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Quantum walled Brauer spin chain
AM Semikhatov
Theory DepartmentLebedev Physics Institute
Mathematics of Conformal Field TheoryCanberra, July 2015
Semikhatov Quantum walled Brauer spin chain
Motivation and excuses
Motivation: The idea of centralizing algebras:
Schur–Weyl duality
Kazhdan–Lusztig duality
Logarithmic Kazhdan–Lusztig duality: kernel of the screenings defines anonsemisimple VOA [FHST]; [Adamovic–Milas], . . . [Tsuchiya–Wood]
The paradigm:screenings (Nichols algebras) vs VOAs/Logarithmic CFTs:
How about screenings vs something else?!Discretized models V b V . . .b V — “spin chains”Discretized versions of (L)CFT models: [Gainutdinov, Reed, Saleur,Vasseur; Gainutdinov, Saleur, Tipuinin, . . . ]Type-A Nichols algebras acting on V b V ˚ . . .V b V ˚, products of afundamental and its dual are centralized by a quotient of qwB.
ApologiesNo LogCFT stuff in this talk
The talk is about this qwB. Work with A. Kiselev and I. Tipunin
Semikhatov Quantum walled Brauer spin chain
Motivation and excuses
Motivation: The idea of centralizing algebras:
Schur–Weyl duality
Kazhdan–Lusztig duality
Logarithmic Kazhdan–Lusztig duality: kernel of the screenings defines anonsemisimple VOA [FHST]; [Adamovic–Milas], . . . [Tsuchiya–Wood]
The paradigm:screenings (Nichols algebras) vs VOAs/Logarithmic CFTs:
How about screenings vs something else?!Discretized models V b V . . .b V — “spin chains”Discretized versions of (L)CFT models: [Gainutdinov, Reed, Saleur,Vasseur; Gainutdinov, Saleur, Tipuinin, . . . ]Type-A Nichols algebras acting on V b V ˚ . . .V b V ˚, products of afundamental and its dual are centralized by a quotient of qwB.
ApologiesNo LogCFT stuff in this talk
The talk is about this qwB. Work with A. Kiselev and I. Tipunin
Semikhatov Quantum walled Brauer spin chain
Motivation and excuses
Motivation: The idea of centralizing algebras:
Schur–Weyl duality
Kazhdan–Lusztig duality
Logarithmic Kazhdan–Lusztig duality: kernel of the screenings defines anonsemisimple VOA [FHST]; [Adamovic–Milas], . . . [Tsuchiya–Wood]
The paradigm:screenings (Nichols algebras) vs VOAs/Logarithmic CFTs:
How about screenings vs something else?!Discretized models V b V . . .b V — “spin chains”Discretized versions of (L)CFT models: [Gainutdinov, Reed, Saleur,Vasseur; Gainutdinov, Saleur, Tipuinin, . . . ]Type-A Nichols algebras acting on V b V ˚ . . .V b V ˚, products of afundamental and its dual are centralized by a quotient of qwB.
ApologiesNo LogCFT stuff in this talk
The talk is about this qwB. Work with A. Kiselev and I. Tipunin
Semikhatov Quantum walled Brauer spin chain
Motivation and excuses
Motivation: The idea of centralizing algebras:
Schur–Weyl duality
Kazhdan–Lusztig duality
Logarithmic Kazhdan–Lusztig duality: kernel of the screenings defines anonsemisimple VOA [FHST]; [Adamovic–Milas], . . . [Tsuchiya–Wood]
The paradigm:screenings (Nichols algebras) vs VOAs/Logarithmic CFTs:
How about screenings vs something else?!Discretized models V b V . . .b V — “spin chains”Discretized versions of (L)CFT models: [Gainutdinov, Reed, Saleur,Vasseur; Gainutdinov, Saleur, Tipuinin, . . . ]Type-A Nichols algebras acting on V b V ˚ . . .V b V ˚, products of afundamental and its dual are centralized by a quotient of qwB.
ApologiesNo LogCFT stuff in this talk
The talk is about this qwB. Work with A. Kiselev and I. Tipunin
Semikhatov Quantum walled Brauer spin chain
Motivation and excuses
Motivation: The idea of centralizing algebras:
Schur–Weyl duality
Kazhdan–Lusztig duality
Logarithmic Kazhdan–Lusztig duality: kernel of the screenings defines anonsemisimple VOA [FHST]; [Adamovic–Milas], . . . [Tsuchiya–Wood]
The paradigm:screenings (Nichols algebras) vs VOAs/Logarithmic CFTs:
How about screenings vs something else?!Discretized models V b V . . .b V — “spin chains”Discretized versions of (L)CFT models: [Gainutdinov, Reed, Saleur,Vasseur; Gainutdinov, Saleur, Tipuinin, . . . ]Type-A Nichols algebras acting on V b V ˚ . . .V b V ˚, products of afundamental and its dual are centralized by a quotient of qwB.
ApologiesNo LogCFT stuff in this talk
The talk is about this qwB. Work with A. Kiselev and I. Tipunin
Semikhatov Quantum walled Brauer spin chain
Motivation and excuses
Motivation: The idea of centralizing algebras:
Schur–Weyl duality
Kazhdan–Lusztig duality
Logarithmic Kazhdan–Lusztig duality: kernel of the screenings defines anonsemisimple VOA [FHST]; [Adamovic–Milas], . . . [Tsuchiya–Wood]
The paradigm:screenings (Nichols algebras) vs VOAs/Logarithmic CFTs:
How about screenings vs something else?!Discretized models V b V . . .b V — “spin chains”Discretized versions of (L)CFT models: [Gainutdinov, Reed, Saleur,Vasseur; Gainutdinov, Saleur, Tipuinin, . . . ]Type-A Nichols algebras acting on V b V ˚ . . .V b V ˚, products of afundamental and its dual are centralized by a quotient of qwB.
ApologiesNo LogCFT stuff in this talk
The talk is about this qwB. Work with A. Kiselev and I. Tipunin
Semikhatov Quantum walled Brauer spin chain
Motivation and excuses
Motivation: The idea of centralizing algebras:
Schur–Weyl duality
Kazhdan–Lusztig duality
Logarithmic Kazhdan–Lusztig duality: kernel of the screenings defines anonsemisimple VOA [FHST]; [Adamovic–Milas], . . . [Tsuchiya–Wood]
The paradigm:screenings (Nichols algebras) vs VOAs/Logarithmic CFTs:
How about screenings vs something else?!Discretized models V b V . . .b V — “spin chains”Discretized versions of (L)CFT models: [Gainutdinov, Reed, Saleur,Vasseur; Gainutdinov, Saleur, Tipuinin, . . . ]Type-A Nichols algebras acting on V b V ˚ . . .V b V ˚, products of afundamental and its dual are centralized by a quotient of qwB.
ApologiesNo LogCFT stuff in this talk
The talk is about this qwB. Work with A. Kiselev and I. Tipunin
Semikhatov Quantum walled Brauer spin chain
Motivation and excuses
Motivation: The idea of centralizing algebras:
Schur–Weyl duality
Kazhdan–Lusztig duality
Logarithmic Kazhdan–Lusztig duality: kernel of the screenings defines anonsemisimple VOA [FHST]; [Adamovic–Milas], . . . [Tsuchiya–Wood]
The paradigm:screenings (Nichols algebras) vs VOAs/Logarithmic CFTs:
How about screenings vs something else?!Discretized models V b V . . .b V — “spin chains”Discretized versions of (L)CFT models: [Gainutdinov, Reed, Saleur,Vasseur; Gainutdinov, Saleur, Tipuinin, . . . ]Type-A Nichols algebras acting on V b V ˚ . . .V b V ˚, products of afundamental and its dual are centralized by a quotient of qwB.
ApologiesNo LogCFT stuff in this talk
The talk is about this qwB. Work with A. Kiselev and I. Tipunin
Semikhatov Quantum walled Brauer spin chain
Motivation and excuses
Motivation: The idea of centralizing algebras:
Schur–Weyl duality
Kazhdan–Lusztig duality
Logarithmic Kazhdan–Lusztig duality: kernel of the screenings defines anonsemisimple VOA [FHST]; [Adamovic–Milas], . . . [Tsuchiya–Wood]
The paradigm:screenings (Nichols algebras) vs VOAs/Logarithmic CFTs:
How about screenings vs something else?!Discretized models V b V . . .b V — “spin chains”Discretized versions of (L)CFT models: [Gainutdinov, Reed, Saleur,Vasseur; Gainutdinov, Saleur, Tipuinin, . . . ]Type-A Nichols algebras acting on V b V ˚ . . .V b V ˚, products of afundamental and its dual are centralized by a quotient of qwB.
ApologiesNo LogCFT stuff in this talk
The talk is about this qwB. Work with A. Kiselev and I. Tipunin
Semikhatov Quantum walled Brauer spin chain
Motivation and excuses
Motivation: The idea of centralizing algebras:
Schur–Weyl duality
Kazhdan–Lusztig duality
Logarithmic Kazhdan–Lusztig duality: kernel of the screenings defines anonsemisimple VOA [FHST]; [Adamovic–Milas], . . . [Tsuchiya–Wood]
The paradigm:screenings (Nichols algebras) vs VOAs/Logarithmic CFTs:
How about screenings vs something else?!Discretized models V b V . . .b V — “spin chains”Discretized versions of (L)CFT models: [Gainutdinov, Reed, Saleur,Vasseur; Gainutdinov, Saleur, Tipuinin, . . . ]Type-A Nichols algebras acting on V b V ˚ . . .V b V ˚, products of afundamental and its dual are centralized by a quotient of qwB.
ApologiesNo LogCFT stuff in this talk
The talk is about this qwB. Work with A. Kiselev and I. Tipunin
Semikhatov Quantum walled Brauer spin chain
Motivation and excuses
Motivation: The idea of centralizing algebras:
Schur–Weyl duality
Kazhdan–Lusztig duality
Logarithmic Kazhdan–Lusztig duality: kernel of the screenings defines anonsemisimple VOA [FHST]; [Adamovic–Milas], . . . [Tsuchiya–Wood]
The paradigm:screenings (Nichols algebras) vs VOAs/Logarithmic CFTs:
How about screenings vs something else?!Discretized models V b V . . .b V — “spin chains”Discretized versions of (L)CFT models: [Gainutdinov, Reed, Saleur,Vasseur; Gainutdinov, Saleur, Tipuinin, . . . ]Type-A Nichols algebras acting on V b V ˚ . . .V b V ˚, products of afundamental and its dual are centralized by a quotient of qwB.
ApologiesNo LogCFT stuff in this talk
The talk is about this qwB. Work with A. Kiselev and I. Tipunin
Semikhatov Quantum walled Brauer spin chain
Motivation and excuses
Motivation: The idea of centralizing algebras:
Schur–Weyl duality
Kazhdan–Lusztig duality
Logarithmic Kazhdan–Lusztig duality: kernel of the screenings defines anonsemisimple VOA [FHST]; [Adamovic–Milas], . . . [Tsuchiya–Wood]
The paradigm:screenings (Nichols algebras) vs VOAs/Logarithmic CFTs:
How about screenings vs something else?!Discretized models V b V . . .b V — “spin chains”Discretized versions of (L)CFT models: [Gainutdinov, Reed, Saleur,Vasseur; Gainutdinov, Saleur, Tipuinin, . . . ]Type-A Nichols algebras acting on V b V ˚ . . .V b V ˚, products of afundamental and its dual are centralized by a quotient of qwB.
