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Affine Matrix Ball Construction Michael Chmutov Pavlo Pylyavskyy Elena Yudovina AMS Meeting #1120 Fargo, ND April 17, 2016 Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April

slide-1
SLIDE 1

Affine Matrix Ball Construction

Michael Chmutov Pavlo Pylyavskyy Elena Yudovina

AMS Meeting #1120 Fargo, ND

April 17, 2016

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 1 / 7

slide-2
SLIDE 2

An analogue of Robinson-Schensted Correspondence

W ֒ → → Ωdom Left-hand side = affine symmetric group Right-hand side

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 2 / 7

slide-3
SLIDE 3

An analogue of Robinson-Schensted Correspondence

W ֒ → → Ωdom Left-hand side = affine symmetric group

Extended: W = Sn = {w : Z ֒ → → Z | w(i + n) = w(i) + n}

Right-hand side

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 2 / 7

slide-4
SLIDE 4

An analogue of Robinson-Schensted Correspondence

W ֒ → → Ωdom Left-hand side = affine symmetric group

Extended: W = Sn = {w : Z ֒ → → Z | w(i + n) = w(i) + n} Non-extended: W = Sn =

  • w ∈

Sn

  • n
  • i=1

w(i) − i = 0

  • Right-hand side

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 2 / 7

slide-5
SLIDE 5

An analogue of Robinson-Schensted Correspondence

W ֒ → → Ωdom Left-hand side = affine symmetric group

Extended: W = Sn = {w : Z ֒ → → Z | w(i + n) = w(i) + n} Non-extended: W = Sn =

  • w ∈

Sn

  • n
  • i=1

w(i) − i = 0

  • Right-hand side

P Q ρ

, ,

                      

Ω =

                       Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 2 / 7

slide-6
SLIDE 6

An analogue of Robinson-Schensted Correspondence

W ֒ → → Ωdom Left-hand side = affine symmetric group

Extended: W = Sn = {w : Z ֒ → → Z | w(i + n) = w(i) + n} Non-extended: W = Sn =

  • w ∈

Sn

  • n
  • i=1

w(i) − i = 0

  • Right-hand side

P Q ρ Ω =

                      

, ,

                      

tabloids of same shape filled with 1 := 1 + nZ, 2, . . . , n.

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 2 / 7

slide-7
SLIDE 7

An analogue of Robinson-Schensted Correspondence

W ֒ → → Ωdom Left-hand side = affine symmetric group

Extended: W = Sn = {w : Z ֒ → → Z | w(i + n) = w(i) + n} Non-extended: W = Sn =

  • w ∈

Sn

  • n
  • i=1

w(i) − i = 0

  • Right-hand side

P Q ρ

tabloids of same shape filled with 1 := 1 + nZ, 2, . . . , n.

Ω =

                      

, ,

                      

Zℓ(sh(P))

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 2 / 7

slide-8
SLIDE 8

An analogue of Robinson-Schensted Correspondence

W ֒ → → Ωdom Left-hand side = affine symmetric group

Extended: W = Sn = {w : Z ֒ → → Z | w(i + n) = w(i) + n} Non-extended: W = Sn =

  • w ∈

Sn

  • n
  • i=1

w(i) − i = 0

  • Right-hand side

P Q ρ

tabloids of same shape filled with 1 := 1 + nZ, 2, . . . , n.

Ω =

                      

, ,

                      

Zℓ(sh(P))

