1 , . . . , µ m )) is a generic braid slice: RP UF ( µ 1 , . . . , µ m ) = RP D for D ∋ v( µ 1 , . . . , µ m ) . 53

D v ( µ 1 ,..., µ m ) 54

D not attained by v( · , . . . , · ) • However, not all D ∈ Ch ( A m ) can be attained by v( · , . . . , · ) . • Hence, { r.p.’s of unfoldings } � { generic braid slices } . • Not all generic braid slices can be realized by the unfolding model. 55

~ ~ ~ K v C 213 C 123 C 231 C 132 v C 321 C 312 RP( v ) = { (213), (231), (321), (312) } 56

D not attained by v( · , . . . , · ) • We have to identify D s which can/cannot be attained by v( · , . . . , · ) . 57

Table of Contents 1. Introduction 2. Unfolding as a braid slice 3. All braid slices 4. Unrealizable braid slices 5. Number of ranking patterns 6. Inequivalent ranking patterns 7. Concluding remarks 58

4 Unrealizable braid slices • We will identify D ∈ Ch ( A m ) which cannot be attained by v( · , . . . , · ) , and thereby identify unrealizable braid slices RP D . 59

Image of v( · , . . . , · ) • We can show im v := { v( µ 1 , . . . , µ m ) : generic µ 1 , . . . , µ m ∈ R m − 2 } is given by ⊔ im v = D D ∈ Ch ( A m ) , D ̸ = − D 1 ,..., − D m where − D 1 , . . . , − D m ∈ Ch ( A m ) are − D i := { v ∈ V : v i < 0 , v j > 0 for all j ̸ = i } . 60

m =3: V : x 1 + x 2 + x 3 =0 3 ( x 3 < 0, x 1 , x 2 > 0) − D 3 x 3 = 0 ( x 1 + x 2 = 0) H { 3 } = H { 1,2 } − D 1 − D 2 ( x 1 < 0, x 2 , x 3 > 0) ( x 2 < 0, x 1 , x 3 > 0) x 2 = 0 ( x 1 + x 3 = 0) x 1 = 0 ( x 2 + x 3 = 0) H { 2 } = H { 1,3 } H { 1 } = H { 2,3 } 61

Unrealizable braid slices • Therefore, ✓ ✏ Theorem: Unrealizable braid slices RP D cannot be realized by the unfolding model if and only if D = − D i for some i = 1 , . . . , m. ✒ ✑ 62

Unrealizable braid slices • That is, exactly RP − D 1 , . . . , RP − D m are unrealizable braid slices. 63

m =3: V : x 1 + x 2 + x 3 =0 3 ( x 3 < 0, x 1 , x 2 > 0) − D 3 x 3 = 0 ( x 1 + x 2 = 0) H { 3 } = H { 1,2 } − D 1 − D 2 ( x 1 < 0, x 2 , x 3 > 0) ( x 2 < 0, x 1 , x 3 > 0) x 2 = 0 ( x 1 + x 3 = 0) x 1 = 0 ( x 2 + x 3 = 0) H { 2 } = H { 1,3 } H { 1 } = H { 2,3 } 64

~ ~ ~ K v C 213 C 123 C 231 C 132 v C 321 C 312 RP( v ) = { (213), (231), (321), (312) } 65

Table of Contents 1. Introduction 2. Unfolding as a braid slice 3. All braid slices 4. Unrealizable braid slices 5. Number of ranking patterns 6. Inequivalent ranking patterns 7. Concluding remarks 66

5 Number of ranking patterns • So far, we have seen { r.p.’s of unfoldings } � { generic braid slices } ↕ { chambers of A m } and { generic braid slices } \ { r.p.’s of unfoldings } = { RP − D 1 , . . . , RP − D m } . 67

Main result 1 ✓ ✏ Theorem: Number of r.p.’s The number q ( m ) of ranking patterns of unfolding models of codimension one with m objects is q ( m ) = |{ chambers of A m }| − m. ✒ ✑ 68

q ( m ) , m ≤ 8 ✓ ✏ Corollary: q ( m ) for m ≤ 8 For m ≤ 8 , we have q (3) = 3 , q (4) = 28 , q (5) = 365 , q (6) = 11286 , q (7) = 1066037 , q (8) = 347326344 . ✒ ✑ 69

Table of Contents 1. Introduction 2. Unfolding as a braid slice 3. All braid slices 4. Unrealizable braid slices 5. Number of ranking patterns 6. Inequivalent ranking patterns 7. Concluding remarks 70

6 Inequivalent ranking patterns • Let’s consider the case where we ignore difference by a permutation of objects. • We consider two ranking patterns are essentially the same if they are just the relabelings of the objects of each other. • What is the number of essentially different ranking patterns of unfolding models of codimension one? 71

