Weighted Hurwitz numbers and hypergeometric τ -functions ∗ J. Harnad Centre de recherches mathématiques Université de Montréal Department of Mathematics and Statistics Concordia University GGI programme Statistical Mechanics, Integrability and Combinatorics Firenze, May 11 - July 3, 2015 ∗ Based in part on joint work with M. Guay-Paquet and A. Yu. Orlov Harnad (CRM and Concordia) Weighted Hurwitz numbers and hypergeometric tau - functions May 11 - 15 , 2015 1 / 39
Classical Hurwitz numbers 1 Group theoretical/combinatorial meaning Geometric meaning: simple Hurwitz numbers Double Hurwitz numbers (Okounkov) 2 KP and 2D Toda τ -functions as generating functions Hirota bilinear relations τ -functions as generating functions for Hurwitz numbers Fermionic representation Composite, signed, weighted and quantum Hurwitz numbers 3 Combinatorial weighted Hurwitz numbers: weighted paths Fermionic representation Weighted Hurwitz numbers Geometric weighted Hurwitz numbers: weighted coverings Example: Belyi curves: strongly monotone paths Example: Composite Hurwitz numbers Example: Signed Hurwitz numbers Quantum Hurwitz numbers Bosonic gases and Planck’s distribution law Harnad (CRM and Concordia) Weighted Hurwitz numbers and hypergeometric tau - functions May 11 - 15 , 2015 2 / 39
Classical Hurwitz numbers Group theoretical/combinatorial meaning Factorization of elements in S n Question: Given a permutation h ∈ S n of cycle type µ = ( µ 1 ≥ µ 2 ≥ · · · ≥ µ ℓ ( µ > 0 ) , what is the number H d ( µ ) of distinct ways it can be written as a product h = ( a 1 b 1 ) · · · ( a d b d ) of d transpositions ? Young diagram of a partition. Example µ = ( 5 , 4 , 4 , 2 ) Harnad (CRM and Concordia) Weighted Hurwitz numbers and hypergeometric tau - functions May 11 - 15 , 2015 3 / 39
Classical Hurwitz numbers Group theoretical/combinatorial meaning Representation theoretic answer (Frobenius): χ λ ( µ ) H d ( µ ) = � ( cont λ ) d z µ h λ λ, | λ | = | µ | � − 1 � 1 where h λ = det is the product of the hook lengths of ( λ i − i + j )! the partition λ = λ 1 ≥ · · · ≥ λ ℓ ( λ > 0 , ℓ ( λ ) λ i ( λ i − 2 i + 1 ) = χ λ (( 2 , ( 1 ) n − 2 ) h λ ( j − i ) = 1 � � cont ( λ ) := 2 z ( 2 , ( 1 ) n − 2 ) ( ij ) ∈ λ i = 1 is the content sum of the associated Young diagram, χ λ ( µ ) is the irreducible character of representation λ evaluated in the conjugacy class µ , and � i m ( µ ) i ( m i ( µ ))! = | aut ( µ ) | z µ := i Harnad (CRM and Concordia) Weighted Hurwitz numbers and hypergeometric tau - functions May 11 - 15 , 2015 4 / 39
Classical Hurwitz numbers Geometric meaning: simple Hurwitz numbers Geometric meaning: simple Hurwitz numbers Hurwitz numbers: Let H ( µ ( 1 ) , . . . , µ ( k ) ) be the number of inequivalent branched n -sheeted covers of the Riemann sphere, with k branch points, and ramification profiles ( µ ( 1 ) , . . . , µ ( k ) ) at these points. The genus of the covering curve is given by the Riemann-Hurwitz d := � l i = 1 ℓ ∗ ( µ ( i ) ) 2 − 2 g = ℓ ( λ ) + ℓ ( µ ) − d , formula : where ℓ ∗ ( µ ) := | µ | − ℓ ( µ ) is the colength of the partition. The Frobenius-Schur formula expresses this in terms of characters: k χ λ ( µ ( i ) ) H ( µ ( 1 ) , . . . , µ ( k ) ) = � h k − 2 � λ z µ ( i ) λ, | λ | = n = | µ ( i ) | i = 1 In particular, choosing only simple ramifications µ ( i ) = ( 2 , ( 1 ) n − 2 ) at d = k − 1 points and one further arbitrary one µ at a single point, say, 0, we have the simple Hurwitz number : H d ( µ ) := H (( 2 , ( 1 ) n − 1 ) , . . . , ( 2 , ( 1 ) n − 1 ) , µ ) . Harnad (CRM and Concordia) Weighted Hurwitz numbers and hypergeometric tau - functions May 11 - 15 , 2015 5 / 39