Offset dominance: ∀i, P, Q ∃r P,Q

i

∈ Z ∪ {−∞} s. t. ρi+1 ρi + r P,Q

i

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 2 / 7

slide-9
SLIDE 9

Matrix Ball Construction

w = 78351a2946

P = Q =

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-10
SLIDE 10

Matrix Ball Construction

w = 78351a2946

1 1 1

P = Q =

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-11
SLIDE 11

Matrix Ball Construction

w = 78351a2946

1 2 1 2 1 2

P = Q =

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-12
SLIDE 12

Matrix Ball Construction

w = 78351a2946

3 2 1 2 3 1 2 3 1

P = Q =

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-13
SLIDE 13

Matrix Ball Construction

w = 78351a2946

4 2 1 2 3 1 2 3 3 1

P = Q =

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-14
SLIDE 14

Matrix Ball Construction

w = 78351a2946

6

* * *

1 2 1 2 3 1 2 3 3 4 1 2 10 1 2 4 6

*

P = Q = 1 2 4 6 1 2 6 10

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-15
SLIDE 15

Matrix Ball Construction

w = 78351a2946

6

* * *

1 2 1 2 3 1 2 3 3 4 1 2 10 1 2 4 6

*

P = Q = 1 2 4 6 1 2 6 10

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-16
SLIDE 16

Matrix Ball Construction

w = 78351a2946

6

* * *

1 2 1 2 3 1 2 3 3 4 1 2 10 1 2 4 6

*

P = Q = 1 2 4 6 1 2 6 10

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-17
SLIDE 17

Matrix Ball Construction

w = 78351a2946

6

* * *

1 2 1 2 3 1 2 3 3 4 1 2 1 2 3 3 1 2 10 1 2 4 6

*

P = Q = 1 2 4 6 1 2 6 10

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-18
SLIDE 18

Matrix Ball Construction

w = 78351a2946

8

* * * * * *

1 2 1 2 3 1 2 3 3 4 1 2 1 2 3 3 1 2 10 4 3 1 2 4 6 6 3 5 9

*

P = Q = 1 2 4 6 1 2 6 10 3 5 9 3 4 8

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-19
SLIDE 19

Matrix Ball Construction

w = 78351a2946

8

* * * * * *

1 2 1 2 3 1 2 3 3 4 1 2 1 2 3 3 1 2 10 4 3 1 2 4 6 6 3 5 9

*

P = Q = 1 2 4 6 1 2 6 10 3 5 9 3 4 8

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-20
SLIDE 20

Matrix Ball Construction

w = 78351a2946

8

* * * * * *

1 2 1 2 3 1 2 3 3 4 1 2 1 2 3 3 1 2 10 4 3 1 2 4 6 6 3 5 9

*

P = Q = 1 2 4 6 1 2 6 10 3 5 9 3 4 8

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-21
SLIDE 21

Matrix Ball Construction

w = 78351a2946

2

* * * * * *

1 2 1 2 3 1 2 3 3 4 1 2 1 2 3 3 1 3 1 2 10 4 3 1 2 4 6 6 3 5 9 8

*

P = Q = 1 2 4 6 1 2 6 10 3 5 9 3 4 8

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-22
SLIDE 22

Matrix Ball Construction

w = 78351a2946

8

* * * * * * * * *

1 2 1 2 3 1 2 3 3 4 1 2 1 2 3 3 1 2 3 1 2 10 5 7 9 4 3 1 2 4 6 6 3 5 9 7 8 10

*

P = Q = 1 2 4 6 1 2 6 10 3 5 9 7 8 10 3 4 8 5 7 9

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-23
SLIDE 23

Matrix Ball Construction

w = 78351a2946

8

* * * * * * * * *

1 2 1 2 3 1 2 3 3 4 1 2 1 2 3 3 1 2 3 1 2 10 5 7 9 4 3 1 2 4 6 6 3 5 9 7 8 10

*

P = Q = 1 2 4 6 1 2 6 10 3 5 9 7 8 10 3 4 8 5 7 9

  • Rmk. From tableaux can tell exact positions of ∗’s.

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

slide-24
SLIDE 24

Proper numberings

Bw = {balls of w}

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-25
SLIDE 25

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-26
SLIDE 26

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b),

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-27
SLIDE 27

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-28
SLIDE 28

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

4 2 2 3 3 1 1 1

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-29
SLIDE 29

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

4 2 2 3 3 1 1 1

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-30
SLIDE 30

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.

4 2 2 3 3 1 1 1

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-31
SLIDE 31

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.

4 2 2 3 3 1 1 1

P = Q = ρ =

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-32
SLIDE 32

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.
  • Rmk. Weight keeps track of ∗ positions

1 2 3 3 4

* * * * *

1 1 2 5 1 10 9 4 6 5 10

P = Q = ρ = 4 5 1 5 ?

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-33
SLIDE 33

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.
  • Rmk. Weight keeps track of ∗ positions

2 3 3 4

* * * * *

1 1 1 2 6 9 4 10 1 5 10 5

P = Q = ρ = 4 5 1 5 ?

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-34
SLIDE 34

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.
  • Rmk. Weight keeps track of ∗ positions

P = Q = ρ = 4 5 1 5 ?

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-35
SLIDE 35

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.
  • Rmk. Weight keeps track of ∗ positions

2 4 1 2 3

P = Q = ρ = 4 5 1 5 ?

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-36
SLIDE 36

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.
  • Rmk. Weight keeps track of ∗ positions

* * * *

1 2 2 3 4 2 6 7 11 2 7 3 8

P = Q = ρ = 4 5 1 5 1 2 2 3 ? ?