ApologiesNo LogCFT stuff in this talk
The talk is about this qwB. Work with A. Kiselev and I. Tipunin
Semikhatov Quantum walled Brauer spin chain
Free Magic
– Do you like card tricks?– No, I hate card tricks.– Well, I’ll just show you this one.
Karl Johnson,The Magician and the Cardsharp:The Search for America’s Greatest
Sleight-of-Hand Artist
Semikhatov Quantum walled Brauer spin chain
Free Magic
– Do you like card tricks?– No, I hate card tricks.– Well, I’ll just show you this one.
Karl Johnson,The Magician and the Cardsharp:The Search for America’s Greatest
Sleight-of-Hand Artist
Semikhatov Quantum walled Brauer spin chain
Free Magic
Take a Young diagram, select a corner:
˚.
Cut the corner off, mark all corners in the new diagram:
˝
˝
˝
, ˚
Mark all possible positions for the corner to join:
˝ ‹
˝ ‹
˝ ‹
‹
,
Semikhatov Quantum walled Brauer spin chain
Free Magic
Take a Young diagram, select a corner:
˚.
Cut the corner off, mark all corners in the new diagram:
˝
˝
˝
, ˚
Mark all possible positions for the corner to join:
˝ ‹
˝ ‹
˝ ‹
‹
,
Semikhatov Quantum walled Brauer spin chain
Free Magic
Take a Young diagram, select a corner:
˚.
Cut the corner off, mark all corners in the new diagram:
˝
˝
˝
, ˚
Mark all possible positions for the corner to join:
˝ ‹
˝ ‹
˝ ‹
‹
,
Semikhatov Quantum walled Brauer spin chain
Free Magic
Mark all possible new positions for the corner to join:
˝ ‹
˝ ‹
˝ ‹
‹
,
Calculate horizontal and vertical distances and writeÿ
‹
ź
‹1‰‹
`
1´ x∆vp‹,‹1qt∆hp‹,‹1
q˘
In this particular example,
`
` `
Semikhatov Quantum walled Brauer spin chain
Free Magic
Mark all possible new positions for the corner to join:
˝ ‹
˝ ‹
˝ ‹
‹
,
Calculate horizontal and vertical distances and writeÿ
‹
ź
‹1‰‹
`
1´ x∆vp‹,‹1qt∆hp‹,‹1
q˘
In this particular example,
p1´ xt´1qp1´ x2t´2qp1´ x3t´4q`p1´ x´1tqp1´ xt´1qp1´ x2t´3q
`p1´ x´2t2qp1´ x´1tqp1´ xt´2q
`p1´ x´3t4qp1´ x´2t3qp1´ x´1t2q
Semikhatov Quantum walled Brauer spin chain
Free Magic
Mark all possible new positions for the corner to join:
˝ ‹
˝ ‹
˝ ‹
‹
,
Calculate horizontal and vertical distances and write
ÿ
‹
ź
˝
`
1´ x∆vp‹,˝qt∆hp‹,˝q˘
ź
‹1‰‹
`
1´ x∆vp‹,‹1qt∆hp‹,‹1
q˘
In this particular example,
p1´ xt´1qp1´ x2t´2qp1´ x3t´4q`p1´ x´1tqp1´ xt´1qp1´ x2t´3q
`p1´ x´2t2qp1´ x´1tqp1´ xt´2q
`p1´ x´3t4qp1´ x´2t3qp1´ x´1t2q
Semikhatov Quantum walled Brauer spin chain
Free Magic
Mark all possible new positions for the corner to join:
˝ ‹
˝ ‹
˝ ‹
‹
,
Calculate horizontal and vertical distances and write
ÿ
‹
ź
˝
`
1´ x∆vp‹,˝qt∆hp‹,˝q˘
ź
‹1‰‹
`
1´ x∆vp‹,‹1qt∆hp‹,‹1
q˘
In this particular example,
p1´ xqp1´ x2t´1qp1´ x3t´3
q
p1´ xt´1qp1´ x2t´2qp1´ x3t´4q`
p1´ tqp1´ xqp1´ x2t´2q
p1´ x´1tqp1´ xt´1qp1´ x2t´3q
`p1´ tqp1´ x´1t2
qp1´ xt´1q
p1´ x´2t2qp1´ x´1tqp1´ xt´2q`
p1´ tqp1´ x´2t4qp1´ x´1t3
q
p1´ x´3t4qp1´ x´2t3qp1´ x´1t2q
Semikhatov Quantum walled Brauer spin chain
Free Magic
Mark all possible new positions for the corner to join:
˝ ‹
˝ ‹
˝ ‹
‹
,
Calculate horizontal and vertical distances and write
ÿ
‹
ź
˝
`
1´ x∆vp‹,˝qt∆hp‹,˝q˘
ź
‹1‰‹
`
1´ x∆vp‹,‹1qt∆hp‹,‹1
q˘“ 1.
In this particular example,
p1´ xqp1´ x2t´1qp1´ x3t´3
q
p1´ xt´1qp1´ x2t´2qp1´ x3t´4q`
p1´ tqp1´ xqp1´ x2t´2q
p1´ x´1tqp1´ xt´1qp1´ x2t´3q
`p1´ tqp1´ x´1t2
qp1´ xt´1q
p1´ x´2t2qp1´ x´1tqp1´ xt´2q`
p1´ tqp1´ x´2t4qp1´ x´1t3
q
p1´ x´3t4qp1´ x´2t3qp1´ x´1t2q“ 1.
Semikhatov Quantum walled Brauer spin chain
Free Magic
More: select in addition a “next-generation” corner, e.g.,
˚1 ˚or
˚
˚1
Then write
ÿ
‹
ź
˝‰˚1
`
1´ x∆vp‹,˝qt∆hp‹,˝q˘
ź
‹1‰‹
`
1´ x∆vp‹,‹1qt∆hp‹,‹1q˘
Semikhatov Quantum walled Brauer spin chain
Free Magic
More: select in addition a “next-generation” corner, e.g.,
˚1 ˚or
˚
˚1
Then write
ÿ
‹
ź
˝‰˚1
`
1´ x∆vp‹,˝qt∆hp‹,˝q˘
ź
‹1‰‹
`
1´ x∆vp‹,‹1qt∆hp‹,‹1q˘
Semikhatov Quantum walled Brauer spin chain
Free Magic
More: select in addition a “next-generation” corner, e.g.,
˚1 ˚or
˚
˚1
Then write
ÿ
‹
ź
˝‰˚1
`
1´ x∆vp‹,˝qt∆hp‹,˝q˘
ź
‹1‰‹
`
1´ x∆vp‹,‹1qt∆hp‹,‹1q˘
p1´ xq�����p1´ x2t´1
qp1´ x3t´3q
p1´ xt´1qp1´ x2t´2qp1´ x3t´4q`
p1´ tq����p1´ xqp1´ x2t´2q
p1´ x´1tqp1´ xt´1qp1´ x2t´3q
`���p1´ tqp1´ x´1t2
qp1´ xt´1q
p1´ x´2t2qp1´ x´1tqp1´ xt´2q`
p1´ tqp1´ x´2t4q�����p1´ x´1t3
q
p1´ x´3t4qp1´ x´2t3qp1´ x´1t2q
Semikhatov Quantum walled Brauer spin chain
Free Magic
More: select in addition a “next-generation” corner, e.g.,
˚1 ˚or
˚
˚1
Then write
ÿ
‹
ź
˝‰˚1
`
1´ x∆vp‹,˝qt∆hp‹,˝q˘
ź
‹1‰‹
`
1´ x∆vp‹,‹1qt∆hp‹,‹1q˘
p1´ xqp1´ x3t´3q
p1´ xt´1qp1´ x2t´2qp1´ x3t´4q`
p1´ tqp1´ x2t´2q
p1´ x´1tqp1´ xt´1qp1´ x2t´3q
`p1´ x´1t2
qp1´ xt´1q
p1´ x´2t2qp1´ x´1tqp1´ xt´2q`
p1´ tqp1´ x´2t4q
p1´ x´3t4qp1´ x´2t3qp1´ x´1t2q
Semikhatov Quantum walled Brauer spin chain
Free Magic
More: select in addition a “next-generation” corner, e.g.,
˚1 ˚or
˚
˚1
Then write
ÿ
‹
ź
˝‰˚1
`
1´ x∆vp‹,˝qt∆hp‹,˝q˘
ź
‹1‰‹
`
1´ x∆vp‹,‹1qt∆hp‹,‹1q˘“ 1.
p1´ xqp1´ x3t´3q
p1´ xt´1qp1´ x2t´2qp1´ x3t´4q`
p1´ tqp1´ x2t´2q
p1´ x´1tqp1´ xt´1qp1´ x2t´3q
`p1´ x´1t2
qp1´ xt´1q
p1´ x´2t2qp1´ x´1tqp1´ xt´2q`
p1´ tqp1´ x´2t4q
p1´ x´3t4qp1´ x´2t3qp1´ x´1t2q“ 1.
Semikhatov Quantum walled Brauer spin chain
Thank you.
Semikhatov Quantum walled Brauer spin chain
The underlying structure: qwB
Hecke
{{TL
##
Hecke
zzBrauer
��walled Brauer
qwBpm, nq
Leduc; Benkart–Chakrabarti–Halverson–Leduc–Lee–Stroomer (1994)Dipper–Doty–Stoll: Schur–Weyl duality on mixed tensor products:
UqpglNq ÝÑ Vbm b V ˚bn ÐÝ qwBpm, nq
Generators: gi (1 ď i ď m ´ 1), E, hj (1 ď i ď n ´ 1).Relations:
gigi “ ´αβ ¨ 1` pα` βqgi , hjhj “ ´αβ ¨ 1` pα` βqhj ,
gihj “ hjgi ,
Egj “ gjE, 2 ď j ď m ´ 1, Ehi “ hiE, 2 ď i ď n ´ 1,
EE “θ ` 1
κpα` βqE,
Eg1E “1
κE, Eh1E “
1
κE,
Eg1h´11 Epg1 ´ h1q “ 0, pg1 ´ h1qEg1h
´11 E “ 0.
Semikhatov Quantum walled Brauer spin chain
The underlying structure: qwB
Hecke
{{TL
##
Hecke
zzBrauer
��walled Brauer
qwBpm, nq
Leduc; Benkart–Chakrabarti–Halverson–Leduc–Lee–Stroomer (1994)Dipper–Doty–Stoll: Schur–Weyl duality on mixed tensor products:
UqpglNq ÝÑ Vbm b V ˚bn ÐÝ qwBpm, nq
Generators: gi (1 ď i ď m ´ 1), E, hj (1 ď i ď n ´ 1).Relations:
gigi “ ´αβ ¨ 1` pα` βqgi , hjhj “ ´αβ ¨ 1` pα` βqhj ,
gihj “ hjgi ,
Egj “ gjE, 2 ď j ď m ´ 1, Ehi “ hiE, 2 ď i ď n ´ 1,
EE “θ ` 1
κpα` βqE,
Eg1E “1
κE, Eh1E “
1
κE,
Eg1h´11 Epg1 ´ h1q “ 0, pg1 ´ h1qEg1h
´11 E “ 0.
