Properties of SLE random fractal curve, dim Hausdorff ( γ ) = 1 + κ 8 for κ ≤ 8 [Beffara 2008] ✞ ☎ ✞ ☎ 4 < κ < 8 0 < κ ≤ 4 ✞ ☎ ✝ ✆ 8 ≤ κ ✝ ✆ ✝ ✆ non-self-crossing curve simple curve random Peano curve touches boundary on a doesn’t touch random Cantor set boundary Correlation functions with hidden quantum group 1. Conformally invariant random curves Kalle Kytölä — Florence, May 2015
Properties of SLE random fractal curve, dim Hausdorff ( γ ) = 1 + κ 8 for κ ≤ 8 [Beffara 2008] ✞ ☎ ✞ ☎ 4 < κ < 8 0 < κ ≤ 4 ✞ ☎ ✝ ✆ 8 ≤ κ ✝ ✆ ✝ ✆ non-self-crossing curve simple curve random Peano curve touches boundary on a doesn’t touch in this talk κ < 8 random Cantor set boundary Correlation functions with hidden quantum group 1. Conformally invariant random curves Kalle Kytölä — Florence, May 2015
Generalizing the Dobrushin boundary conditions Critical Ising model with Dobrushin boundary conditions [simulation and picture by Eveliina Peltola] Correlation functions with hidden quantum group 1. Conformally invariant random curves Kalle Kytölä — Florence, May 2015
Generalizing the Dobrushin boundary conditions Critical Ising model with alternating boundary conditions [simulation and picture by Eveliina Peltola] Correlation functions with hidden quantum group 1. Conformally invariant random curves Kalle Kytölä — Florence, May 2015
Classification problem of multiple SLEs · · · a 3 Random curves ( γ ( i ) ) N i = 1 in domain D connecting boundary D a 2 points a 1 , a 2 , . . . , a 2 N a 2 N a 1 · · · ✞ ☎ law P D ; a 1 ,..., a 2 N [Dubédat 2007] ✝ ✆ [Bauer & Bernard & K. 2005] Can we give a classification? ◮ Conformal invariance ◮ Domain Markov property (w.r.t. all initial segments) - initial segments absolutely continuous w.r.t. chordal SLE κ ! Convex set of multiple-SLE κ ’s ( P D ; a 1 ,..., a 2 N ) Correlation functions with hidden quantum group 1. Conformally invariant random curves Kalle Kytölä — Florence, May 2015
2. M ULTIPLE S CHRAMM -L OEWNER E VOLUTIONS GROWTH PROCESSES Correlation functions with hidden quantum group 2. Local multiple SLEs Kalle Kytölä — Florence, May 2015
Overview of classification of multiple SLEs local multiple SLEs Möbius covariant with curves starting from ← → positive solutions Z 2 N boundary points to a system of PDEs [pictures by Eveliina Peltola] Correlation functions with hidden quantum group 2. Local multiple SLEs Kalle Kytölä — Florence, May 2015
Overview of classification of multiple SLEs local multiple SLEs Möbius covariant with curves starting from ← → positive solutions Z 2 N boundary points to a system of PDEs vector space of solutions convex set of probability measures (finite dimensional) (finite dimensional) [pictures by Eveliina Peltola] Correlation functions with hidden quantum group 2. Local multiple SLEs Kalle Kytölä — Florence, May 2015
Overview of classification of multiple SLEs local multiple SLEs Möbius covariant with curves starting from ← → positive solutions Z 2 N boundary points to a system of PDEs vector space of solutions convex set of probability measures (finite dimensional) (finite dimensional) extremal points: deterministic connectivity pattern (a planar pair partition α ∈ PPP N ) [pictures by Eveliina Peltola] Correlation functions with hidden quantum group 2. Local multiple SLEs Kalle Kytölä — Florence, May 2015
Overview of classification of multiple SLEs local multiple SLEs Möbius covariant with curves starting from ← → positive solutions Z 2 N boundary points to a system of PDEs vector space of solutions convex set of probability measures (finite dimensional) (finite dimensional) extremal points: deterministic connectivity pattern (a planar pair partition α ∈ PPP N ) 1 � 2 N � PPP = � N ∈ N PPP N , # PPP N = C N = N + 1 N Correlation functions with hidden quantum group 2. Local multiple SLEs Kalle Kytölä — Florence, May 2015
Overview of classification of multiple SLEs local multiple SLEs Möbius covariant with curves starting from ← → positive solutions Z 2 N boundary points to a system of PDEs vector space of solutions convex set of probability measures (finite dimensional) (finite dimensional) dim = C N [Flores & Kleban 2014] extremal points: deterministic connectivity pattern (a planar pair partition α ∈ PPP N ) 1 � 2 N � PPP = � N ∈ N PPP N , # PPP N = C N = N + 1 N Correlation functions with hidden quantum group 2. Local multiple SLEs Kalle Kytölä — Florence, May 2015