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-37
SLIDE 37

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.
  • Rmk. Weight keeps track of ∗ positions

* * * *

1 2 2 3 4 2 6 7 11 2 7 3 8

P = Q = ρ = 4 5 1 5 1 2 2 3 ? ?

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-38
SLIDE 38

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.
  • Rmk. Weight keeps track of ∗ positions

P = Q = ρ = 4 5 1 5 1 2 2 3 ? ?

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-39
SLIDE 39

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.
  • Rmk. Weight keeps track of ∗ positions

1

P = Q = ρ = 4 5 1 5 1 2 2 3 ? ?

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-40
SLIDE 40

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.
  • Rmk. Weight keeps track of ∗ positions

* *

1 8 13 4 9

P = Q = ρ = 4 5 1 5 ? 1 2 2 3 ? ? 4 3

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-41
SLIDE 41

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.
  • Rmk. Weight keeps track of ∗ positions

P = Q = ρ = 4 5 1 5 ? 1 2 2 3 ? ? 4 3

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-42
SLIDE 42

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.
  • Rmk. Weight keeps track of ∗ positions
  • Thm. P,Q don’t depend on choices.

P = Q = ρ = 4 5 1 5 ? 1 2 2 3 ? ? 4 3

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-43
SLIDE 43

Proper numberings

Bw = {balls of w}

  • Def. d : Bw → Z is proper if

if a NW of b, then d(a) < d(b), ∀b ∃a NW of b w/ d(a) = d(b) − 1.

  • Thm. Proper numberings exist (but may

not be unique).

  • Thm. Proper numberings are periodic.
  • Rmk. Weight keeps track of ∗ positions
  • Thm. P,Q don’t depend on choices.

P = Q = ρ = 4 5 1 5 1 2 2 3 1 4 3 1

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

slide-44
SLIDE 44

Channel numberings

  • Def. Channel = densest

periodic collection of balls in negative slope

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 5 / 7

slide-45
SLIDE 45

Channel numberings

  • Def. Channel = densest

periodic collection of balls in negative slope

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 5 / 7

slide-46
SLIDE 46

Channel numberings

  • Def. Channel = densest

periodic collection of balls in negative slope

  • Thm. Proper numbering

numbers any channel by consecutive integers

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 5 / 7

slide-47
SLIDE 47

Channel numberings

  • Def. Channel = densest

periodic collection of balls in negative slope

  • Thm. Proper numbering

numbers any channel by consecutive integers

4 6 5 1 2 3

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 5 / 7

slide-48
SLIDE 48

Channel numberings

  • Def. Channel = densest

periodic collection of balls in negative slope

  • Thm. Proper numbering

numbers any channel by consecutive integers

6 3 4 2 5 1

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 5 / 7

slide-49
SLIDE 49

Channel numberings

  • Def. Channel = densest

periodic collection of balls in negative slope

  • Thm. Proper numbering

numbers any channel by consecutive integers

1

2 1 6 3 4 5

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 5 / 7

slide-50
SLIDE 50

Channel numberings

  • Def. Channel = densest

periodic collection of balls in negative slope

  • Thm. Proper numbering

numbers any channel by consecutive integers

2 1

6 3 4 2 5 1

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 5 / 7

slide-51
SLIDE 51

Channel numberings

  • Def. Channel = densest

periodic collection of balls in negative slope

  • Thm. Proper numbering

numbers any channel by consecutive integers

2 3 1

5 3 4 2 6 1

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 5 / 7

slide-52
SLIDE 52

Channel numberings

  • Def. Channel = densest

periodic collection of balls in negative slope

  • Thm. Proper numbering

numbers any channel by consecutive integers

  • Def. Channel numbering:

d(b) = max

paths from b(lower bound)

2 3 1

5 3 4 2 6 1

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 5 / 7

slide-53
SLIDE 53

Channel numberings

  • Def. Channel = densest

periodic collection of balls in negative slope

  • Thm. Proper numbering

numbers any channel by consecutive integers

  • Def. Channel numbering:

d(b) = max

paths from b(lower bound)

  • Thm. d(b) is finite

2 3 1

5 3 4 2 6 1

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 5 / 7

slide-54
SLIDE 54

Channel numberings

  • Def. Channel = densest

periodic collection of balls in negative slope

  • Thm. Proper numbering

numbers any channel by consecutive integers

  • Def. Channel numbering:

d(b) = max

paths from b(lower bound)

  • Thm. d(b) is finite
  • Rmk. No need to consider

paths with translates, so process is finite.

