Affine Matrix Ball Construction Michael Chmutov Pavlo Pylyavskyy - PowerPoint PPT Presentation

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

An analogue of Robinson-Schensted Correspondence W ֒ → → Ω dom 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

An analogue of Robinson-Schensted Correspondence W ֒ → → Ω dom Left-hand side = affine symmetric group Extended: W = � S n = { 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

An analogue of Robinson-Schensted Correspondence W ֒ → → Ω dom Left-hand side = affine symmetric group Extended: W = � S n = { w : Z ֒ → → Z | w ( i + n ) = w ( i ) + n } � � � � n � Non-extended: W = � w ∈ � S n = w ( i ) − i = 0 S n i =1 Right-hand side Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 2 / 7

An analogue of Robinson-Schensted Correspondence W ֒ → → Ω dom Left-hand side = affine symmetric group Extended: W = � S n = { w : Z ֒ → → Z | w ( i + n ) = w ( i ) + n } � � � � n � Non-extended: W = � w ∈ � S n = w ( i ) − i = 0 S n i =1 Right-hand side                    Q   P  Ω =  ρ               , ,          Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 2 / 7

An analogue of Robinson-Schensted Correspondence W ֒ → → Ω dom Left-hand side = affine symmetric group Extended: W = � S n = { w : Z ֒ → → Z | w ( i + n ) = w ( i ) + n } � � � � n � Non-extended: W = � w ∈ � S n = w ( i ) − i = 0 S n i =1 Right-hand side                    Q   P  Ω =  ρ               , ,          tabloids of same shape filled with 1 := 1 + n Z , 2 , . . . , n . Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 2 / 7

An analogue of Robinson-Schensted Correspondence W ֒ → → Ω dom Left-hand side = affine symmetric group Extended: W = � S n = { w : Z ֒ → → Z | w ( i + n ) = w ( i ) + n } � � � � n � Non-extended: W = � w ∈ � S n = w ( i ) − i = 0 S n i =1 Right-hand side                    Q   P  Ω =  ρ               , ,          Z ℓ ( sh ( P )) tabloids of same shape filled with 1 := 1 + n Z , 2 , . . . , n . Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 2 / 7

An analogue of Robinson-Schensted Correspondence W ֒ → → Ω dom Left-hand side = affine symmetric group Extended: W = � S n = { w : Z ֒ → → Z | w ( i + n ) = w ( i ) + n } � � � � n � Non-extended: W = � w ∈ � S n = w ( i ) − i = 0 S n i =1 Right-hand side                    Q   P  Ω =  ρ               , ,          Z ℓ ( sh ( P )) tabloids of same shape filled with 1 := 1 + n Z , 2 , . . . , n . Offset dominance: ∀ i , P , Q ∃ r P , Q ∈ Z ∪ {−∞} s. t. ρ i +1 � ρ i + r P , Q i i Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 2 / 7

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

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

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

Matrix Ball Construction w = 78351a2946 1 2 1 2 1 3 2 3 3 P = Q = Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

Matrix Ball Construction w = 78351a2946 1 2 1 2 1 3 2 3 3 4 P = Q = Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

Matrix Ball Construction w = 78351a2946 1 2 4 6 1 1 * 2 2 * 1 2 1 6 3 * 2 3 3 10 4 * 1 2 4 6 1 2 6 10 P = Q = Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

Matrix Ball Construction w = 78351a2946 1 2 4 6 1 1 * 2 2 * 1 2 1 6 3 * 2 3 3 10 4 * 1 2 4 6 1 2 6 10 P = Q = Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

Matrix Ball Construction w = 78351a2946 1 2 4 6 1 1 * 2 2 * 1 2 1 6 3 * 2 3 3 10 4 * 1 2 4 6 1 2 6 10 P = Q = Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

Matrix Ball Construction w = 78351a2946 1 2 4 6 1 1 * 2 2 * 1 1 2 2 1 1 6 3 * 2 2 3 3 3 3 10 4 * 1 2 4 6 1 2 6 10 P = Q = Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

Matrix Ball Construction w = 78351a2946 1 2 4 6 3 5 9 1 1 * 2 2 * 1 3 1 * 4 2 * 2 1 1 6 3 * 2 2 3 8 * 3 3 3 10 4 * 1 2 4 6 1 2 6 10 P = 3 5 9 Q = 3 4 8 Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

Matrix Ball Construction w = 78351a2946 1 2 4 6 3 5 9 1 1 * 2 2 * 1 3 1 * 4 2 * 2 1 1 6 3 * 2 2 3 8 * 3 3 3 10 4 * 1 2 4 6 1 2 6 10 P = 3 5 9 Q = 3 4 8 Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

Matrix Ball Construction w = 78351a2946 1 2 4 6 3 5 9 1 1 * 2 2 * 1 3 1 * 4 2 * 2 1 1 6 3 * 2 2 3 8 * 3 3 3 10 4 * 1 2 4 6 1 2 6 10 P = 3 5 9 Q = 3 4 8 Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

Matrix Ball Construction w = 78351a2946 1 2 4 6 3 5 9 1 1 * 2 2 * 1 3 1 * 4 2 * 2 1 1 1 6 3 * 2 2 2 3 8 * 3 3 3 3 10 4 * 1 2 4 6 1 2 6 10 P = 3 5 9 Q = 3 4 8 Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

Matrix Ball Construction w = 78351a2946 1 2 4 6 3 5 9 7 8 10 1 1 * 2 2 * 1 3 1 * 4 2 * 2 5 1 1 1 * 6 3 * 2 2 7 2 * 3 8 * 3 9 3 3 * 3 10 4 * 1 2 4 6 1 2 6 10 P = 3 5 9 Q = 3 4 8 7 8 10 5 7 9 Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 3 / 7

Matrix Ball Construction w = 78351a2946 1 2 4 6 3 5 9 7 8 10 1 1 * 2 2 * 1 3 1 * 4 2 * 2 5 1 1 1 * 6 3 * 2 2 7 2 * 3 8 * 3 9 3 3 * 3 10 4 * 1 2 4 6 1 2 6 10 P = 3 5 9 Q = 3 4 8 7 8 10 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

Proper numberings B w = { balls of w } Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

Proper numberings B w = { balls of w } Def. d : B w → Z is proper if Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina (AMS Meeting #1120 Fargo, ND) Affine Matrix Ball Construction April 17, 2016 4 / 7

Proper numberings B w = { balls of w } Def. d : B w → 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