Overview of classification of multiple SLEs local multiple SLEs Möbius covariant with curves starting from ← → positive solutions Z 2 N boundary points to a system of PDEs vector space of solutions convex set of probability measures (finite dimensional) (finite dimensional) dim = C N [Flores & Kleban 2014] extremal points: deterministic solutions Z α with particular connectivity pattern (a planar pair asymptotic behavior partition α ∈ PPP N ) 1 � 2 N � PPP = � N ∈ N PPP N , # PPP N = C N = N + 1 N Correlation functions with hidden quantum group 2. Local multiple SLEs Kalle Kytölä — Florence, May 2015
Role of the partition function Local multiple SLE κ classification: H Z ”partition function“ defined on � � X 2 N = x 1 < x 2 < · · · < x 2 N x 1 x 2 x 3 x 4 x 5 x 6 Correlation functions with hidden quantum group 2. Local multiple SLEs Kalle Kytölä — Florence, May 2015
Role of the partition function Local multiple SLE κ classification: H Z ”partition function“ defined on � � X 2 N = x 1 < x 2 < · · · < x 2 N ( h = h 1 , 2 = 6 − κ 2 κ ) x 1 x 2 x 3 x 4 x 5 x 6 Z specifies Girsanov transforms w.r.t. chordal SLE κ : k � = j g ′ ( x k ) h × Z d ( j :th curve ) d ( chordal SLE κ ) ∝ � � g ( x 1 ) , . . . , g ( tip ) , . . . , g ( x 2 N )) . where g : H \ ( j :th curve ) → H is conformal s.t. g ( z ) = z + o ( 1 ) . Correlation functions with hidden quantum group 2. Local multiple SLEs Kalle Kytölä — Florence, May 2015
Role of the partition function Local multiple SLE κ classification: H Z ”partition function“ defined on � � X 2 N = x 1 < x 2 < · · · < x 2 N ( h = h 1 , 2 = 6 − κ 2 κ ) x 1 x 2 x 3 x 4 x 5 x 6 (PDE) D j Z = 0 for all j = 1 , . . . , 2 N , where � � ∂ 2 D j = κ 2 ∂ 2 h j + � ∂ x i − . i � = j x i − x j 2 ∂ x 2 ( x i − x j ) 2 Z specifies Girsanov transforms w.r.t. chordal SLE κ : k � = j g ′ ( x k ) h × Z d ( j :th curve ) d ( chordal SLE κ ) ∝ � � g ( x 1 ) , . . . , g ( tip ) , . . . , g ( x 2 N )) . where g : H \ ( j :th curve ) → H is conformal s.t. g ( z ) = z + o ( 1 ) . Correlation functions with hidden quantum group 2. Local multiple SLEs Kalle Kytölä — Florence, May 2015
Role of the partition function Local multiple SLE κ classification: H Z ”partition function“ defined on � � X 2 N = x 1 < x 2 < · · · < x 2 N ( h = h 1 , 2 = 6 − κ 2 κ ) x 1 x 2 x 3 x 4 x 5 x 6 (PDE) D j Z = 0 for all j = 1 , . . . , 2 N , where � � ∂ 2 D j = κ 2 ∂ 2 h j + � ∂ x i − . i � = j x i − x j 2 ∂ x 2 ( x i − x j ) 2 (COV) For µ : H → H Möbius s.t. µ ( x 1 ) < · · · < µ ( x 2 N ) we have j = 1 µ ′ ( x j ) h × Z = � 2 N � � � � Z x 1 , . . . , x 2 N µ ( x 1 ) , . . . , µ ( x 2 N ) . Z specifies Girsanov transforms w.r.t. chordal SLE κ : k � = j g ′ ( x k ) h × Z d ( j :th curve ) d ( chordal SLE κ ) ∝ � � g ( x 1 ) , . . . , g ( tip ) , . . . , g ( x 2 N )) . where g : H \ ( j :th curve ) → H is conformal s.t. g ( z ) = z + o ( 1 ) . Correlation functions with hidden quantum group 2. Local multiple SLEs Kalle Kytölä — Florence, May 2015
Collapsing marked points Suppose Z ( x 1 ,..., x 2 N ) lim x j , x j + 1 → ξ ( x j + 1 − x j ) − 2 h = ˆ 1 2 3 4 5 6 7 8 Z ( x 1 , . . . , x j − 1 , x j + 2 , . . . , x 2 N ) . Then the law of the curves other than j , j + 1 under local 2 N -SLE κ defined by Z tends to the local ( 2 N − 2 ) -SLE κ defined 1 2 3 4 5 6 by ˆ Z as x j , x j + 1 → ξ . For pure partition functions Z α , α ∈ PPP, thus require � Z α/ { j , j + 1 } if { j , j + 1 } ∈ α Z α (ASY) lim ( x j + 1 − x j ) − 2 h = 0 if { j , j + 1 } / ∈ α x j , x j + 1 → ξ Correlation functions with hidden quantum group 2. Local multiple SLEs Kalle Kytölä — Florence, May 2015
Multiple SLEs pure partition function problem � � Z α α ∈ PPP � � for α ∈ PPP N , function Z α on X 2 N = x 1 < x 2 < · · · < x 2 N s.t. ∂ 2 D j = κ 2 ∂ 2 h � � (PDE) D j Z α = 0 � − where + ∂ x 2 ( x i − x j ) 2 2 x i − x j ∂ x i j i � = j 2 N (COV) Z � µ ′ ( x j ) h ×Z � � � � x 1 , . . . , x 2 N = µ ( x 1 ) , . . . , µ ( x 2 N ) j = 1 � Z α Z α/ { j , j + 1 } if { j , j + 1 } ∈ α (ASY) lim ( x j + 1 − x j ) − 2 h = if { j , j + 1 } / ∈ α x j , x j + 1 → ξ 0 Correlation functions with hidden quantum group 2. Local multiple SLEs Kalle Kytölä — Florence, May 2015