Semikhatov Quantum walled Brauer spin chain
The underlying structure: qwB
Hecke
{{TL
##
Hecke
zzBrauer
��walled Brauer
qwBpm, nq
Leduc; Benkart–Chakrabarti–Halverson–Leduc–Lee–Stroomer (1994)Dipper–Doty–Stoll: Schur–Weyl duality on mixed tensor products:
UqpglNq ÝÑ Vbm b V ˚bn ÐÝ qwBpm, nq
Generators: gi (1 ď i ď m ´ 1), E, hj (1 ď i ď n ´ 1).Relations:
gigi “ ´αβ ¨ 1` pα` βqgi , hjhj “ ´αβ ¨ 1` pα` βqhj ,
gihj “ hjgi ,
Egj “ gjE, 2 ď j ď m ´ 1, Ehi “ hiE, 2 ď i ď n ´ 1,
EE “θ ` 1
κpα` βqE,
Eg1E “1
κE, Eh1E “
1
κE,
Eg1h´11 Epg1 ´ h1q “ 0, pg1 ´ h1qEg1h
´11 E “ 0.
Semikhatov Quantum walled Brauer spin chain
The underlying structure: qwB
qwBpm, nq
Leduc; Benkart–Chakrabarti–Halverson–Leduc–Lee–Stroomer (1994)Dipper–Doty–Stoll: Schur–Weyl duality on mixed tensor products:
UqpglNq ÝÑ Vbm b V ˚bn ÐÝ qwBpm, nq
Murakami; Kosuda; Halverson; Enyang; Rui, Song;(Cox, De Visscher, Doty, Martin; Brundan, Stroppel; Stoll, Werth)class.
Generators: gi (1 ď i ď m ´ 1), E, hj (1 ď i ď n ´ 1).Relations:
gigi “ ´αβ ¨ 1` pα` βqgi , hjhj “ ´αβ ¨ 1` pα` βqhj ,
gihj “ hjgi ,
Egj “ gjE, 2 ď j ď m ´ 1, Ehi “ hiE, 2 ď i ď n ´ 1,
EE “θ ` 1
κpα` βqE,
Eg1E “1
κE, Eh1E “
1
κE,
Eg1h´11 Epg1 ´ h1q “ 0, pg1 ´ h1qEg1h
´11 E “ 0.
Semikhatov Quantum walled Brauer spin chain
The underlying structure: qwB
qwBpm, nq
Leduc; Benkart–Chakrabarti–Halverson–Leduc–Lee–Stroomer (1994)Dipper–Doty–Stoll: Schur–Weyl duality on mixed tensor products:
UqpglNq ÝÑ Vbm b V ˚bn ÐÝ qwBpm, nq
Murakami; Kosuda; Halverson; Enyang; Rui, Song;(Cox, De Visscher, Doty, Martin; Brundan, Stroppel; Stoll, Werth)class.
Generators: gi (1 ď i ď m ´ 1), E, hj (1 ď i ď n ´ 1).Relations:
gigi “ ´αβ ¨ 1` pα` βqgi , hjhj “ ´αβ ¨ 1` pα` βqhj ,
gihj “ hjgi ,
Egj “ gjE, 2 ď j ď m ´ 1, Ehi “ hiE, 2 ď i ď n ´ 1,
EE “θ ` 1
κpα` βqE,
Eg1E “1
κE, Eh1E “
1
κE,
Eg1h´11 Epg1 ´ h1q “ 0, pg1 ´ h1qEg1h
´11 E “ 0.
Semikhatov Quantum walled Brauer spin chain
The underlying structure: qwB
qwBpm, nq
Leduc; Benkart–Chakrabarti–Halverson–Leduc–Lee–Stroomer (1994)Dipper–Doty–Stoll: Schur–Weyl duality on mixed tensor products:
UqpglNq ÝÑ Vbm b V ˚bn ÐÝ qwBpm, nq
Generators: gi (1 ď i ď m ´ 1), E, hj (1 ď i ď n ´ 1).
E “
‚ ‚ ‚ ˝ ˝ ˝� � �‚ ‚ ‚ ˝ ˝ ˝
pg3, h1q “
‚ ‚ ‚ ‚ ˝ ˝ ˝
‚ ‚ ‚ ‚ ˝ ˝ ˝
Relations:
gigi “ ´αβ ¨ 1` pα` βqgi , hjhj “ ´αβ ¨ 1` pα` βqhj ,
gihj “ hjgi ,
Egj “ gjE, 2 ď j ď m ´ 1, Ehi “ hiE, 2 ď i ď n ´ 1,
EE “θ ` 1
κpα` βqE,
Eg1E “1
κE, Eh1E “
1
κE,
Eg1h´11 Epg1 ´ h1q “ 0, pg1 ´ h1qEg1h
´11 E “ 0.
Semikhatov Quantum walled Brauer spin chain
The underlying structure: qwB
qwBpm, nq
Leduc; Benkart–Chakrabarti–Halverson–Leduc–Lee–Stroomer (1994)Dipper–Doty–Stoll: Schur–Weyl duality on mixed tensor products:
UqpglNq ÝÑ Vbm b V ˚bn ÐÝ qwBpm, nq
Generators: gi (1 ď i ď m ´ 1), E, hj (1 ď i ď n ´ 1).Relations:
gigi “ ´αβ ¨ 1` pα` βqgi , hjhj “ ´αβ ¨ 1` pα` βqhj ,
gihj “ hjgi ,
Egj “ gjE, 2 ď j ď m ´ 1, Ehi “ hiE, 2 ď i ď n ´ 1,
EE “θ ` 1
κpα` βqE,
Eg1E “1
κE, Eh1E “
1
κE,
Eg1h´11 Epg1 ´ h1q “ 0, pg1 ´ h1qEg1h
´11 E “ 0.
Semikhatov Quantum walled Brauer spin chain
qwB from a braided category
[Ram, Leduc, . . . ]
A braided monoidal category generated by two simple objects,˝ and ‚
The braiding assumed to satisfty the Hecke relation
˝ ˝
˝ ˝
“ ´αβ
˝ ˝
˝ ˝
` pα` βq
˝ ˝
˝ ˝
.
The category is rigid: evaluation and coevaluation morphisms‚ ˝, ˝ ‚,��
‚ ˝and ��
˝ ‚.
Then also
‚ ‚
‚ ‚
“ ´αβ
‚ ‚
‚ ‚
` pα` βq
‚ ‚
‚ ‚
.
Semikhatov Quantum walled Brauer spin chain
qwB from a braided category
[Ram, Leduc, . . . ]
A braided monoidal category generated by two simple objects,˝ and ‚
The braiding assumed to satisfty the Hecke relation
˝ ˝
˝ ˝
“ ´αβ
˝ ˝
˝ ˝
` pα` βq
˝ ˝
˝ ˝
.
The category is rigid: evaluation and coevaluation morphisms‚ ˝, ˝ ‚,��
‚ ˝and ��
˝ ‚.
Then also
‚ ‚
‚ ‚
“ ´αβ
‚ ‚
‚ ‚
` pα` βq
‚ ‚
‚ ‚
.
Semikhatov Quantum walled Brauer spin chain
qwB from a braided category
[Ram, Leduc, . . . ]
A braided monoidal category generated by two simple objects,˝ and ‚
The braiding assumed to satisfty the Hecke relation
˝ ˝
˝ ˝
“ ´αβ
˝ ˝
˝ ˝
` pα` βq
˝ ˝
˝ ˝
.
The category is rigid: evaluation and coevaluation morphisms‚ ˝, ˝ ‚,��
‚ ˝and ��
˝ ‚.
Then also
‚ ‚
‚ ‚
“ ´αβ
‚ ‚
‚ ‚
` pα` βq
‚ ‚
‚ ‚
.
Semikhatov Quantum walled Brauer spin chain
qwB from a braided category
[Ram, Leduc, . . . ]
A braided monoidal category generated by two simple objects,˝ and ‚
The braiding assumed to satisfty the Hecke relation
˝ ˝
˝ ˝
“ ´αβ
˝ ˝
˝ ˝
` pα` βq
˝ ˝
˝ ˝
.
The category is rigid: evaluation and coevaluation morphisms‚ ˝, ˝ ‚,��
‚ ˝and ��
˝ ‚.
Then also
‚ ‚
‚ ‚
“ ´αβ
‚ ‚
‚ ‚
` pα` βq
‚ ‚
‚ ‚
.
Semikhatov Quantum walled Brauer spin chain
qwB from a braided category
The category is ribbon (two constants κ and κ1)
‚ ��
‚
“1
κ
‚
‚
and
˝ ��
˝
“1
κ1
˝
˝
Then also‚ ˝
‚ ˝
“ ´1
αβ
‚ ˝
‚ ˝
`α` β
αβκ
‚ ˝� � �‚ ˝
and
˝ ‚
˝ ‚
“ ´1
αβ
˝ ‚
˝ ‚
`α` β
αβκ1
˝ ‚� � �˝ ‚
.
Semikhatov Quantum walled Brauer spin chain
qwB from a braided category
The category is ribbon (two constants κ and κ1)
‚ ��
‚
“1
κ
‚
‚
and
˝ ��
˝
“1
κ1
˝
˝
Then also‚ ˝
‚ ˝
“ ´1
αβ
‚ ˝
‚ ˝
`α` β
αβκ
‚ ˝� � �‚ ˝
and
˝ ‚
˝ ‚
“ ´1
αβ
˝ ‚
˝ ‚
`α` β
αβκ1
˝ ‚� � �˝ ‚
.
Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
qwBpm, nq “ End ‚ ¨ ¨ ¨ ‚loomoon
m
˝ ¨ ¨ ¨ ˝loomoon
n
Generators: gj (1 ď j ď m ´ 1), hi (1 ď i ď n ´ 1), and E:
E “
‚ ‚ ‚ ˝ ˝ ˝� � �‚ ‚ ‚ ˝ ˝ ˝
pg3, h1q “
‚ ‚ ‚ ‚ ˝ ˝ ˝
‚ ‚ ‚ ‚ ˝ ˝ ˝
Relations: θ “ αβκκ1
The Hecke relations for gj and hi , plus
EE “θ ` 1
κpα` βqE,
Eg1E “1
κE, Eh1E “
1
κE,
Egj “ gjE, 2 ď j ď m ´ 1, Ehi “ hiE, 2 ď i ď n ´ 1,
Eg1h´11 Epg1 ´ h1q “ 0, pg1 ´ h1qEg1h
´11 E “ 0.
Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
qwBpm, nq “ End ‚ ¨ ¨ ¨ ‚loomoon
m
˝ ¨ ¨ ¨ ˝loomoon
n
Generators: gj (1 ď j ď m ´ 1), hi (1 ď i ď n ´ 1), and E:
E “
‚ ‚ ‚ ˝ ˝ ˝� � �‚ ‚ ‚ ˝ ˝ ˝
pg3, h1q “
‚ ‚ ‚ ‚ ˝ ˝ ˝
‚ ‚ ‚ ‚ ˝ ˝ ˝
Relations: θ “ αβκκ1
The Hecke relations for gj and hi , plus
EE “θ ` 1
κpα` βqE,
Eg1E “1
κE, Eh1E “
1
κE,
Egj “ gjE, 2 ď j ď m ´ 1, Ehi “ hiE, 2 ď i ď n ´ 1,
Eg1h´11 Epg1 ´ h1q “ 0, pg1 ´ h1qEg1h
´11 E “ 0.
Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
qwBpm, nq “ End ‚ ¨ ¨ ¨ ‚loomoon
m
˝ ¨ ¨ ¨ ˝loomoon
n
Generators: gj (1 ď j ď m ´ 1), hi (1 ď i ď n ´ 1), and E:
E “
‚ ‚ ‚ ˝ ˝ ˝� � �‚ ‚ ‚ ˝ ˝ ˝
pg3, h1q “
‚ ‚ ‚ ‚ ˝ ˝ ˝
‚ ‚ ‚ ‚ ˝ ˝ ˝
Relations: θ “ αβκκ1
The Hecke relations for gj and hi , plus
EE “θ ` 1
κpα` βqE,
Eg1E “1
κE, Eh1E “
1
κE,
Egj “ gjE, 2 ď j ď m ´ 1, Ehi “ hiE, 2 ď i ď n ´ 1,
Eg1h´11 Epg1 ´ h1q “ 0, pg1 ´ h1qEg1h
´11 E “ 0.
Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
“Why” these relations?The relations Eg1E “
1
κE, Eh1E “
1
κE are just
‚ ‚ ˝����
‚ ‚ ˝
“1
κ
‚ ‚ ˝��
‚ ‚ ˝
,
‚ ˝ ˝����‚ ˝ ˝
“1
κ
‚ ˝ ˝��‚ ˝ ˝
.
Relations Eg1h´11 Epg1 ´ h1q “ 0, pg1 ´ h1qEg1h
´11 E “ 0 hold
because
Eg1h´11 E “
‚ ‚ ˝ ˝� �� ���‚ ‚ ˝ ˝
, so
‚ ‚ ˝ ˝
� � “
‚ ‚ ˝ ˝
� �Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
“Why” these relations?The relations Eg1E “
1
κE, Eh1E “
1
κE are just
‚ ‚ ˝����
‚ ‚ ˝
“1
κ
‚ ‚ ˝��
‚ ‚ ˝
,
‚ ˝ ˝����‚ ˝ ˝
“1
κ
‚ ˝ ˝��‚ ˝ ˝
.
Relations Eg1h´11 Epg1 ´ h1q “ 0, pg1 ´ h1qEg1h
´11 E “ 0 hold
because
Eg1h´11 E “
‚ ‚ ˝ ˝� �� ���‚ ‚ ˝ ˝
, so
‚ ‚ ˝ ˝
� � “
‚ ‚ ˝ ˝
� �Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
Crystal/impurity interpretation:
Isomorphic algebra: endomorphisms of ‚ ‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝ ‚ ‚ . . .m “ n, uniform distribution of impurities
‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝looooomooooon
m
is expected to produce good CFT as mÑ8.
My personal qwBpm, nq story
(quotients of) qwBpm, nq are the centralizers of An Nichols algebrason products of “fundamental/antifundamental” Yetter–Drinfeldmodules.The case of A2:
the bosonized (nonbraided) algebra is Uqs`p2|1q;V and V ˚ are three-dimensional (“spin 1”)The kernel in qwBpm, nq can be described, so far for generic q.. . .integrable versions of the t–J model.
Today:
more about qwB per se.
Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
Crystal/impurity interpretation:
Isomorphic algebra: endomorphisms of ‚ ‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝ ‚ ‚ . . .m “ n, uniform distribution of impurities
‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝looooomooooon
m
is expected to produce good CFT as mÑ8.
My personal qwBpm, nq story
(quotients of) qwBpm, nq are the centralizers of An Nichols algebrason products of “fundamental/antifundamental” Yetter–Drinfeldmodules.The case of A2:
the bosonized (nonbraided) algebra is Uqs`p2|1q;V and V ˚ are three-dimensional (“spin 1”)The kernel in qwBpm, nq can be described, so far for generic q.. . .integrable versions of the t–J model.
Today:
more about qwB per se.
Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
Crystal/impurity interpretation:
Isomorphic algebra: endomorphisms of ‚ ‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝ ‚ ‚ . . .m “ n, uniform distribution of impurities
‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝looooomooooon
m
is expected to produce good CFT as mÑ8.
My personal qwBpm, nq story
(quotients of) qwBpm, nq are the centralizers of An Nichols algebrason products of “fundamental/antifundamental” Yetter–Drinfeldmodules.The case of A2:
the bosonized (nonbraided) algebra is Uqs`p2|1q;V and V ˚ are three-dimensional (“spin 1”)The kernel in qwBpm, nq can be described, so far for generic q.. . .integrable versions of the t–J model.
Today:
more about qwB per se.
Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
Crystal/impurity interpretation:
Isomorphic algebra: endomorphisms of ‚ ‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝ ‚ ‚ . . .m “ n, uniform distribution of impurities
‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝looooomooooon
m
is expected to produce good CFT as mÑ8.
My personal qwBpm, nq story
(quotients of) qwBpm, nq are the centralizers of An Nichols algebrason products of “fundamental/antifundamental” Yetter–Drinfeldmodules.The case of A2:
the bosonized (nonbraided) algebra is Uqs`p2|1q;V and V ˚ are three-dimensional (“spin 1”)The kernel in qwBpm, nq can be described, so far for generic q.. . .integrable versions of the t–J model.
Today:
more about qwB per se.
Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
Crystal/impurity interpretation:
Isomorphic algebra: endomorphisms of ‚ ‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝ ‚ ‚ . . .m “ n, uniform distribution of impurities
‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝looooomooooon
m
is expected to produce good CFT as mÑ8.
My personal qwBpm, nq story
(quotients of) qwBpm, nq are the centralizers of An Nichols algebrason products of “fundamental/antifundamental” Yetter–Drinfeldmodules.The case of A2:
the bosonized (nonbraided) algebra is Uqs`p2|1q;V and V ˚ are three-dimensional (“spin 1”)The kernel in qwBpm, nq can be described, so far for generic q.. . .integrable versions of the t–J model.
Today:
more about qwB per se.
Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
Crystal/impurity interpretation:
Isomorphic algebra: endomorphisms of ‚ ‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝ ‚ ‚ . . .m “ n, uniform distribution of impurities
‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝looooomooooon
m
is expected to produce good CFT as mÑ8.
My personal qwBpm, nq story
(quotients of) qwBpm, nq are the centralizers of An Nichols algebrason products of “fundamental/antifundamental” Yetter–Drinfeldmodules.The case of A2:
the bosonized (nonbraided) algebra is Uqs`p2|1q;V and V ˚ are three-dimensional (“spin 1”)The kernel in qwBpm, nq can be described, so far for generic q.. . .integrable versions of the t–J model.
Today:
more about qwB per se.
Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
Crystal/impurity interpretation:
Isomorphic algebra: endomorphisms of ‚ ‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝ ‚ ‚ . . .m “ n, uniform distribution of impurities
‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝looooomooooon
m
is expected to produce good CFT as mÑ8.
My personal qwBpm, nq story
(quotients of) qwBpm, nq are the centralizers of An Nichols algebrason products of “fundamental/antifundamental” Yetter–Drinfeldmodules.The case of A2:
the bosonized (nonbraided) algebra is Uqs`p2|1q;V and V ˚ are three-dimensional (“spin 1”)The kernel in qwBpm, nq can be described, so far for generic q.. . .integrable versions of the t–J model.
Today:
more about qwB per se.
Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
Crystal/impurity interpretation:
Isomorphic algebra: endomorphisms of ‚ ‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝ ‚ ‚ . . .m “ n, uniform distribution of impurities
‚ ˝ ‚ ¨ ¨ ¨ ‚ ˝looooomooooon
m
is expected to produce good CFT as mÑ8.
My personal qwBpm, nq story
(quotients of) qwBpm, nq are the centralizers of An Nichols algebrason products of “fundamental/antifundamental” Yetter–Drinfeldmodules.The case of A2:
the bosonized (nonbraided) algebra is Uqs`p2|1q;V and V ˚ are three-dimensional (“spin 1”)The kernel in qwBpm, nq can be described, so far for generic q.. . .integrable versions of the t–J model.
Today:
more about qwB per se.
Semikhatov Quantum walled Brauer spin chain
The algebra qwBpm, nq
My personal qwBpm, nq story
(quotients of) qwBpm, nq are the centralizers of An Nichols algebrason products of “fundamental/antifundamental” Yetter–Drinfeldmodules.The case of A2:
the bosonized (nonbraided) algebra is Uqs`p2|1q;V and V ˚ are three-dimensional (“spin 1”)The kernel in qwBpm, nq can be described, so far for generic q.. . .integrable versions of the t–J model.
Today:
more about qwB per se.
Semikhatov Quantum walled Brauer spin chain
Baxterization
Hecke relations can be equivalently stated as the Yang–Baxter equations
gi pwqgi`1pzwqgi pzq “ gi`1pzqgi pzwqgi`1pwq
or
‚ ‚ ‚
z
zw
w
‚ ‚ ‚
“
‚ ‚ ‚
w
zw
z
‚ ‚ ‚
,
for
gi pzq “ gi `α` β
z ´ 1¨ 1, or
‚ ‚
z‚ ‚
“
‚ ‚
‚ ‚
`α` β
z ´ 1
‚ ‚
‚ ‚
z P C.
Semikhatov Quantum walled Brauer spin chain
Baxterization
What about the “mixed” relations, e.g.,
‚ ˝ ‚
z
zw
w
‚ ˝ ‚
“
‚ ˝ ‚
w
zw
z
‚ ˝ ‚
.
The “mixed braidings”‚ ˝
‚ ˝
are NOT elements of qwB.
Baxterization uses the evaluation and coevaluation morphisms fromthe category (NOT elements of qwB!)
‚ ˝
z
˝ ‚
“
‚ ˝
˝ ‚
`α` β
αβ
u
z ´ u
‚ ˝��˝ ‚
,
˝ ‚
z
‚ ˝
“
˝ ‚
‚ ˝
`α` β
αβ
v
z ´ v
˝ ‚��‚ ˝
uv “ θ ” αβκκ1
Semikhatov Quantum walled Brauer spin chain
Baxterization
What about the “mixed” relations, e.g.,
‚ ˝ ‚
z
zw
w
‚ ˝ ‚
“
‚ ˝ ‚
w
zw
z
‚ ˝ ‚
.
The “mixed braidings”‚ ˝
‚ ˝
are NOT elements of qwB.
Baxterization uses the evaluation and coevaluation morphisms fromthe category (NOT elements of qwB!)
‚ ˝
z
˝ ‚
“
‚ ˝
˝ ‚
`α` β
αβ
u
z ´ u
‚ ˝��˝ ‚
,
˝ ‚
z
‚ ˝
“
˝ ‚
‚ ˝
`α` β
αβ
v
z ´ v
˝ ‚��‚ ˝
uv “ θ ” αβκκ1
Semikhatov Quantum walled Brauer spin chain
Baxterization
What about the “mixed” relations, e.g.,
‚ ˝ ‚
z
zw
w
‚ ˝ ‚
“
‚ ˝ ‚
w
zw
z
‚ ˝ ‚
.
The “mixed braidings”‚ ˝
‚ ˝
are NOT elements of qwB.
Baxterization uses the evaluation and coevaluation morphisms fromthe category (NOT elements of qwB!)
‚ ˝
z
˝ ‚
“
‚ ˝
˝ ‚
`α` β
αβ
u
z ´ u
‚ ˝��˝ ‚
,
˝ ‚
z
‚ ˝
“
˝ ‚
‚ ˝
`α` β
αβ
v
z ´ v
˝ ‚��‚ ˝
uv “ θ ” αβκκ1
Semikhatov Quantum walled Brauer spin chain
“Conservation laws”—a commutative familyDefine
κpα` βqAm,npz ,wq “
� �m n˝ ‚ ‚ ˝
z
z
w
w
˝ ‚ ‚ ˝ (“2-row transfer matrix”).
Max pole: pz ´ 1q´mpz ´ uq´npw ´ vq´npw ´ 1q´m.
Theorem:
Am,npz ,wq P qwBpm, nqpz ,wq.For a fixed ρ P CP1, set Am,npzq “ Am,npz , ρzq. Then Am,npzq is agenerating function for a commutative family of elements ofqwBpm, nq:
Am,npzqAm,npwq ´Am,npwqAm,npzq “ 0.