Equivalence of ranking patterns • Let S m be the symmetric group on m letters (all bijections σ : { 1 , . . . , m } → { 1 , . . . , m } ). • We will say RP D and RP D ′ are equivalent : RP D ∼ RP D ′ if and only if RP D = σ RP D ′ , ∃ σ ∈ S m , where σ RP D ′ := { ( σ ( i 1 ) · · · σ ( i m )) : ( i 1 · · · i m ) ∈ RP D ′ } . 72

Action of S m on Ch ( A m ) • Consider the action of S m on Ch ( A m ) : S m × Ch ( A m ) ∋ ( σ, D ) �− → σD ∈ Ch ( A m ) , where σD := { σ v : v ∈ D } with σ v := ( v σ − 1 (1) , . . . , v σ − 1 ( m ) ) T for v = ( v 1 , . . . , v m ) T . 73

⇒ O ( D ) = O ( D ′ ) RP D ∼ RP D ′ ⇐ • We can check RP σD = σ RP D . • Thus, ⇒ D = σD ′ , ∃ σ ∈ S m RP D ∼ RP D ′ ⇐ (i.e., O ( D ) = O ( D ′ ) ), where O ( D ) := { σD : σ ∈ S m } is the S m -orbit containing D . 74

⇒ O ( D ) = O ( D ′ ) RP D ∼ RP D ′ ⇐ • Therefore, the number of inequivalent braid slices RP D , D ∈ Ch ( A m ) , is equal to |{ S m -orbits of Ch ( A m ) }| . 75

Orbit of excluded chambers • On the other hand, the excluded m chambers − D 1 , . . . , − D m constitute one orbit O ( − D m ) = {− D 1 , . . . , − D m } . 76

Number of inequivalent ranking patterns • Therefore, ✓ ✏ Proposition: # of inequivalent r.p.’s The number of inequivalent ranking pat- terns of unfolding models of codimension one is |{ S m -orbits of Ch ( A m ) }| − 1 . ✒ ✑ 77

|{ S m -orbits of Ch ( A m ) }| • Our job is to count |{ S m -orbits of Ch ( A m ) }| . 78

{ F ∈ Ch ( A m ∪ B m ) : F ⊂ C 1 ··· m } • There is a one-to-one correspondence { F ∈ Ch ( A m ∪ B m ) : F ⊂ C 1 ··· m } ← → { S m -orbits of Ch ( A m ) } given by F �− → O ( D F ) , where D F := unique D ∈ Ch ( A m ) such that F ⊂ D for F ∈ Ch ( A m ∪ B m ) . 79

F ↔ O ( D F ) , m = 3 m =3: m =3: 3 x 2 = x 1 x 2 = x 1 3 C 123 3 D F 3 F F x 3 = x 1 x 2 = x 3 x 3 = x 1 x 2 = x 3 F ’ x 3 =0 ( x 1 + x 2 = 0) x 3 =0 ( x 1 + x 2 = 0) x 3 =0 ( x 2 =0 ( x 1 + x 3 = 0) x 2 =0 ( x 1 + x 3 = 0) x 1 =0 ( x 2 + x 3 = 0) x 1 =0 ( x 2 + x 3 = 0) ( D F ) | ( D F ) | =3 F { S m -orbits of Ch ( A m ) } = {O ( D F ) , O ( D F ′ ) } . 80

|{ S m -orbits of Ch ( A m ) }| • Since | Ch ( B m ) | = | S m | = m ! , we obtain |{ S m -orbits of Ch ( A m ) }| = |{ F ∈ Ch ( A m ∪ B m ) : F ⊂ C 1 ··· m }| = | Ch ( A m ∪ B m ) | . m ! 81

Main result 2 ✓ ✏ Theorem: # of inequivalent r.p.’s The number q IE ( m ) of inequivalent rank- ing patterns of unfolding models of codi- mension one is q IE ( m ) = | Ch ( A m ∪ B m ) | − 1 . m ! ✒ ✑ 82

q IE ( m ) , m ≤ 9 ✓ ✏ Corollary: q IE ( m ) for m ≤ 9 For m ≤ 9 , we have q IE (3) = 1 , q IE (4) = 3 , q IE (5) = 11 , q IE (6) = 55 , q IE (7) = 575 , q IE (8) = 16639 , q IE (9) = 1681099 . ✒ ✑ 83

Illustration: m = 4 • We illustrate our results when m = 4 . 84

F ′ ⊂ C 1234 F, F ′ , ˜ F, ˜ • Chamber C 1234 : x 1 > x 2 > x 3 > x 4 of B 4 contains exactly 4 chambers F ′ ∈ Ch ( A 4 ∪ B 4 ) . F, F ′ , ˜ F, ˜ 85