3. S OLUTION OF PURE PARTITION FUNCTIONS BY A HIDDEN QUANTUM GROUP Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Overview of the quantum group method Correspondence: vectors in an n -fold tensor product representation ← → functions of n variables of a quantum group highest weight vectors ← → solutions to partial of subrepresentations differential equations vectors in the ← → Möbius covariant trivial subrepresentation functions ← → prescribed projections prescribed asymptotic to subrepresentations behavior Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Idea: Integral solutions to PDEs How to solve the PDEs? ( D j Z )( x 1 , . . . , x 2 N ) = 0, � � ∂ 2 where D j = κ 2 ∂ 2 h j + � ∂ x i − . i � = j 2 ∂ x 2 x i − x j ( x i − x j ) 2 Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Idea: Integral solutions to PDEs How to solve the PDEs? ( D j Z )( x 1 , . . . , x 2 N ) = 0, � � ∂ 2 where D j = κ 2 ∂ 2 h j + � ∂ x i − . i � = j 2 ∂ x 2 x i − x j ( x i − x j ) 2 � f ( x 1 , . . . , x 2 N ; w 1 , . . . , w ℓ ) d w 1 · · · d w ℓ ? Z ( x 1 , . . . , x 2 N ) = Γ Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Idea: Integral solutions to PDEs How to solve the PDEs? ( D j Z )( x 1 , . . . , x 2 N ) = 0, � � ∂ 2 where D j = κ 2 ∂ 2 h j + � ∂ x i − . i � = j 2 ∂ x 2 x i − x j ( x i − x j ) 2 � f ( x 1 , . . . , x 2 N ; w 1 , . . . , w ℓ ) d w 1 · · · d w ℓ ? Z ( x 1 , . . . , x 2 N ) = Γ ◮ find appropriate f ( x 1 , . . . , x 2 N ; w 1 , . . . , w ℓ ) Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Idea: Integral solutions to PDEs How to solve the PDEs? ( D j Z )( x 1 , . . . , x 2 N ) = 0, � � ∂ 2 where D j = κ 2 ∂ 2 h j + � ∂ x i − . i � = j 2 ∂ x 2 x i − x j ( x i − x j ) 2 � f ( x 1 , . . . , x 2 N ; w 1 , . . . , w ℓ ) d w 1 · · · d w ℓ ? Z ( x 1 , . . . , x 2 N ) = Γ ◮ find appropriate f ( x 1 , . . . , x 2 N ; w 1 , . . . , w ℓ ) ◮ ( D j f ) d w 1 · · · d w ℓ is an exact ℓ -form Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Idea: Integral solutions to PDEs How to solve the PDEs? ( D j Z )( x 1 , . . . , x 2 N ) = 0, � � ∂ 2 where D j = κ 2 ∂ 2 h j + � ∂ x i − . i � = j 2 ∂ x 2 x i − x j ( x i − x j ) 2 � f ( x 1 , . . . , x 2 N ; w 1 , . . . , w ℓ ) d w 1 · · · d w ℓ ? Z ( x 1 , . . . , x 2 N ) = Γ ◮ find appropriate f ( x 1 , . . . , x 2 N ; w 1 , . . . , w ℓ ) ◮ ( D j f ) d w 1 · · · d w ℓ is an exact ℓ -form ◮ if ∂ Γ = ∅ , then the integral Z solves ( D j Z )( z ) = 0 Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Idea: Integral solutions to PDEs How to solve the PDEs? ( D j Z )( x 1 , . . . , x 2 N ) = 0, � � ∂ 2 where D j = κ 2 ∂ 2 h j + � ∂ x i − . i � = j 2 ∂ x 2 x i − x j ( x i − x j ) 2 � f ( x 1 , . . . , x 2 N ; w 1 , . . . , w ℓ ) d w 1 · · · d w ℓ ? Z ( x 1 , . . . , x 2 N ) = Γ ◮ find appropriate f ( x 1 , . . . , x 2 N ; w 1 , . . . , w ℓ ) [Dotsenko-Fateev 1984] − 4 2 8 ◮ f = � κ × � κ × � i < j ( x j − x i ) i , r ( w r − x i ) r < s ( w s − w r ) κ ◮ ( D j f ) d w 1 · · · d w ℓ is an exact ℓ -form ◮ if ∂ Γ = ∅ , then the integral Z solves ( D j Z )( z ) = 0 Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Idea: Integral solutions to PDEs How to solve the PDEs? ( D j Z )( x 1 , . . . , x 2 N ) = 0, � � ∂ 2 where D j = κ 2 ∂ 2 h j + � ∂ x i − . i � = j 2 ∂ x 2 x i − x j ( x i − x j ) 2 � f ( x 1 , . . . , x 2 N ; w 1 , . . . , w ℓ ) d w 1 · · · d w ℓ ? Z ( x 1 , . . . , x 2 N ) = Γ ◮ find appropriate f ( x 1 , . . . , x 2 N ; w 1 , . . . , w ℓ ) [Dotsenko-Fateev 1984] − 4 2 8 ◮ f = � κ × � κ × � i < j ( x j − x i ) i , r ( w r − x i ) r < s ( w s − w r ) κ ◮ ( D j f ) d w 1 · · · d w ℓ is an exact ℓ -form ◮ if ∂ Γ = ∅ , then the integral Z solves ( D j Z )( z ) = 0 ◮ find appropriate Γ to solve PDEs with boundary conditions? Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Idea: Integral solutions to PDEs How to solve the PDEs? ( D j Z )( x 1 , . . . , x 2 N ) = 0, � � ∂ 2 where D j = κ 2 ∂ 2 h j + � ∂ x i − . i � = j 2 ∂ x 2 x i − x j ( x i − x j ) 2 � f ( x 1 , . . . , x 2 N ; w 1 , . . . , w ℓ ) d w 1 · · · d w ℓ ? Z ( x 1 , . . . , x 2 N ) = Γ ◮ find appropriate f ( x 1 , . . . , x 2 N ; w 1 , . . . , w ℓ ) [Dotsenko-Fateev 1984] − 4 2 8 ◮ f = � κ × � κ × � i < j ( x j − x i ) i , r ( w r − x i ) r < s ( w s − w r ) κ ◮ ( D j f ) d w 1 · · · d w ℓ is an exact ℓ -form ◮ if ∂ Γ = ∅ , then the integral Z solves ( D j Z )( z ) = 0 ◮ find appropriate Γ to solve PDEs with boundary conditions? quantum group U q ( sl 2 ) acts on Γ [Felder & Wieczerkowski 1991, Peltola & K. 2014] Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The quantum group and its representations Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The quantum group and its representations q = e i π 4 /κ (assume κ / ∈ Q ) Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The quantum group and its representations q = e i π 4 /κ (assume κ / ∈ Q ) ◮ Algebra U q ( sl 2 ) : gen. E , F , K , K − 1 and q -Chevalley rel. Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The quantum group and its representations q = e i π 4 /κ (assume κ / ∈ Q ) ◮ Algebra U q ( sl 2 ) : gen. E , F , K , K − 1 and q -Chevalley rel. KK − 1 = K − 1 K = 1 KE = q 2 EK , KF = q − 2 FK , 1 � K − K − 1 � EF − FE = q − q − 1 Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The quantum group and its representations q = e i π 4 /κ (assume κ / ∈ Q ) ◮ Algebra U q ( sl 2 ) : gen. E , F , K , K − 1 and q -Chevalley rel. KK − 1 = K − 1 K = 1 KE = q 2 EK , KF = q − 2 FK , 1 � K − K − 1 � EF − FE = q − q − 1 ◮ Irreducible rep. M d of dimension d : basis e 0 , e 1 , . . . , e d − 1 Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The quantum group and its representations q = e i π 4 /κ (assume κ / ∈ Q ) ◮ Algebra U q ( sl 2 ) : gen. E , F , K , K − 1 and q -Chevalley rel. KK − 1 = K − 1 K = 1 KE = q 2 EK , KF = q − 2 FK , 1 � K − K − 1 � EF − FE = q − q − 1 ◮ Irreducible rep. M d of dimension d : basis e 0 , e 1 , . . . , e d − 1 K . e j = q d − 1 − 2 j e j , F . e j = e j + 1 , E . e j = [ j ] [ d − j ] e j − 1 where [ n ] = q n − q − n q − q − 1 are ” q -integers“ Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The quantum group and its representations q = e i π 4 /κ (assume κ / ∈ Q ) ◮ Algebra U q ( sl 2 ) : gen. E , F , K , K − 1 and q -Chevalley rel. KK − 1 = K − 1 K = 1 KE = q 2 EK , KF = q − 2 FK , 1 � K − K − 1 � EF − FE = q − q − 1 ◮ Irreducible rep. M d of dimension d : basis e 0 , e 1 , . . . , e d − 1 K . e j = q d − 1 − 2 j e j , F . e j = e j + 1 , E . e j = [ j ] [ d − j ] e j − 1 where [ n ] = q n − q − n q − q − 1 are ” q -integers“ ◮ Tensor products of representations: for v ⊗ w ∈ V ⊗ W set Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The quantum group and its representations q = e i π 4 /κ (assume κ / ∈ Q ) ◮ Algebra U q ( sl 2 ) : gen. E , F , K , K − 1 and q -Chevalley rel. KK − 1 = K − 1 K = 1 KE = q 2 EK , KF = q − 2 FK , 1 � K − K − 1 � EF − FE = q − q − 1 ◮ Irreducible rep. M d of dimension d : basis e 0 , e 1 , . . . , e d − 1 K . e j = q d − 1 − 2 j e j , F . e j = e j + 1 , E . e j = [ j ] [ d − j ] e j − 1 where [ n ] = q n − q − n q − q − 1 are ” q -integers“ ◮ Tensor products of representations: for v ⊗ w ∈ V ⊗ W set K . ( v ⊗ w ) = K . v ⊗ K . w , E . ( v ⊗ w ) = E . v ⊗ K . w + v ⊗ E . w , F . ( v ⊗ w ) = F . v ⊗ w + K − 1 . v ⊗ F . w Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The quantum group and its representations q = e i π 4 /κ (assume κ / ∈ Q ) ◮ Algebra U q ( sl 2 ) : gen. E , F , K , K − 1 and q -Chevalley rel. KK − 1 = K − 1 K = 1 KE = q 2 EK , KF = q − 2 FK , 1 � K − K − 1 � EF − FE = q − q − 1 ◮ Irreducible rep. M d of dimension d : basis e 0 , e 1 , . . . , e d − 1 K . e j = q d − 1 − 2 j e j , F . e j = e j + 1 , E . e j = [ j ] [ d − j ] e j − 1 where [ n ] = q n − q − n q − q − 1 are ” q -integers“ ◮ Tensor products of representations: for v ⊗ w ∈ V ⊗ W set K . ( v ⊗ w ) = K . v ⊗ K . w , E . ( v ⊗ w ) = E . v ⊗ K . w + v ⊗ E . w , F . ( v ⊗ w ) = F . v ⊗ w + K − 1 . v ⊗ F . w ◮ Semisimple tensor products of the irreps: Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The quantum group and its representations q = e i π 4 /κ (assume κ / ∈ Q ) ◮ Algebra U q ( sl 2 ) : gen. E , F , K , K − 1 and q -Chevalley rel. KK − 1 = K − 1 K = 1 KE = q 2 EK , KF = q − 2 FK , 1 � K − K − 1 � EF − FE = q − q − 1 ◮ Irreducible rep. M d of dimension d : basis e 0 , e 1 , . . . , e d − 1 K . e j = q d − 1 − 2 j e j , F . e j = e j + 1 , E . e j = [ j ] [ d − j ] e j − 1 where [ n ] = q n − q − n q − q − 1 are ” q -integers“ ◮ Tensor products of representations: for v ⊗ w ∈ V ⊗ W set K . ( v ⊗ w ) = K . v ⊗ K . w , E . ( v ⊗ w ) = E . v ⊗ K . w + v ⊗ E . w , F . ( v ⊗ w ) = F . v ⊗ w + K − 1 . v ⊗ F . w ◮ Semisimple tensor products of the irreps: M d 2 ⊗ M d 1 ∼ = M d 1 + d 2 − 1 ⊕ M d 1 + d 2 − 3 ⊕ · · · ⊕ M | d 1 − d 2 | + 1 Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The correspondence theorem (special case) κ ∈ ( 0 , 8 ) \ Q , q = e i π 4 /κ Theorem (K. & Peltola) F ( x 0 ) : M ⊗ 2 N → { functions on X ( x 0 ) − 2 N } 2 Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The correspondence theorem (special case) ◮ X ( x 0 ) x 0 X ( x 0 ) 2 N = { ( x j ) 2 N � x 0 < x 1 < · · · < x 2 N } , � X 2 N = � j = 1 2 N κ ∈ ( 0 , 8 ) \ Q , q = e i π 4 /κ Theorem (K. & Peltola) F ( x 0 ) : M ⊗ 2 N → { functions on X ( x 0 ) − 2 N } 2 Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The correspondence theorem (special case) ◮ X ( x 0 ) x 0 X ( x 0 ) 2 N = { ( x j ) 2 N � x 0 < x 1 < · · · < x 2 N } , � X 2 N = � j = 1 2 N κ ∈ ( 0 , 8 ) \ Q , q = e i π 4 /κ Theorem (K. & Peltola) F ( x 0 ) : M ⊗ 2 N → { functions on X ( x 0 ) − 2 N } 2 ( X ) If E . v = 0, then F ( x 0 ) [ v ]: X ( x 0 ) → C is independent of x 0 , n thus defines a function F [ v ]: X 2 N → C . Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The correspondence theorem (special case) ◮ X ( x 0 ) x 0 X ( x 0 ) 2 N = { ( x j ) 2 N � x 0 < x 1 < · · · < x 2 N } , � X 2 N = � j = 1 2 N κ ∈ ( 0 , 8 ) \ Q , q = e i π 4 /κ Theorem (K. & Peltola) F ( x 0 ) : M ⊗ 2 N → { functions on X ( x 0 ) − 2 N } 2 ( X ) If E . v = 0, then F ( x 0 ) [ v ]: X ( x 0 ) → C is independent of x 0 , n thus defines a function F [ v ]: X 2 N → C . (PDE) If E . v = 0, then Z = F [ v ] satisfies (PDE). Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The correspondence theorem (special case) ◮ X ( x 0 ) x 0 X ( x 0 ) 2 N = { ( x j ) 2 N � x 0 < x 1 < · · · < x 2 N } , � X 2 N = � j = 1 2 N κ ∈ ( 0 , 8 ) \ Q , q = e i π 4 /κ Theorem (K. & Peltola) F ( x 0 ) : M ⊗ 2 N → { functions on X ( x 0 ) − 2 N } 2 ( X ) If E . v = 0, then F ( x 0 ) [ v ]: X ( x 0 ) → C is independent of x 0 , n thus defines a function F [ v ]: X 2 N → C . (PDE) If E . v = 0, then Z = F [ v ] satisfies (PDE). (COV) If E . v = 0 and K . v = v , then Z = F [ v ] satisfies (COV). Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The correspondence theorem (special case) ◮ X ( x 0 ) x 0 X ( x 0 ) 2 N = { ( x j ) 2 N � x 0 < x 1 < · · · < x 2 N } , � X 2 N = � j = 1 2 N κ ∈ ( 0 , 8 ) \ Q , q = e i π 4 /κ Theorem (K. & Peltola) F ( x 0 ) : M ⊗ 2 N → { functions on X ( x 0 ) − 2 N } 2 ( X ) If E . v = 0, then F ( x 0 ) [ v ]: X ( x 0 ) → C is independent of x 0 , n thus defines a function F [ v ]: X 2 N → C . (PDE) If E . v = 0, then Z = F [ v ] satisfies (PDE). (COV) If E . v = 0 and K . v = v , then Z = F [ v ] satisfies (COV). F ( x 0 ) [ v ] ( x j + 1 − x j ) − 2 h = F ( x 0 ) [ˆ (ASY) lim x j , x j + 1 → ξ π j ( v )] Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The correspondence theorem (special case) ◮ M 2 ⊗ M 2 ∼ = M 1 ⊕ M 3 ◮ X ( x 0 ) x 0 X ( x 0 ) 2 N = { ( x j ) 2 N � x 0 < x 1 < · · · < x 2 N } , � X 2 N = � j = 1 2 N κ ∈ ( 0 , 8 ) \ Q , q = e i π 4 /κ Theorem (K. & Peltola) F ( x 0 ) : M ⊗ 2 N → { functions on X ( x 0 ) − 2 N } 2 ( X ) If E . v = 0, then F ( x 0 ) [ v ]: X ( x 0 ) → C is independent of x 0 , n thus defines a function F [ v ]: X 2 N → C . (PDE) If E . v = 0, then Z = F [ v ] satisfies (PDE). (COV) If E . v = 0 and K . v = v , then Z = F [ v ] satisfies (COV). F ( x 0 ) [ v ] ( x j + 1 − x j ) − 2 h = F ( x 0 ) [ˆ (ASY) lim x j , x j + 1 → ξ π j ( v )] Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The correspondence theorem (special case) ◮ M 2 ⊗ M 2 ∼ = M 1 ⊕ M 3 , proj. to M 1 ∼ = C is ˆ π : M 2 ⊗ M 2 → C . ◮ X ( x 0 ) x 0 X ( x 0 ) 2 N = { ( x j ) 2 N � x 0 < x 1 < · · · < x 2 N } , � X 2 N = � j = 1 2 N κ ∈ ( 0 , 8 ) \ Q , q = e i π 4 /κ Theorem (K. & Peltola) F ( x 0 ) : M ⊗ 2 N → { functions on X ( x 0 ) − 2 N } 2 ( X ) If E . v = 0, then F ( x 0 ) [ v ]: X ( x 0 ) → C is independent of x 0 , n thus defines a function F [ v ]: X 2 N → C . (PDE) If E . v = 0, then Z = F [ v ] satisfies (PDE). (COV) If E . v = 0 and K . v = v , then Z = F [ v ] satisfies (COV). F ( x 0 ) [ v ] ( x j + 1 − x j ) − 2 h = F ( x 0 ) [ˆ (ASY) lim x j , x j + 1 → ξ π j ( v )] Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
The correspondence theorem (special case) ◮ M 2 ⊗ M 2 ∼ = M 1 ⊕ M 3 , proj. to M 1 ∼ = C is ˆ π : M 2 ⊗ M 2 → C . → M ⊗ 2 ( N − 1 ) π j : M ⊗ 2 N ◮ ˆ , projection ˆ π in factors j and j + 1 2 2 ◮ X ( x 0 ) x 0 X ( x 0 ) 2 N = { ( x j ) 2 N � x 0 < x 1 < · · · < x 2 N } , � X 2 N = � j = 1 2 N κ ∈ ( 0 , 8 ) \ Q , q = e i π 4 /κ Theorem (K. & Peltola) F ( x 0 ) : M ⊗ 2 N → { functions on X ( x 0 ) − 2 N } 2 ( X ) If E . v = 0, then F ( x 0 ) [ v ]: X ( x 0 ) → C is independent of x 0 , n thus defines a function F [ v ]: X 2 N → C . (PDE) If E . v = 0, then Z = F [ v ] satisfies (PDE). (COV) If E . v = 0 and K . v = v , then Z = F [ v ] satisfies (COV). F ( x 0 ) [ v ] ( x j + 1 − x j ) − 2 h = F ( x 0 ) [ˆ (ASY) lim x j , x j + 1 → ξ π j ( v )] Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Translation of the multiple SLE problem ◮ M 2 ⊗ M 2 ∼ = M 1 ⊕ M 3 , proj. to M 1 ∼ = C is ˆ π : M 2 ⊗ M 2 → C . → M ⊗ 2 ( N − 1 ) π j : M ⊗ 2 N ◮ ˆ , projection ˆ π in factors j and j + 1 2 2 The translation: If ( v α ) α ∈ PPP N satisfies (SING) K . v α = v α , E . v α = 0, ( F . v α = 0 ) � v α/ { j , j + 1 } if { j , j + 1 } ∈ α (PROJ) ˆ π j ( v α ) = ∀ j if { j , j + 1 } / ∈ α 0 then the functions Z α = F [ v α ] satisfy (PDE), (COV), (ASY). Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Translation of the multiple SLE problem ◮ M 2 ⊗ M 2 ∼ = M 1 ⊕ M 3 , proj. to M 1 ∼ = C is ˆ π : M 2 ⊗ M 2 → C . → M ⊗ 2 ( N − 1 ) π j : M ⊗ 2 N ◮ ˆ , projection ˆ π in factors j and j + 1 2 2 The translation: If ( v α ) α ∈ PPP N satisfies (SING) K . v α = v α , E . v α = 0, ( F . v α = 0 ) � v α/ { j , j + 1 } if { j , j + 1 } ∈ α (PROJ) ˆ π j ( v α ) = ∀ j if { j , j + 1 } / ∈ α 0 then the functions Z α = F [ v α ] satisfy (PDE), (COV), (ASY). Trivial subrepresentation: dim { v ∈ M ⊗ 2 N � (SING) } = C N = 1 � 2 N � � 2 N + 1 N Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Translation of the multiple SLE problem ◮ M 2 ⊗ M 2 ∼ = M 1 ⊕ M 3 , proj. to M 1 ∼ = C is ˆ π : M 2 ⊗ M 2 → C . → M ⊗ 2 ( N − 1 ) π j : M ⊗ 2 N ◮ ˆ , projection ˆ π in factors j and j + 1 2 2 The translation: If ( v α ) α ∈ PPP N satisfies (SING) K . v α = v α , E . v α = 0, ( F . v α = 0 ) � v α/ { j , j + 1 } if { j , j + 1 } ∈ α (PROJ) ˆ π j ( v α ) = ∀ j if { j , j + 1 } / ∈ α 0 then the functions Z α = F [ v α ] satisfy (PDE), (COV), (ASY). Trivial subrepresentation: dim { v ∈ M ⊗ 2 N � (SING) } = C N = 1 � 2 N � � 2 N + 1 N Uniqueness of solutions: The only solution of the π j ( v ) = 0 ∀ j & (SING), is v = 0. homogeneous problem, ˆ Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Explicit solution for the maximally nested case Rainbow configuration: ⋓ N = {{ 1 , 2 N } , { 2 , 2 N − 1 } , . . . , { N , N + 1 }} Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Explicit solution for the maximally nested case Rainbow configuration: ⋓ N = {{ 1 , 2 N } , { 2 , 2 N − 1 } , . . . , { N , N + 1 }} Note : for rainbow configuration ⋓ N ∈ PPP N , (PROJ) becomes π N ( v ⋓ N ) = v ⋓ N − 1 and ˆ ˆ π j ( v ⋓ N ) = 0 for j � = N Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Explicit solution for the maximally nested case Rainbow configuration: ⋓ N = {{ 1 , 2 N } , { 2 , 2 N − 1 } , . . . , { N , N + 1 }} Note : for rainbow configuration ⋓ N ∈ PPP N , (PROJ) becomes π N ( v ⋓ N ) = v ⋓ N − 1 and ˆ ˆ π j ( v ⋓ N ) = 0 for j � = N Explicit formula : The solution for rainbow configurations is N ( − 1 ) k q k ( N − k − 1 ) × ( F k . ( e ⊗ N � )) ⊗ ( F N − k . ( e ⊗ N v ⋓ N = const . × )) . 0 0 k = 0 Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
� � � � � � � � � � � � � � � � � � Recursive solution on the poset of configurations � � � Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
� � � � � � � � � � � � � � � � � � Recursive solution on the poset of configurations Tying operation ℘ j : PPP N → PPP N : - connect j and j + 1 - connect the points to which j and j + 1 were previously connected α l 2 l 1 j j + 1 ℘ j α ℘ j l 2 l 1 j j + 1 � � � Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
� � � � � � � � � � � � � � � � � � Recursive solution on the poset of configurations Tying operation ℘ j : PPP N → PPP N : - connect j and j + 1 - connect the points to which j and j + 1 were previously connected Recursion based on formula: if { j , j + 1 } ∈ ̺ ∈ PPP N , then ( id − π j ) ( v ̺ ) = − 1 � v β [ 2 ] β ∈ ℘ − 1 ( ̺ ) \{ ̺ } j � � � Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Summary: solution of pure partition functions Theorem (K. & Peltola) Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Summary: solution of pure partition functions Theorem (K. & Peltola) ◮ With v ∅ = 1, there is a unique collection ( v α ) α ∈ PPP solving the system (SING) & (PROJ). Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Summary: solution of pure partition functions Theorem (K. & Peltola) ◮ With v ∅ = 1, there is a unique collection ( v α ) α ∈ PPP solving the system (SING) & (PROJ). ◮ The vectors ( v α ) α ∈ PPP N span the C N -dimensional trivial subrepresentation { v ∈ W ⊗ 2 N � (SING) } � 2 Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
Summary: solution of pure partition functions Theorem (K. & Peltola) ◮ With v ∅ = 1, there is a unique collection ( v α ) α ∈ PPP solving the system (SING) & (PROJ). ◮ The vectors ( v α ) α ∈ PPP N span the C N -dimensional trivial subrepresentation { v ∈ W ⊗ 2 N � (SING) } � 2 ◮ The functions Z α = F [ v α ] , span c N -dimensional solution spaces of the system � � ∂ 2 (PDE) D j Z α = 0, D j = κ 2 ∂ 2 h j + � ∂ x i − ∂ x 2 i � = j ( x i − x j ) 2 2 x i − x j j = 1 µ ′ ( x j ) h × Z = � 2 N � � � � (COV) Z x 1 , . . . µ ( x 1 ) , . . . and their asymptotic behavior as x j , x j + 1 → ξ is � Z α/ { j , j + 1 } if { j , j + 1 } ∈ α Z α (ASY) lim ( x j + 1 − x j ) − 2 h = . 0 if { j , j + 1 } / ∈ α Correlation functions with hidden quantum group 3. Quantum group solutions for multiple SLEs Kalle Kytölä — Florence, May 2015