Am,npz ,wq “ ζpz ,wq ¨ 1` pzw ` θqp1´ zwqAm,npz ,wq,Am,npz ,wq regular at w “ 1{z and at w “ ´θ{z .
Semikhatov Quantum walled Brauer spin chain
“Conservation laws”—a commutative familyDefine
κpα` βqAm,npz ,wq “
� �m n˝ ‚ ‚ ˝
z
z
w
w
˝ ‚ ‚ ˝ .
Max pole: pz ´ 1q´mpz ´ uq´npw ´ vq´npw ´ 1q´m.
Theorem:
Am,npz ,wq P qwBpm, nqpz ,wq.For a fixed ρ P CP1, set Am,npzq “ Am,npz , ρzq. Then Am,npzq is agenerating function for a commutative family of elements ofqwBpm, nq:
Am,npzqAm,npwq ´Am,npwqAm,npzq “ 0.
Am,npz ,wq “ ζpz ,wq ¨ 1` pzw ` θqp1´ zwqAm,npz ,wq,Am,npz ,wq regular at w “ 1{z and at w “ ´θ{z .
Semikhatov Quantum walled Brauer spin chain
“Conservation laws”—a commutative familyDefine
κpα` βqAm,npz ,wq “
� �m n˝ ‚ ‚ ˝
z
z
w
w
˝ ‚ ‚ ˝ .
Max pole: pz ´ 1q´mpz ´ uq´npw ´ vq´npw ´ 1q´m.
Theorem:
Am,npz ,wq P qwBpm, nqpz ,wq.For a fixed ρ P CP1, set Am,npzq “ Am,npz , ρzq. Then Am,npzq is agenerating function for a commutative family of elements ofqwBpm, nq:
Am,npzqAm,npwq ´Am,npwqAm,npzq “ 0.
Am,npz ,wq “ ζpz ,wq ¨ 1` pzw ` θqp1´ zwqAm,npz ,wq,Am,npz ,wq regular at w “ 1{z and at w “ ´θ{z .
Semikhatov Quantum walled Brauer spin chain
“Conservation laws”—a commutative familyDefine
κpα` βqAm,npz ,wq “
� �m n˝ ‚ ‚ ˝
z
z
w
w
˝ ‚ ‚ ˝ .
Max pole: pz ´ 1q´mpz ´ uq´npw ´ vq´npw ´ 1q´m.
Theorem:
Am,npz ,wq P qwBpm, nqpz ,wq.For a fixed ρ P CP1, set Am,npzq “ Am,npz , ρzq. Then Am,npzq is agenerating function for a commutative family of elements ofqwBpm, nq:
Am,npzqAm,npwq ´Am,npwqAm,npzq “ 0.
Am,npz ,wq “ ζpz ,wq ¨ 1` pzw ` θqp1´ zwqAm,npz ,wq,Am,npz ,wq regular at w “ 1{z and at w “ ´θ{z .
Semikhatov Quantum walled Brauer spin chain
Conservation laws — a commutative family
Proof that Am,npz ,wq P qwBpm, nqpz ,wq: by recursion relations.
Identity� �˝ ‚ ‚ ˝
z��w
˝ ‚ ‚ ˝ “ ´αβ
‚ ˝
‚ ˝
`αβκpα` βq
θ
zw ` θ
pz ´ 1qpw ´ 1q
‚ ˝� � �‚ ˝
implies that
Am,npz ,wq “ ´αβAm´1,npz ,wq
`αβ
θ
zw ` θ
pz ´ 1qpw ´ 1qp´αβqm´n´1Jm,npz ,wq, m ě 1,
Semikhatov Quantum walled Brauer spin chain
Conservation laws — a commutative family
Proof that Am,npz ,wq P qwBpm, nqpz ,wq: by recursion relations.
Am,npz ,wq “ ´αβAm´1,npz ,wq
`αβ
θ
zw ` θ
pz ´ 1qpw ´ 1qp´αβqm´n´1Jm,npz ,wq, m ě 1,
where
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w
‚ ‚ ˝
Assume Jm,npz ,wq known and deal with A0,npz ,wq.
Semikhatov Quantum walled Brauer spin chain
Conservation laws — a commutative family
Proof that Am,npz ,wq P qwBpm, nqpz ,wq: by recursion relations.
Am,npz ,wq “ ´αβAm´1,npz ,wq
`αβ
θ
zw ` θ
pz ´ 1qpw ´ 1qp´αβqm´n´1Jm,npz ,wq, m ě 1,
where
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w
‚ ‚ ˝
Assume Jm,npz ,wq known and deal with A0,npz ,wq.
Semikhatov Quantum walled Brauer spin chain
Conservation laws — a commutative family
Proof that Am,npz ,wq P qwBpm, nqpz ,wq: by recursion relations.
Another (very simple) recursion:
A0,npz ,wq “ ´1
αβA0,n´1pz ,wq `
1
αβ
zw ´ 1
pz ´ uqpw ´ vqrJ0,npz ,wq,
where
rJ0,npz ,wq “`
r
rhn´1pwq . . .r
rh1pwq˘`
rh1pzq . . . rhn´1pzq˘
,
rhi pzq “˝ ˝
˝ ˝
`pα` βqu
αβ
1
z ´ u
˝ ˝
˝ ˝
,
r
rhi pzq “˝ ˝
˝ ˝
`pα` βqv
αβ
1
z ´ v
˝ ˝
˝ ˝
.
So A0,npz ,wq P qwBp˚, nq if we show that Jm,npz ,wq P qwBp˚, nq.
Semikhatov Quantum walled Brauer spin chain
Conservation laws — a commutative family
Proof that Am,npz ,wq P qwBpm, nqpz ,wq: by recursion relations.
Another (very simple) recursion:
A0,npz ,wq “ ´1
αβA0,n´1pz ,wq `
1
αβ
zw ´ 1
pz ´ uqpw ´ vqrJ0,npz ,wq,
where
rJ0,npz ,wq “`
r
rhn´1pwq . . .r
rh1pwq˘`
rh1pzq . . . rhn´1pzq˘
,
rhi pzq “˝ ˝
˝ ˝
`pα` βqu
αβ
1
z ´ u
˝ ˝
˝ ˝
,
r
rhi pzq “˝ ˝
˝ ˝
`pα` βqv
αβ
1
z ´ v
˝ ˝
˝ ˝
.
So A0,npz ,wq P qwBp˚, nq if we show that Jm,npz ,wq P qwBp˚, nq.
Semikhatov Quantum walled Brauer spin chain
Conservation laws — a commutative family
Proof that Am,npz ,wq P qwBpm, nqpz ,wq: by recursion relations.
Return to
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w‚ ‚ ˝
Two more relations:
Jm,npz ,wq “ ´1
αβgm´1pwqJm´1,npz ,wqgm´1pzq, m ě 2,
J1,npz ,wq “ J1,n´1pz ,wq ´α` β
αβ
1´ zw
pz ´ uqpw ´ vqκp´αβqnU1,npz ,wq,
where U1,npz ,wq “r
rhn´1pwq . . .r
rh1pwqE rh1pzq . . . rhn´1pzq.ùñ Am,npz ,wq is expressed in terms of qwBpm, nq generators withcoefficients rational in z ,w .
Semikhatov Quantum walled Brauer spin chain
Conservation laws — a commutative family
Proof that Am,npz ,wq P qwBpm, nqpz ,wq: by recursion relations.
Return to
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w‚ ‚ ˝
Two more relations:
Jm,npz ,wq “ ´1
αβgm´1pwqJm´1,npz ,wqgm´1pzq, m ě 2,
J1,npz ,wq “ J1,n´1pz ,wq ´α` β
αβ
1´ zw
pz ´ uqpw ´ vqκp´αβqnU1,npz ,wq,
where U1,npz ,wq “r
rhn´1pwq . . .r
rh1pwqE rh1pzq . . . rhn´1pzq.ùñ Am,npz ,wq is expressed in terms of qwBpm, nq generators withcoefficients rational in z ,w .
Semikhatov Quantum walled Brauer spin chain
Conservation laws — a commutative family
Proof that Am,npz ,wq P qwBpm, nqpz ,wq: by recursion relations.
Return to
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w‚ ‚ ˝
Two more relations:
Jm,npz ,wq “ ´1
αβgm´1pwqJm´1,npz ,wqgm´1pzq, m ě 2,
J1,npz ,wq “ J1,n´1pz ,wq ´α` β
αβ
1´ zw
pz ´ uqpw ´ vqκp´αβqnU1,npz ,wq,
where U1,npz ,wq “r
rhn´1pwq . . .r
rh1pwqE rh1pzq . . . rhn´1pzq.ùñ Am,npz ,wq is expressed in terms of qwBpm, nq generators withcoefficients rational in z ,w .
Semikhatov Quantum walled Brauer spin chain
Conservation laws — a commutative family
Proof of a commutative family:
By a version of the “train argument.”
Semikhatov Quantum walled Brauer spin chain
Conservation laws — a commutative family
Solving the recursion: Relatively explicit expressions
Jm,npz ,wq “ p´αβq´m`1gm´1pwq . . . g1pwq
´
1`p1´ zwqpα` βqκ
pz ´ uqpw ´ vq
nÿ
s“1
p´αβqs´1rrhs´1pwq . . .
r
rh1pwqErh1pzq . . . rhs´1pzq¯
g1pzq . . . gm´1pzq,
Am,npz ,wq “ ζpz ,wq ¨ 1´zw ` θ
θpz ´ 1qpw ´ 1qp´αβqm´n
mÿ
j“1
Jj ,npz ,wq
`1´ zw
pz ´ uqpw ´ vq
n´1ÿ
i“1
p´αβqm´iJi ,n´1pz ,wq,
It follows that
Am,npz ,wq “ ζpz ,wq ¨ 1´ pzw ` θqp1´ zwqAm,npz ,wq,
with Am,npz ,wq P qwBpm, nqpz ,wq regular at w “ 1{z and at w “ ´θ{z .
Semikhatov Quantum walled Brauer spin chain
Conservation laws — a commutative family
Solving the recursion: Relatively explicit expressions
Jm,npz ,wq “ p´αβq´m`1gm´1pwq . . . g1pwq
´
1`p1´ zwqpα` βqκ
pz ´ uqpw ´ vq
nÿ
s“1
p´αβqs´1rrhs´1pwq . . .
r
rh1pwqErh1pzq . . . rhs´1pzq¯
g1pzq . . . gm´1pzq,
Am,npz ,wq “ ζpz ,wq ¨ 1´zw ` θ
θpz ´ 1qpw ´ 1qp´αβqm´n
mÿ
j“1
Jj ,npz ,wq
`1´ zw
pz ´ uqpw ´ vq
n´1ÿ
i“1
p´αβqm´iJi ,n´1pz ,wq,
where Ji ,n´1pz ,wq “`
rhi pzq . . . rhn´1pzq˘`
r
rhn´1pwq . . .r
rhi pwq˘
.
It follows that
Am,npz ,wq “ ζpz ,wq ¨ 1´ pzw ` θqp1´ zwqAm,npz ,wq,
with Am,npz ,wq P qwBpm, nqpz ,wq regular at w “ 1{z and at w “ ´θ{z .