~ ’ ∪ ~ ’ ’ ~ F ′ ⊂ C 1234 F, F ′ , ˜ F, ˜ x 1 > x 2 x 1 > x 2 4 4 4 x 2 > x 4 x 2 > x 4 x 1 + x 3 > 0 x 1 + x 2 > 0 x 3 > 0 ~ F ’ F C 1234 x 3 > x 4 x 3 > x 4 F x 1 > x 3 x 1 > x 3 F x 4 > 0 x 2 > x 3 x 2 > x 3 x 1 + x 4 > 0 x 1 > 0 x 2 > 0 x 1 > x 4 x 1 > x 4 C 1234 : x 1 > x 2 > x 3 > x 4 F , , , ⊂ C 1234 F F F (intersection with the unit sphere S 2 in V ≃ R 3 ) 86

~ ’ x 1 > x 2 x 2 > x 4 x 1 + x 3 > 0 x 1 + x 2 > 0 x 3 > 0 ~ F ’ F x 3 > x 4 F x 1 > x 3 F x 4 > 0 x 2 > x 3 x 1 + x 4 > 0 x 1 > 0 x 2 > 0 x 1 > x 4 A 4 ∪ B 4 (intersection with the unit sphere S 2 in V ≃ R 3 ) 87

F ′ ∈ Ch ( A 4 ) D F , D F ′ , D ˜ F , D ˜ • The chambers of A 4 containing F, F ′ , ˜ F, ˜ F ′ are D F = − D 4 : x 1 > 0 , x 2 > 0 , x 3 > 0 , D F ′ = D 1 : x 4 < 0 , x 3 < 0 , x 2 < 0 , D ˜ F = E : x 2 + x 3 > 0 , x 1 + x 3 > 0 , x 3 < 0 , F ′ = E ′ : x 3 + x 2 < 0 , x 4 + x 2 < 0 , x 2 > 0 . D ˜ 88

~ ~ ~ ~ F ′ ∈ Ch ( A 4 ) D F , D F ′ , D ˜ F , D ˜ x 4 + x 2 < 0 x 2 > 0 x 2 + x 3 > 0 x 1 + x 3 > 0 F F ’ � x 3 < 0 x 3 + x 2 < 0 D =E ’ D =E F ’ � F x 3 > 0 x 0 F 4 < x 0 x 2 > 0 2 < x 1 > 0 F ’ x 0 3 < D = － D 4 D = D 1 F F ’ 89

Cross section • These D F , D F ′ , D ˜ F , D ˜ F ′ constitute a complete set of representatives of the orbits: { S m -orbits of Ch ( A m ) } = {O ( D F ) , O ( D F ′ ) , O ( D ˜ F ) , O ( D ˜ F ′ ) } . 90

Four inequivalent braid slices • Four inequivalent braid slices: RP D F = RP − D 4 , RP D F ′ = RP D 1 , RP D ˜ F = RP E , RP D ˜ F ′ = RP E ′ . – RP − D 4 : not realizable. – RP D 1 , RP E , RP E ′ : realizable. 91

Three inequivalent r.p.’s of unfoldings • There are exactly three inequivalent ranking patterns RP D 1 , RP E , RP E ′ of unfolding models. 92

Unrealizable RP − D 4 • Unrealizable braid slice: RP − D 4 = P 4 \ { (4123) , (4132) , (4213) , (4231) , (4312) , (4321) } . (“object 4 is never ranked first”) 93

Realizable RP D 1 , RP E , RP E ′ • Realizable braid slices: RP D 1 = P 4 \ { (2341) , (2431) , (3241) , (3421) , (4231) , (4321) } , (“object 1 is never ranked last”) RP E = P 4 \ { (3412) , (3421) , (4312) , (4321) , (4132) , (4231) } , RP E ′ = P 4 \ { (3412) , (4312) , (3421) , (4321) , (3241) , (4231) } . 94

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Realizing unfolding models • The unfolding models realizing RP D 1 , RP E , RP E ′ are illustrated in the following figures ( µ 1 , µ 2 , µ 3 , µ 4 are written as 1 , 2 , 3 , 4 in the figures). 96

RP D 1 (4213) (2413) (4123) (2143) (1423) Inadmissible (4132) (1243) rankings: (2134) (2341) (1432) (2431) 4 (1234) (3241) 1 (1342) (3421) (1324) (4231) 3 2 (4321) (2314) (4312) 1 (3142) 3 (3124) 4 (3214) 3 (3412) 4 4 1 2 2 3 2 97 1

RP E 3 2 3 (3241) (3214) 4 (2341) (3124) (2431) 3 Inadmissible 3 (2314) 1 rankings: (1234) (2134) (3412) (3142) (2413) (3421) (2143) (1324) 2 (4312) (1243) (4321) (1342) 1 4 2 (4132) 4 (4231) (4213) (1432) (1423) (4123) 2 1 4 1 98

RP E ′ (4132) (4213) (4123) Inadmissible 4 4 rankings: 1 (1243) (2413) (1432) (3412) (2143) (2431) 4 (1423) (4312) 3 (1234) (2341) 1 (2134) (3421) (1342) (1324) (4321) 2 3 (3241) 1 (3142) (4231) (2314) 3 (3124) 4 2 (3214) 1 2 2 3 99