4. G ENERAL QUANTUM GROUP METHOD AND SOME DETAILS Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Overview of the quantum group method (again) Correspondence: vectors in an n -fold tensor product representation ← → functions of n variables of a quantum group highest weight vectors ← → solutions to partial of subrepresentations differential equations vectors in the ← → Möbius covariant trivial subrepresentation functions ← → prescribed projections prescribed asymptotic to subrepresentations behavior Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Integral solutions to PDEs of CFTs Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
The correspondence theorem (general case) F ( x 0 ) d 1 ,..., d n : � n → { functions on X ( x 0 ) Theorem (K. & Peltola) j = 1 M d j − } n Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
The correspondence theorem (general case) ( v highest weight vector ⇔ E . v = 0) ( v in trivial subrepresentation ⇔ E . v = 0 and K . v = v ) F ( x 0 ) d 1 ,..., d n : � n → { functions on X ( x 0 ) Theorem (K. & Peltola) j = 1 M d j − } n Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
The correspondence theorem (general case) ( v highest weight vector ⇔ E . v = 0) ( v in trivial subrepresentation ⇔ E . v = 0 and K . v = v ) F ( x 0 ) d 1 ,..., d n : � n → { functions on X ( x 0 ) Theorem (K. & Peltola) j = 1 M d j − } n ( X n ) If v is a highest weight vector, then F ( x 0 ) [ v ]: X ( x 0 ) → C is n independent of x 0 , thus defines a function F [ v ]: X n → C . Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
The correspondence theorem (general case) ( v highest weight vector ⇔ E . v = 0) ( v in trivial subrepresentation ⇔ E . v = 0 and K . v = v ) F ( x 0 ) d 1 ,..., d n : � n → { functions on X ( x 0 ) Theorem (K. & Peltola) j = 1 M d j − } n ( X n ) If v is a highest weight vector, then F ( x 0 ) [ v ]: X ( x 0 ) → C is n independent of x 0 , thus defines a function F [ v ]: X n → C . (PDE) If v is a highest weight vector, then F [ v ]: X n → C satisfies n linear homogeneous PDEs of orders d 1 , . . . , d n . Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
The correspondence theorem (general case) ( v highest weight vector ⇔ E . v = 0) ( v in trivial subrepresentation ⇔ E . v = 0 and K . v = v ) F ( x 0 ) d 1 ,..., d n : � n → { functions on X ( x 0 ) Theorem (K. & Peltola) j = 1 M d j − } n ( X n ) If v is a highest weight vector, then F ( x 0 ) [ v ]: X ( x 0 ) → C is n independent of x 0 , thus defines a function F [ v ]: X n → C . (PDE) If v is a highest weight vector, then F [ v ]: X n → C satisfies n linear homogeneous PDEs of orders d 1 , . . . , d n . (COV) F ( x 0 ) [ v ]: X ( x 0 ) → C is n - translation invariant - homogeneous, if v is K -eigenvector - Möbius covariant, if v is in trivial subrepresentation Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
The correspondence theorem (general case) ( v highest weight vector ⇔ E . v = 0) ( v in trivial subrepresentation ⇔ E . v = 0 and K . v = v ) F ( x 0 ) d 1 ,..., d n : � n → { functions on X ( x 0 ) Theorem (K. & Peltola) j = 1 M d j − } n ( X n ) If v is a highest weight vector, then F ( x 0 ) [ v ]: X ( x 0 ) → C is n independent of x 0 , thus defines a function F [ v ]: X n → C . (PDE) If v is a highest weight vector, then F [ v ]: X n → C satisfies n linear homogeneous PDEs of orders d 1 , . . . , d n . (COV) F ( x 0 ) [ v ]: X ( x 0 ) → C is n - translation invariant - homogeneous, if v is K -eigenvector - Möbius covariant, if v is in trivial subrepresentation (ASY) M d j + 1 ⊗ M d j ∼ d M d induces a decomp. of � n = � j = 1 M d j . � � � � � � If v ∈ i > j + 1 M d i ⊗ M d ⊗ i < j M d i , then F ( x 0 ) ..., d j , d j + 1 ,... [ v ] ∼ ( x j + 1 − x j ) ∆ d × F ( x 0 ) ..., d ,... [ v ] . Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Sketch: definition of the correspondence * anchor x 0 , chamber X ( x 0 ) = { x 0 < x 1 < x 2 < · · · < x n } ⊂ R n n Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Sketch: definition of the correspondence * anchor x 0 , chamber X ( x 0 ) = { x 0 < x 1 < x 2 < · · · < x n } ⊂ R n n * parameters d 1 , d 2 , . . . , d n ∈ N Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Sketch: definition of the correspondence * anchor x 0 , chamber X ( x 0 ) = { x 0 < x 1 < x 2 < · · · < x n } ⊂ R n n * parameters d 1 , d 2 , . . . , d n ∈ N n F ( x 0 ) : → { functions on X ( x 0 ) � M d j − } n j = 1 Informally, F ( x 0 ) [ v ]( x ) = “ � Γ[ v ] f ( x ; w ) d w ”, where the integration surface Γ[ v ] depends on v ∈ � n j = 1 M d j . Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Sketch: definition of the correspondence * anchor x 0 , chamber X ( x 0 ) = { x 0 < x 1 < x 2 < · · · < x n } ⊂ R n n * parameters d 1 , d 2 , . . . , d n ∈ N n F ( x 0 ) : → { functions on X ( x 0 ) � M d j − } n j = 1 Informally, F ( x 0 ) [ v ]( x ) = “ � Γ[ v ] f ( x ; w ) d w ”, where the integration surface Γ[ v ] depends on v ∈ � n j = 1 M d j . F ( x 0 ) [ e l n ⊗ · · · ⊗ e l 1 ] = ϕ ( x 0 ) l 1 ,..., l n (below), extend linearly x 0 x 1 x 2 x n . . . l 1 l 2 l n ϕ ( x 0 ) l 1 ,..., l n ( x ) = Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Sketch: definition of the correspondence * anchor x 0 , chamber X ( x 0 ) = { x 0 < x 1 < x 2 < · · · < x n } ⊂ R n n * parameters d 1 , d 2 , . . . , d n ∈ N n F ( x 0 ) : → { functions on X ( x 0 ) � M d j − } n j = 1 Informally, F ( x 0 ) [ v ]( x ) = “ � Γ[ v ] f ( x ; w ) d w ”, where the integration surface Γ[ v ] depends on v ∈ � n j = 1 M d j . F ( x 0 ) [ e l n ⊗ · · · ⊗ e l 1 ] = ϕ ( x 0 ) l 1 ,..., l n (below), extend linearly x 0 x 1 x 2 x n . . . l 1 l 2 l n ϕ ( x 0 ) l 1 ,..., l n ( x ) = κ ( d i − 1 )( d j − 1 ) × � ( w s − w r ) 2 κ × � ( w r − x i ) − 4 8 κ ( d i − 1 ) f ∝ � ( x j − x i ) Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Sketch: asymptotics with subrepresentations M d ֒ → M d j + 1 ⊗ M d j d = d j + d j + 1 − 1 − 2 m Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Sketch: asymptotics with subrepresentations ( d ; d j , d j + 1 ) M d ֒ → M d j + 1 ⊗ M d j , e 0 �→ τ , d = d j + d j + 1 − 1 − 2 m 0 Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Sketch: asymptotics with subrepresentations ( d ; d j , d j + 1 ) M d ֒ → M d j + 1 ⊗ M d j , e 0 �→ τ , d = d j + d j + 1 − 1 − 2 m 0 k ( − 1 ) k [ d j − 1 − k ] ! [ d j + 1 − 1 − m + k ] ! ( d ; d j , d j + 1 ) q k ( d 1 − k ) τ ∝ � ( q − q − 1 ) m ( e k ⊗ e m − k ) 0 [ k ]! [ d j − 1 ] ![ m − k ]![ d 2 − 1 ]! Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Sketch: asymptotics with subrepresentations ( d ; d j , d j + 1 ) M d ֒ → M d j + 1 ⊗ M d j , e 0 �→ τ , d = d j + d j + 1 − 1 − 2 m 0 k ( − 1 ) k [ d j − 1 − k ] ! [ d j + 1 − 1 − m + k ] ! ( d ; d j , d j + 1 ) q k ( d 1 − k ) τ ∝ � ( q − q − 1 ) m ( e k ⊗ e m − k ) 0 [ k ]! [ d j − 1 ] ![ m − k ]![ d 2 − 1 ]! Calculation for v = e l n ⊗ · · · ⊗ e l j + 2 ⊗ ( F l .τ 0 ) ⊗ e l j − 1 ⊗ · · · ⊗ e l 1 Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Sketch: asymptotics with subrepresentations ( d ; d j , d j + 1 ) M d ֒ → M d j + 1 ⊗ M d j , e 0 �→ τ , d = d j + d j + 1 − 1 − 2 m 0 k ( − 1 ) k [ d j − 1 − k ] ! [ d j + 1 − 1 − m + k ] ! ( d ; d j , d j + 1 ) q k ( d 1 − k ) τ ∝ � ( q − q − 1 ) m ( e k ⊗ e m − k ) 0 [ k ]! [ d j − 1 ] ![ m − k ]![ d 2 − 1 ]! Calculation for v = e l n ⊗ · · · ⊗ e l j + 2 ⊗ ( F l .τ 0 ) ⊗ e l j − 1 ⊗ · · · ⊗ e l 1 x j x j +1 x 0 x 1 x n . . . . . . l l 1 l n m F ( x 0 ) [ v ]( x ) = Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Sketch: asymptotics with subrepresentations ( d ; d j , d j + 1 ) M d ֒ → M d j + 1 ⊗ M d j , e 0 �→ τ , d = d j + d j + 1 − 1 − 2 m 0 k ( − 1 ) k [ d j − 1 − k ] ! [ d j + 1 − 1 − m + k ] ! ( d ; d j , d j + 1 ) q k ( d 1 − k ) τ ∝ � ( q − q − 1 ) m ( e k ⊗ e m − k ) 0 [ k ]! [ d j − 1 ] ![ m − k ]![ d 2 − 1 ]! Calculation for v = e l n ⊗ · · · ⊗ e l j + 2 ⊗ ( F l .τ 0 ) ⊗ e l j − 1 ⊗ · · · ⊗ e l 1 x j x j +1 x 0 x 1 x n . . . . . . l l 1 l n m F ( x 0 ) [ v ]( x ) = dominated convergence: ( x 0 ) F ..., dj , dj + 1 ,... [ v ]( ... ) x j , x j + 1 → ξ F ( x 0 ) − → ..., d ,... [ v ]( . . . , ξ, . . . ) dj , dj + 1 ( x j + 1 − x j ) ∆ d 2 ( 1 + d 2 − d 2 j − d 2 j + 1 )+ κ ( d j + d j + 1 − d − 1 ) d j , d j + 1 where ∆ = d 2 κ Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Sketch: anchor point independence Write ϕ ( x 0 ) l 1 ,..., l n ( x ) in terms of α ( x 0 ) m 1 ,..., m n ( x ) x 0 x 1 x 2 x n . . . l 1 l 2 l n ϕ ( x 0 ) l 1 ,..., l n ( x ) = x n − 1 x 0 x 1 x 2 x n . . . α ( x 0 ) m 1 ,..., m n ( x ) = m 1 m 2 m n Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
Sketch: anchor point independence Write ϕ ( x 0 ) l 1 ,..., l n ( x ) in terms of α ( x 0 ) m 1 ,..., m n ( x ) x 0 x 1 x 2 x n . . . l 1 l 2 l n ϕ ( x 0 ) l 1 ,..., l n ( x ) = x n − 1 x 0 x 1 x 2 x n . . . α ( x 0 ) m 1 ,..., m n ( x ) = m 1 m 2 m n Highest weight vectors: If E . v = 0, then in F ( x 0 ) [ v ]( x ) , the coefficient of α ( x 0 ) m 1 ,..., m n ( x ) vanishes whenever m 1 � = 0. Correlation functions with hidden quantum group 4. Details about the quantum group method Kalle Kytölä — Florence, May 2015
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