Semikhatov Quantum walled Brauer spin chain
Conservation laws — a commutative family
Solving the recursion: Relatively explicit expressions
Jm,npz ,wq “ p´αβq´m`1gm´1pwq . . . g1pwq
´
1`p1´ zwqpα` βqκ
pz ´ uqpw ´ vq
nÿ
s“1
p´αβqs´1rrhs´1pwq . . .
r
rh1pwqErh1pzq . . . rhs´1pzq¯
g1pzq . . . gm´1pzq,
Am,npz ,wq “ ζpz ,wq ¨ 1´zw ` θ
θpz ´ 1qpw ´ 1qp´αβqm´n
mÿ
j“1
Jj ,npz ,wq
`1´ zw
pz ´ uqpw ´ vq
n´1ÿ
i“1
p´αβqm´iJi ,n´1pz ,wq,
where Ji ,n´1pz ,wq “`
rhi pzq . . . rhn´1pzq˘`
r
rhn´1pwq . . .r
rhi pwq˘
.
It follows that
Am,npz ,wq “ ζpz ,wq ¨ 1´ pzw ` θqp1´ zwqAm,npz ,wq,
with Am,npz ,wq P qwBpm, nqpz ,wq regular at w “ 1{z and at w “ ´θ{z .
Semikhatov Quantum walled Brauer spin chain
Commutative families
Hamiltonians: two parameters, u and ρ.
Am,npz , ρ2zq “ p. . . q ¨ 1` 2pz ´
1
ρqHp1qm,npρq ` . . . .
Then
Hp1qm,npρq “ ´ρ2
p1´ ρuqpρ´ vq
n´1ÿ
i“1
p´αβqm´iJi,n´1p1
ρ, ρq
´pα` βqp1` θqρ3
θpρ´ 1q2p1´ ρuqpρ´ vqκ
mÿ
i“1
nÿ
j“1
p´αβqm´n´i`jEi,jp1
ρ, ρq
´
mÿ
k“2
k´1ÿ
j“1
p´1qj´k`1p´αβqm´k`1´n ρ3pβ ` αρqk´1´j
pα` βρqk´1´j
θpρ´ 1q2p1´j`kq
ˆ gk´1pρq . . . gk´j`1pρqgk´jp1
ρq . . . gk´1p
1
ρq
where
Ei,jpz ,wq “r
rhj´1pwq . . .r
rh1pwqgi´1pwq . . . g1pwqEg1pzq . . . gi´1pzqrh1pzq . . . rhj´1pzq.
Semikhatov Quantum walled Brauer spin chain
Commutative families
Hamiltonians: two parameters, u and ρ.
Compare with the t–J modelsTo be done.
Diagonalizing the commutative family:
By Bethe ansatzTo be done.
“Instead:” diagonalize Jucys–Murphy elments
Take z ,w Ñ8.Generic values of the parameters of the algebra.
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w
‚ ‚ ˝
Jm,n “
‚ ‚ ˝
‚ ‚ ˝Semikhatov Quantum walled Brauer spin chain
Commutative families
Hamiltonians: two parameters, u and ρ.
Compare with the t–J modelsTo be done.
Diagonalizing the commutative family:
By Bethe ansatzTo be done.
“Instead:” diagonalize Jucys–Murphy elments
Take z ,w Ñ8.Generic values of the parameters of the algebra.
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w
‚ ‚ ˝
Jm,n “
‚ ‚ ˝
‚ ‚ ˝Semikhatov Quantum walled Brauer spin chain
Commutative families
Hamiltonians: two parameters, u and ρ.
Compare with the t–J modelsTo be done.
Diagonalizing the commutative family:
By Bethe ansatzTo be done.
“Instead:” diagonalize Jucys–Murphy elments
Take z ,w Ñ8.Generic values of the parameters of the algebra.
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w
‚ ‚ ˝
Jm,n “
‚ ‚ ˝
‚ ‚ ˝Semikhatov Quantum walled Brauer spin chain
Commutative families
Hamiltonians: two parameters, u and ρ.
Compare with the t–J modelsTo be done.
Diagonalizing the commutative family:
By Bethe ansatzTo be done.
“Instead:” diagonalize Jucys–Murphy elments
Take z ,w Ñ8.Generic values of the parameters of the algebra.
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w
‚ ‚ ˝
Jm,n “
‚ ‚ ˝
‚ ‚ ˝Semikhatov Quantum walled Brauer spin chain
Commutative families
Hamiltonians: two parameters, u and ρ.
Compare with the t–J modelsTo be done.
Diagonalizing the commutative family:
By Bethe ansatzTo be done.
“Instead:” diagonalize Jucys–Murphy elments
Take z ,w Ñ8.Generic values of the parameters of the algebra.
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w
‚ ‚ ˝
Jm,n “
‚ ‚ ˝
‚ ‚ ˝Semikhatov Quantum walled Brauer spin chain
Commutative families
Hamiltonians: two parameters, u and ρ.
Compare with the t–J modelsTo be done.
Diagonalizing the commutative family:
By Bethe ansatzTo be done.
“Instead:” diagonalize Jucys–Murphy elments
Take z ,w Ñ8.Generic values of the parameters of the algebra.
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w
‚ ‚ ˝
Jm,n “
‚ ‚ ˝
‚ ‚ ˝Semikhatov Quantum walled Brauer spin chain
Commutative families
Hamiltonians: two parameters, u and ρ.
Compare with the t–J modelsTo be done.
Diagonalizing the commutative family:
By Bethe ansatzTo be done.
“Instead:” diagonalize Jucys–Murphy elments
Take z ,w Ñ8.Generic values of the parameters of the algebra.
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w
‚ ‚ ˝
Jm,n “
‚ ‚ ˝
‚ ‚ ˝Semikhatov Quantum walled Brauer spin chain
Commutative families
“Instead:” diagonalize Jucys–Murphy elments
Take z ,w Ñ8.Generic values of the parameters of the algebra.
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w
‚ ‚ ˝
Jm,n “
‚ ‚ ˝
‚ ‚ ˝
Semikhatov Quantum walled Brauer spin chain
Commutative families
“Instead:” diagonalize Jucys–Murphy elments
Take z ,w Ñ8.Generic values of the parameters of the algebra.
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w
‚ ‚ ˝
Jm,n “
‚ ‚ ˝
‚ ‚ ˝
Semikhatov Quantum walled Brauer spin chain
Commutative families
“Instead:” diagonalize Jucys–Murphy elments
Take z ,w Ñ8.Generic values of the parameters of the algebra.
Jm,npz ,wq “
‚ ‚ ˝z
z
w
w
‚ ‚ ˝
Jm,n “
‚ ‚ ˝
‚ ‚ ˝
Semikhatov Quantum walled Brauer spin chain
Jucys–Murphy elements
Define m ` n ´ 1 elements
Jm,n “ p´αβqn´m`1
‚ ‚ ‚ ˝ ˝ ˝
‚ ‚ ‚ ˝ ˝ ˝
, Jm´1,n “ p´αβqn´m`2
‚ ‚ ‚ ˝ ˝ ˝
‚ ‚ ‚ ˝ ˝ ˝
,
p´αβq´n`1
‚ ‚ ‚ ˝ ˝ ˝
‚ ‚ ‚ ˝ ˝ ˝
, . . . , p´αβq´1
‚ ‚ ‚ ˝ ˝ ˝
‚ ‚ ‚ ˝ ˝ ˝
Semikhatov Quantum walled Brauer spin chain
Jucys–Murphy elements
The m ` n ´ 1 elements relabeled:
Jpnqm`n “ p´αβqn´m`1
‚ ‚ ‚ ˝ ˝ ˝
‚ ‚ ‚ ˝ ˝ ˝
, Jpnqm`n´1 “ p´αβqn´m`2
‚ ‚ ‚ ˝ ˝ ˝
‚ ‚ ‚ ˝ ˝ ˝
,
Jpnqn “ p´αβq´n`1
‚ ‚ ‚ ˝ ˝ ˝
‚ ‚ ‚ ˝ ˝ ˝
, . . . , Jpnq2 “ p´αβq´1
‚ ‚ ‚ ˝ ˝ ˝
‚ ‚ ‚ ˝ ˝ ˝
Semikhatov Quantum walled Brauer spin chain
Jucys–Murphy elements
The m ` n ´ 1 elements are in qwBpm, nq,
they commute:
JpnqiJpnqj “ JpnqjJpnqi , 2 ď j , j ď m ` n,
and have the explicit form
Jpnqn`j “ p´αβq1´jˆ
gj´1,1
´
1´ pα` βqκnÿ
i“1
p´αβqi´1h´1i´1,1 E h´1
1,i´1
¯
g1,j´1,
where g˘1i ,j and h˘1
i ,j are contiguous products,
gm,n “
#
gmgm`1 . . . gn, m ď n,
gmgm´1 . . . gn, m ą n,g´1m,n “
#
g´1m g´1
m`1 . . . g´1n , m ď n,
g´1m g´1
m´1 . . . g´1n , m ą n.
Semikhatov Quantum walled Brauer spin chain
Jucys–Murphy elements
The m ` n ´ 1 elements are in qwBpm, nq,
they commute:
JpnqiJpnqj “ JpnqjJpnqi , 2 ď j , j ď m ` n,
and have the explicit form
Jpnqn`j “ p´αβq1´jˆ
gj´1,1
´
1´ pα` βqκnÿ
i“1
p´αβqi´1h´1i´1,1 E h´1
1,i´1
¯
g1,j´1,
where g˘1i ,j and h˘1
i ,j are contiguous products,
gm,n “
#
gmgm`1 . . . gn, m ď n,
gmgm´1 . . . gn, m ą n,g´1m,n “
#
g´1m g´1
m`1 . . . g´1n , m ď n,
g´1m g´1
m´1 . . . g´1n , m ą n.
Semikhatov Quantum walled Brauer spin chain
Jucys–Murphy elements
The m ` n ´ 1 elements are in qwBpm, nq,
they commute:
JpnqiJpnqj “ JpnqjJpnqi , 2 ď j , j ď m ` n,
and have the explicit form
Jpnqn`j “ p´αβq1´jˆ
gj´1,1
´
1´ pα` βqκnÿ
i“1
p´αβqi´1h´1i´1,1 E h´1
1,i´1
¯
g1,j´1,
where g˘1i ,j and h˘1
i ,j are contiguous products,
gm,n “
#
gmgm`1 . . . gn, m ď n,
gmgm´1 . . . gn, m ą n,g´1m,n “
#
g´1m g´1
m`1 . . . g´1n , m ď n,
g´1m g´1
m´1 . . . g´1n , m ą n.
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
An irrep V λ1,λ2 is characterized by pλ1, λ2q
— two Young diagrams such that
m ´ |λ1| “ n ´ |λ2| “: f ě 0.
Fix pλ1, λ2q.
Let Y`f pλ2q be the set of all Young diagrams obtained by adding fboxes to λ2.For any λ1 P Y`f pλ2q, these f boxes form a skew shape λ1{λ2, suchthat |λ1| ` |λ
1{λ2| “ n.A hypertableau is a filling of pλ1, λ
1{λ2, λ1q in accordance with the
following rules:
1 λ1 is filled with 1, . . . , n into a standard tableau.2 The disjoint union λ1 \ pλ
1{λ2q is filled with 11, . . . , m1 suchthat λ1 is made into a standard tableau and λ1{λ2, into anantistandard tableau.
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
An irrep V λ1,λ2 is characterized by pλ1, λ2q
— two Young diagrams such that
m ´ |λ1| “ n ´ |λ2| “: f ě 0.
Fix pλ1, λ2q.