2 3 1

5 3 4 2 6 1

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 5 / 7

slide-55
SLIDE 55

Motivation and History

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND)

Affine Matrix Ball Construction April 17, 2016 6 / 7

slide-56
SLIDE 56

Motivation and History

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Standard basis {Tw}w∈W ; Kazhdan-Lusztig basis {Cw}w∈W

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 6 / 7

slide-57
SLIDE 57

Motivation and History

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Standard basis {Tw}w∈W ; Kazhdan-Lusztig basis {Cw}w∈W

Transition matrix entries are Kazhdan-Lusztig polynomials Pv,w(q)

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 6 / 7

slide-58
SLIDE 58

Motivation and History

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Standard basis {Tw}w∈W ; Kazhdan-Lusztig basis {Cw}w∈W

Transition matrix entries are Kazhdan-Lusztig polynomials Pv,w(q) H acts on itself by left multiplication

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 6 / 7

slide-59
SLIDE 59

Motivation and History

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Standard basis {Tw}w∈W ; Kazhdan-Lusztig basis {Cw}w∈W

Transition matrix entries are Kazhdan-Lusztig polynomials Pv,w(q) H acts on itself by left multiplication Cells: subsets L of W s. t. {Cw}w∈L carries subquotient rep.

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 6 / 7

slide-60
SLIDE 60

Motivation and History

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Standard basis {Tw}w∈W ; Kazhdan-Lusztig basis {Cw}w∈W

Transition matrix entries are Kazhdan-Lusztig polynomials Pv,w(q) H acts on itself by left multiplication Cells: subsets L of W s. t. {Cw}w∈L carries subquotient rep. In type A, cells given by RS correspondence: LQ = {w ∈ Sn | Q(w) = Q}.

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 6 / 7

slide-61
SLIDE 61

Motivation and History

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Standard basis {Tw}w∈W ; Kazhdan-Lusztig basis {Cw}w∈W

Transition matrix entries are Kazhdan-Lusztig polynomials Pv,w(q) H acts on itself by left multiplication Cells: subsets L of W s. t. {Cw}w∈L carries subquotient rep. In type A, cells given by RS correspondence: LQ = {w ∈ Sn | Q(w) = Q}.

  • Thm. Same is true for the affine RS correspondence from AMBC.

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 6 / 7

slide-62
SLIDE 62

Motivation and History

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Standard basis {Tw}w∈W ; Kazhdan-Lusztig basis {Cw}w∈W

Transition matrix entries are Kazhdan-Lusztig polynomials Pv,w(q) H acts on itself by left multiplication Cells: subsets L of W s. t. {Cw}w∈L carries subquotient rep. In type A, cells given by RS correspondence: LQ = {w ∈ Sn | Q(w) = Q}.

  • Thm. Same is true for the affine RS correspondence from AMBC.

History:

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 6 / 7

slide-63
SLIDE 63

Motivation and History

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Standard basis {Tw}w∈W ; Kazhdan-Lusztig basis {Cw}w∈W

Transition matrix entries are Kazhdan-Lusztig polynomials Pv,w(q) H acts on itself by left multiplication Cells: subsets L of W s. t. {Cw}w∈L carries subquotient rep. In type A, cells given by RS correspondence: LQ = {w ∈ Sn | Q(w) = Q}.

  • Thm. Same is true for the affine RS correspondence from AMBC.

History:

Shi (1991): w → P(w) (not bijection; not similar to RS)

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 6 / 7

slide-64
SLIDE 64

Motivation and History

(W , S) – Coxeter system, ground ring: Z[q±1/2] H = Ts, s ∈ S TsTtTs · · · = TtTsTt . . . (Ts + 1)(Ts − q) = 0

  • Standard basis {Tw}w∈W ; Kazhdan-Lusztig basis {Cw}w∈W

Transition matrix entries are Kazhdan-Lusztig polynomials Pv,w(q) H acts on itself by left multiplication Cells: subsets L of W s. t. {Cw}w∈L carries subquotient rep. In type A, cells given by RS correspondence: LQ = {w ∈ Sn | Q(w) = Q}.

  • Thm. Same is true for the affine RS correspondence from AMBC.

History:

Shi (1991): w → P(w) (not bijection; not similar to RS) Honeywill (2005): extension to bijection (not similar to RS)

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 6 / 7

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SLIDE 65

Thank you!

Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 7 / 7