Let Y`f pλ2q be the set of all Young diagrams obtained by adding fboxes to λ2.For any λ1 P Y`f pλ2q, these f boxes form a skew shape λ1{λ2, suchthat |λ1| ` |λ
1{λ2| “ n.A hypertableau is a filling of pλ1, λ
1{λ2, λ1q in accordance with the
following rules:
1 λ1 is filled with 1, . . . , n into a standard tableau.2 The disjoint union λ1 \ pλ
1{λ2q is filled with 11, . . . , m1 suchthat λ1 is made into a standard tableau and λ1{λ2, into anantistandard tableau.
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
An irrep V λ1,λ2 is characterized by pλ1, λ2q
— two Young diagrams such that
m ´ |λ1| “ n ´ |λ2| “: f ě 0.
Fix pλ1, λ2q.
Let Y`f pλ2q be the set of all Young diagrams obtained by adding fboxes to λ2.For any λ1 P Y`f pλ2q, these f boxes form a skew shape λ1{λ2, suchthat |λ1| ` |λ
1{λ2| “ n.A hypertableau is a filling of pλ1, λ
1{λ2, λ1q in accordance with the
following rules:
1 λ1 is filled with 1, . . . , n into a standard tableau.2 The disjoint union λ1 \ pλ
1{λ2q is filled with 11, . . . , m1 suchthat λ1 is made into a standard tableau and λ1{λ2, into anantistandard tableau.
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
An irrep V λ1,λ2 is characterized by pλ1, λ2q
— two Young diagrams such that
m ´ |λ1| “ n ´ |λ2| “: f ě 0.
Fix pλ1, λ2q.
Let Y`f pλ2q be the set of all Young diagrams obtained by adding fboxes to λ2.For any λ1 P Y`f pλ2q, these f boxes form a skew shape λ1{λ2, suchthat |λ1| ` |λ
1{λ2| “ n.A hypertableau is a filling of pλ1, λ
1{λ2, λ1q in accordance with the
following rules:
1 λ1 is filled with 1, . . . , n into a standard tableau.2 The disjoint union λ1 \ pλ
1{λ2q is filled with 11, . . . , m1 suchthat λ1 is made into a standard tableau and λ1{λ2, into anantistandard tableau.
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
An irrep V λ1,λ2 is characterized by pλ1, λ2q
— two Young diagrams such that
m ´ |λ1| “ n ´ |λ2| “: f ě 0.
Fix pλ1, λ2q.
Let Y`f pλ2q be the set of all Young diagrams obtained by adding fboxes to λ2.For any λ1 P Y`f pλ2q, these f boxes form a skew shape λ1{λ2, suchthat |λ1| ` |λ
1{λ2| “ n.A hypertableau is a filling of pλ1, λ
1{λ2, λ1q in accordance with the
following rules:
1 λ1 is filled with 1, . . . , n into a standard tableau.2 The disjoint union λ1 \ pλ
1{λ2q is filled with 11, . . . , m1 suchthat λ1 is made into a standard tableau and λ1{λ2, into anantistandard tableau.
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
An irrep V λ1,λ2 is characterized by pλ1, λ2q
— two Young diagrams such that
m ´ |λ1| “ n ´ |λ2| “: f ě 0.
Fix pλ1, λ2q.
Let Y`f pλ2q be the set of all Young diagrams obtained by adding fboxes to λ2.For any λ1 P Y`f pλ2q, these f boxes form a skew shape λ1{λ2, suchthat |λ1| ` |λ
1{λ2| “ n.A hypertableau is a filling of pλ1, λ
1{λ2, λ1q in accordance with the
following rules:
1 λ1 is filled with 1, . . . , n into a standard tableau.2 The disjoint union λ1 \ pλ
1{λ2q is filled with 11, . . . , m1 suchthat λ1 is made into a standard tableau and λ1{λ2, into anantistandard tableau.
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
An irrep V λ1,λ2 is characterized by pλ1, λ2q
— two Young diagrams such that
m ´ |λ1| “ n ´ |λ2| “: f ě 0.
Fix pλ1, λ2q.
Let Y`f pλ2q be the set of all Young diagrams obtained by adding fboxes to λ2.For any λ1 P Y`f pλ2q, these f boxes form a skew shape λ1{λ2, suchthat |λ1| ` |λ
1{λ2| “ n.A hypertableau is a filling of pλ1, λ
1{λ2, λ1q in accordance with the
following rules:
1 λ1 is filled with 1, . . . , n into a standard tableau.2 The disjoint union λ1 \ pλ
1{λ2q is filled with 11, . . . , m1 suchthat λ1 is made into a standard tableau and λ1{λ2, into anantistandard tableau.
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
Example:
If m “ 6, n “ 6, and
λ1 “ , λ2 “ , f “ 4,
then a possible hypertableau is
pB,S ,W q “
˜
11
41,
51
61 31 21,
1 2 4
3 5 6
¸
.
Another example is
pB,S ,W q “
˜
21
51,
41
61 11
31
,1 3 4
2 6
5
¸
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
Hypertableaux to weights: pB, S ,W q ÞÑ µ P Cm`n´1.
Position: ktpλu “ position of k in pλ, as in 1,1 1,2 1,3. . .2,1 2,2
3,1
.
Number of the diagonal: Γpti , juq “ i ´ j .The weight: µ “ pµ1, . . . , µn´1
loooooomoooooon
n´1
; µn, . . . , µm`n´1loooooooomoooooooon
m
q,
where
µi “´
´β
α
¯Γppi`1qtW uq
, 1 ď i ď n ´ 1,
µn´1`i 1 “
$
’
&
’
%
´
´β
α
¯Γpi 1tBuq, i 1 P B,
´θ´
´β
α
¯´Γpi 1tSuq, i 1 P S .
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
Hypertableaux to weights: pB, S ,W q ÞÑ µ P Cm`n´1.
Position: ktpλu “ position of k in pλ, as in 1,1 1,2 1,3. . .2,1 2,2
3,1
.
Number of the diagonal: Γpti , juq “ i ´ j .The weight: µ “ pµ1, . . . , µn´1
loooooomoooooon
n´1
; µn, . . . , µm`n´1loooooooomoooooooon
m
q,
where
µi “´
´β
α
¯Γppi`1qtW uq
, 1 ď i ď n ´ 1,
µn´1`i 1 “
$
’
&
’
%
´
´β
α
¯Γpi 1tBuq, i 1 P B,
´θ´
´β
α
¯´Γpi 1tSuq, i 1 P S .
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
Hypertableaux to weights: pB, S ,W q ÞÑ µ P Cm`n´1.
Position: ktpλu “ position of k in pλ, as in 1,1 1,2 1,3. . .2,1 2,2
3,1
.
Number of the diagonal: Γpti , juq “ i ´ j .The weight: µ “ pµ1, . . . , µn´1
loooooomoooooon
n´1
; µn, . . . , µm`n´1loooooooomoooooooon
m
q,
where
µi “´
´β
α
¯Γppi`1qtW uq
, 1 ď i ď n ´ 1,
µn´1`i 1 “
$
’
&
’
%
´
´β
α
¯Γpi 1tBuq, i 1 P B,
´θ´
´β
α
¯´Γpi 1tSuq, i 1 P S .
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
Hypertableaux to weights: pB, S ,W q ÞÑ µ P Cm`n´1.
Position: ktpλu “ position of k in pλ, as in 1,1 1,2 1,3. . .2,1 2,2
3,1
.
Number of the diagonal: Γpti , juq “ i ´ j .The weight: µ “ pµ1, . . . , µn´1
loooooomoooooon
n´1
; µn, . . . , µm`n´1loooooooomoooooooon
m
q,
where
µi “´
´β
α
¯Γppi`1qtW uq
, 1 ď i ď n ´ 1,
µn´1`i 1 “
$
’
&
’
%
´
´β
α
¯Γpi 1tBuq, i 1 P B,
´θ´
´β
α
¯´Γpi 1tSuq, i 1 P S .
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
Hypertableaux to weights: pB, S ,W q ÞÑ µ P Cm`n´1.
Position: ktpλu “ position of k in pλ, as in 1,1 1,2 1,3. . .2,1 2,2
3,1
.
Number of the diagonal: Γpti , juq “ i ´ j .The weight: µ “ pµ1, . . . , µn´1
loooooomoooooon
n´1
; µn, . . . , µm`n´1loooooooomoooooooon
m
q,
where
µi “´
´β
α
¯Γppi`1qtW uq
, 1 ď i ď n ´ 1,
µn´1`i 1 “
$
’
&
’
%
´
´β
α
¯Γpi 1tBuq, i 1 P B,
´θ´
´β
α
¯´Γpi 1tSuq, i 1 P S .
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nqExample:
If pB,S ,W q “
˜
11
41,
51
61 31 21,
1 2 4
3 5 6
¸
,
then µ “´
´α
β, ´
β
α,α2
β2 , 1, ´α
β; 1,
βθ
α, ´θ, ´
β
α, ´
β2θ
α2 ,αθ
β
¯
Example:
pB,S ,W q “
˜
21
51,
41
61 11
31
,1 3 4
2 6
5
¸
µ “´
´β
α, ´
α
β,α2
β2 ,β2
α2 , 1; ´θ, 1, ´α2θ
β2 , ´β2θ
α2 , ´β
α,αθ
β
¯
Given λ1 and λ2, each µ is the list of eigenvalues of pJpnq2, . . . , Jpnqn`mqon the vector in V λ1,λ2 associated with the hypertableau.All hypertableaux ùñ all eigenvalues in V λ1,λ2 .
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nqExample:
If pB,S ,W q “
˜
11
41,
51
61 31 21,
1 2 4
3 5 6
¸
,
then µ “´
´α
β, ´
β
α,α2
β2 , 1, ´α
β; 1,
βθ
α, ´θ, ´
β
α, ´
β2θ
α2 ,αθ
β
¯
Example:
pB,S ,W q “
˜
21
51,
41
61 11
31
,1 3 4
2 6
5
¸
µ “´
´β
α, ´
α
β,α2
β2 ,β2
α2 , 1; ´θ, 1, ´α2θ
β2 , ´β2θ
α2 , ´β
α,αθ
β
¯
Given λ1 and λ2, each µ is the list of eigenvalues of pJpnq2, . . . , Jpnqn`mqon the vector in V λ1,λ2 associated with the hypertableau.All hypertableaux ùñ all eigenvalues in V λ1,λ2 .
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nqExample:
If pB,S ,W q “
˜
11
41,
51
61 31 21,
1 2 4
3 5 6
¸
,
then µ “´
´α
β, ´
β
α,α2
β2 , 1, ´α
β; 1,
βθ
α, ´θ, ´
β
α, ´
β2θ
α2 ,αθ
β
¯
Example:
pB,S ,W q “
˜
21
51,
41
61 11
31
,1 3 4
2 6
5
¸
µ “´
´β
α, ´
α
β,α2
β2 ,β2
α2 , 1; ´θ, 1, ´α2θ
β2 , ´β2θ
α2 , ´β
α,αθ
β
¯
Given λ1 and λ2, each µ is the list of eigenvalues of pJpnq2, . . . , Jpnqn`mqon the vector in V λ1,λ2 associated with the hypertableau.All hypertableaux ùñ all eigenvalues in V λ1,λ2 .
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nqhypertableaux Q H ÞÑ µpHq
V λ1,λ2 “à
HPhypertableaux
VµpHq,
dimVµpHq “ 1,
Jpnqi`1VµpHq “ µi pHqVµpHq, 1 ď i ď m ` n ´ 1.
How to prove:
By defining a qwBpm, nq action on the set of hypertableaux.
hi ...pB,S ,W q “ pB, S , hi ...W q, 1 ď i ď n ´ 1:
hn´i ...W “
$
’
&
’
%
αW , i and i ` 1 are in the same row,
βW , i and i ` 1 are in the same column,
ρiW ` ρ1iWpi ,i`1q otherwise.
ρi “pα` βqµi
µi ´ µi´1, ρ1i “
c
´pαµi ` βµi´1qpαµi´1 ` βµi q
pµi ´ µi´1q2.
gj ...pB,S ,W q “ pgj ...pB, Sq,W q, 1 ď j ď m ´ 1,
gj ...pB, Sq “$
’
&
’
%
α ¨ pB,Sq, j and j ` 1 are in the same row of B or S ,
β ¨ pB,Sq, j and j ` 1 are in the same column of B or S ,
σj ¨ pB,Sq ` σ1j ¨ pB, Sqpj ,j`1q, otherwise.
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nqhypertableaux Q H ÞÑ µpHq
V λ1,λ2 “à
HPhypertableaux
VµpHq,
dimVµpHq “ 1,
Jpnqi`1VµpHq “ µi pHqVµpHq, 1 ď i ď m ` n ´ 1.
How to prove:
By defining a qwBpm, nq action on the set of hypertableaux.
hi ...pB,S ,W q “ pB, S , hi ...W q, 1 ď i ď n ´ 1:
hn´i ...W “
$
’
&
’
%
αW , i and i ` 1 are in the same row,
βW , i and i ` 1 are in the same column,
ρiW ` ρ1iWpi ,i`1q otherwise.
ρi “pα` βqµi
µi ´ µi´1, ρ1i “
c
´pαµi ` βµi´1qpαµi´1 ` βµi q
pµi ´ µi´1q2.
gj ...pB,S ,W q “ pgj ...pB, Sq,W q, 1 ď j ď m ´ 1,
gj ...pB, Sq “$
’
&
’
%
α ¨ pB,Sq, j and j ` 1 are in the same row of B or S ,
β ¨ pB,Sq, j and j ` 1 are in the same column of B or S ,
σj ¨ pB,Sq ` σ1j ¨ pB, Sqpj ,j`1q, otherwise.
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nqhypertableaux Q H ÞÑ µpHq
V λ1,λ2 “à
HPhypertableaux
VµpHq,
dimVµpHq “ 1,
Jpnqi`1VµpHq “ µi pHqVµpHq, 1 ď i ď m ` n ´ 1.
How to prove:
By defining a qwBpm, nq action on the set of hypertableaux.
hi ...pB,S ,W q “ pB, S , hi ...W q, 1 ď i ď n ´ 1:
hn´i ...W “
$
’
&
’
%
αW , i and i ` 1 are in the same row,
βW , i and i ` 1 are in the same column,
ρiW ` ρ1iWpi ,i`1q otherwise.
ρi “pα` βqµi
µi ´ µi´1, ρ1i “
c
´pαµi ` βµi´1qpαµi´1 ` βµi q
pµi ´ µi´1q2.
gj ...pB,S ,W q “ pgj ...pB, Sq,W q, 1 ď j ď m ´ 1,
gj ...pB, Sq “$
’
&
’
%
α ¨ pB,Sq, j and j ` 1 are in the same row of B or S ,
β ¨ pB,Sq, j and j ` 1 are in the same column of B or S ,
σj ¨ pB,Sq ` σ1j ¨ pB, Sqpj ,j`1q, otherwise.
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nqHow to prove:
By defining a qwBpm, nq action on the set of hypertableaux.
hi ...pB,S ,W q “ pB,S , hi ...W q, 1 ď i ď n ´ 1:
hn´i ...W “
$
’
&
’
%
αW , i and i ` 1 are in the same row,
βW , i and i ` 1 are in the same column,
ρiW ` ρ1iWpi ,i`1q otherwise.
ρi “pα` βqµi
µi ´ µi´1, ρ1i “
c
´pαµi ` βµi´1qpαµi´1 ` βµi q
pµi ´ µi´1q2.
gj ...pB,S ,W q “ pgj ...pB, Sq,W q, 1 ď j ď m ´ 1,
gj ...pB, Sq “$
’
&
’
%
α ¨ pB,Sq, j and j ` 1 are in the same row of B or S ,
β ¨ pB,Sq, j and j ` 1 are in the same column of B or S ,
σj ¨ pB,Sq ` σ1j ¨ pB, Sqpj ,j`1q, otherwise.
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
How to prove:
By defining a qwBpm, nq action on the set of hypertableaux.
gj ...pB,S ,W q “ pgj ...pB, Sq,W q, 1 ď j ď m ´ 1,
gj ...pB, Sq “$
’
&
’
%
α ¨ pB,Sq, j and j ` 1 are in the same row of B or S ,
β ¨ pB,Sq, j and j ` 1 are in the same column of B or S ,
σj ¨ pB,Sq ` σ1j ¨ pB, Sqpj ,j`1q, otherwise.
σj “pα` βqµn`j
µn`j ´ µn`j´1, σ1j “
c
´pαµn`j ` βµn`j´1qpαµn`j´1 ` βµn`jq
pµn`j ´ µn`j´1q2
E ...pB,S ,W q “?
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
How to prove:
By defining a qwBpm, nq action on the set of hypertableaux.
gj ...pB,S ,W q “ pgj ...pB, Sq,W q, 1 ď j ď m ´ 1,
gj ...pB, Sq “$
’
&
’
%
α ¨ pB,Sq, j and j ` 1 are in the same row of B or S ,
β ¨ pB,Sq, j and j ` 1 are in the same column of B or S ,
σj ¨ pB,Sq ` σ1j ¨ pB, Sqpj ,j`1q, otherwise.
σj “pα` βqµn`j
µn`j ´ µn`j´1, σ1j “
c
´pαµn`j ` βµn`j´1qpαµn`j´1 ` βµn`jq
pµn`j ´ µn`j´1q2
E ...pB,S ,W q “?
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
How to prove:
By defining a qwBpm, nq action on the set of hypertableaux.
E ...pB,S ,W q “?
Mobile elements: Black numbers talk to white.Superimpose S on W :Mobile element:a box with the “white” (unprimed) numer n and the black 11.It can travel!
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
How to prove:
By defining a qwBpm, nq action on the set of hypertableaux.
E ...pB,S ,W q “?
Mobile elements:a Black numbers talk to white.Superimpose S on W :Mobile element:a box with the “white” (unprimed) numer n and the black 11.It can travel!
aBarbara McClintock: the 1983 Nobel Prize in Physiology and Medicine “forher discovery of mobile genetic elements.”J.A. Shapiro Mobile Genetic Elements, Academic Press, New York (1983).
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
How to prove:
By defining a qwBpm, nq action on the set of hypertableaux.
E ...pB,S ,W q “?
Mobile elements:a Black numbers talk to white.Superimpose S on W :Mobile element:a box with the “white” (unprimed) numer n and the black 11.It can travel!
aBarbara McClintock: the 1983 Nobel Prize in Physiology and Medicine “forher discovery of mobile genetic elements.”J.A. Shapiro Mobile Genetic Elements, Academic Press, New York (1983).
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
How to prove:
By defining a qwBpm, nq action on the set of hypertableaux.
E ...pB,S ,W q “?
Mobile elements: Black numbers talk to white.Superimpose S on W :
pB,S ,W q “
˜
11
41,
51
61 31 21,
1 2 4
3 5 6
¸
.
S _ W “1 2 451
361 531 621
Mobile element:a box with the “white” (unprimed) numer n and the black 11.It can travel!
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nqHow to prove:
By defining a qwBpm, nq action on the set of hypertableaux.
E ...pB,S ,W q “?
Mobile elements: Black numbers talk to white.Superimpose S on W :
pB, S ,W q “
˜
21
51,
41
61 11
31
,1 3 4
2 6
5
¸
S _ W “1 3 441
261 611
531
Mobile element:a box with the “white” (unprimed) numer n and the black 11.It can travel!
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
How to prove:
E ...pB,S ,W q “?
Mobile elements: Black numbers talk to white.Superimpose S on W :
pB, S ,W q “
˜
21
51,
41
61 11
31
,1 3 4
2 6
5
¸
S _ W “1 3 441
261 611
531
Mobile element:a box with the “white” (unprimed) numer n and the black 11.It can travel!
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
How to prove:
E ...pB,S ,W q “?
Mobile elements: Black numbers talk to white.Superimpose S on W :
pB, S ,W q “
˜
21
51,
41
61 11
31
,1 3 4
2 6
5
¸
S _ W “1 3 441
261 611
531
Mobile element:a box with the “white” (unprimed) numer n and the black 11.It can travel!
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
How to prove:
By defining a qwBpm, nq action on the set of hypertableaux.
E ...pB,S ,W q “?
Semikhatov Quantum walled Brauer spin chain
The spectrum of Jpnqj in irreps of qwBpm, nq
How to prove:
By defining a qwBpm, nq action on the set of hypertableaux.
E ...pBi ,Si ,Wi q “
$
’
&
’
%
0, no mobile element in pSi , Wi q,ÿ
jPOrbpSi ,Wi q
ci ,jpBi ,Sj ,Wjq otherwise,
where OrbpSi , Wi q “Ů
jpSj ,Wjq and
ci ,j “1
κpα` βq
´
1`θ
p´β{αqΓpnrWi sq
¯
ś
˝PcornerspWjnnq
xxΓp˝, nrWj sqyy
ś
∆POrb1pSj ,Wj q
xxΓpnr∆s, nrWj sqyy
Position of the mobile element: nrWi s;Hook distance: Γpti1, j1u, ti2, j2uq “ i1 ´ i2 ´ j1 ` j2;
“p´β{αq-integers”: xxkyy “ 1´´
´β
α
¯k.
Semikhatov Quantum walled Brauer spin chain
Demystification:
1st identity ùñ EE “ θ`1κpα`βqE,
2nd identity ùñ Eg1E “1κE and Eh1E “
1κE.
Semikhatov Quantum walled Brauer spin chain
Demystification:
1st identity ùñ EE “ θ`1κpα`βqE,
2nd identity ùñ Eg1E “1κE and Eh1E “
1κE.
Semikhatov Quantum walled Brauer spin chain
Program
comparison with t–J models
“bosonization”: split E “‚ ˝����‚ ˝
into pieces. Vertex operators?
specializations(!)“Logarithmic” featureslimit m, nÑ8
Semikhatov Quantum walled Brauer spin chain
Program
comparison with t–J models
“bosonization”: split E “‚ ˝����‚ ˝
into pieces. Vertex operators?
specializations(!)“Logarithmic” featureslimit m, nÑ8
Semikhatov Quantum walled Brauer spin chain
Program
comparison with t–J models
“bosonization”: split E “‚ ˝����‚ ˝
into pieces. Vertex operators?
specializations(!)“Logarithmic” featureslimit m, nÑ8
Semikhatov Quantum walled Brauer spin chain
Program
comparison with t–J models
“bosonization”: split E “‚ ˝����‚ ˝
into pieces. Vertex operators?
specializations(!)“Logarithmic” featureslimit m, nÑ8
Semikhatov Quantum walled Brauer spin chain
Program
comparison with t–J models
“bosonization”: split E “‚ ˝����‚ ˝
into pieces. Vertex operators?
specializations(!)“Logarithmic” featureslimit m, nÑ8
Semikhatov Quantum walled Brauer spin chain
Thank you.
Semikhatov Quantum walled Brauer spin chain
Semikhatov Quantum walled Brauer spin chain