Classical Hurwitz numbers Geometric meaning: simple Hurwitz numbers 3 -sheeted branched cover with ramification profiles ( 3 ) and ( 2 , 1 ) Harnad (CRM and Concordia) Weighted Hurwitz numbers and hypergeometric tau - functions May 11 - 15 , 2015 6 / 39
Classical Hurwitz numbers Double Hurwitz numbers (Okounkov) Double Hurwitz numbers Double Hurwitz numbers: The double Hurwitz number (Okounkov (2000)), defined as Cov d ( µ, ν ) = H d exp ( µ, ν )) := H (( 2 , ( 1 ) n − 1 ) , . . . , ( 2 , ( 1 ) n − 1 ) , µ, ν ) . has the ramification type ( µ, ν ) at two points, say ( 0 , ∞ ) , and simple ramification µ ( i ) = ( 2 , ( 1 ) n − 2 ) at d other branch points. Combinatorially : This equals the number of d -step paths in the Cayley graph of S n generated by transpositions, starting at an element h ∈ C µ and ending in the conjugacy class C ν . Here { C µ , | µ | = n ∈ C [ S n ]) } is defined to be the basis of the group algebra C [ S n ] consisting of the sums over all elements h in the various conjugacy classes of cycle type µ . � C µ = h . h ∈ conj ( µ ) Harnad (CRM and Concordia) Weighted Hurwitz numbers and hypergeometric tau - functions May 11 - 15 , 2015 7 / 39
Classical Hurwitz numbers Double Hurwitz numbers (Okounkov) Example: Cayley graph for S 4 generated by all transpositions Transpositioncayleyons4.png 867 × 779 pixels 14-08-23 10:17 PM Harnad (CRM and Concordia) Weighted Hurwitz numbers and hypergeometric tau - functions May 11 - 15 , 2015 8 / 39
KP and 2D Toda τ -functions as generating functions τ -function generating functions for Hurwitz numbers Define r ( u , z ) τ mKP ( u , z ) ( N , t ) := � ( N ) h − 1 λ S λ ( t ) λ λ r ( u , z ) τ 2 DToda ( u , z ) ( N , t , s ) := � ( N ) S λ ( t ) S λ ( s ) λ λ r ( u , z ) r ( u , z ) r ( u , z ) � := ue jz where ( N ) := N + j − i , λ j ( ij ) ∈ λ and t = ( t 1 , t 2 , . . . ) , s = ( s 1 , s 2 , . . . ) are the KP and 2D Toda flow variables. Harnad (CRM and Concordia) Weighted Hurwitz numbers and hypergeometric tau - functions May 11 - 15 , 2015 9 / 39
KP and 2D Toda τ -functions as generating functions Hirota bilinear relations mKP Hirota bilinear relations for τ mKP ( N , t ) , t := ( t 1 , t 2 , . . . ) , N ∈ Z g � z N − N ′ e − ξ ( δ t , z ) τ mKP ( N , t − [ z − 1 ]) τ mKP ( N ′ , t + δ t + [ z − 1 ]) = 0 g g z = ∞ ∞ [ z − 1 ] i := 1 � δ t i z i , i z − i , ξ ( δ t , z ) := identically in δ t = ( δ t 1 , δ t 2 , . . . ) i = 1 2 D Toda Hirota bilinear relations for τ 2 Toda ( N , t , s ) , s := ( s 1 , s 2 , . . . ) g � z N − N ′ e − ξ ( δ t , z ) τ 2 Toda ( N , t − [ z − 1 ] , s ) τ 2 Toda ( N ′ , t + δ t + [ z − 1 ] , s ) = g g z = ∞ � ( N ′ − 1 , t , s + δ s + [ z ]) z N − N ′ e − ξ ( δ s , z ) τ 2 Toda ( N + 1 , t , s − [ z ]) τ 2 Toda g g z = 0 [ z ] i := 1 i z i , identically in δ t = ( δ t 1 , δ t 2 , . . . ) , δ s := ( δ s 1 , δ s 2 , . . . ) Harnad (CRM and Concordia) Weighted Hurwitz numbers and hypergeometric tau - functions May 11 - 15 , 2015 10 / 39
KP and 2D Toda τ -functions as generating functions τ -functions as generating functions for Hurwitz numbers Hypergeometric τ -functions as generating functions for Hurwitz numbers For N = 0, we have r ( u , z ) ( 0 ) = u | λ | e z cont ( λ ) λ Using the Frobenius character formula : χ λ ( µ ) � S λ ( t ) = P µ ( t ) Z µ µ, | µ | = | λ | where we restrict to is i := p ′ it i := p i , i and the P µ ’s are the power sum symmetric functions ℓ ( µ ) n n � � x i p ′ � y i P µ = p µ i , p i := a , i := a , i = 1 a = 1 a = 1 Harnad (CRM and Concordia) Weighted Hurwitz numbers and hypergeometric tau - functions May 11 - 15 , 2015 11 / 39
KP and 2D Toda τ -functions as generating functions τ -functions as generating functions for Hurwitz numbers Hypergeometric τ -functions as generating functions for Hurwitz numbers r ( u , z ) := r ( u , z ) ( 0 ) = u | λ | e z cont ( λ ) λ λ τ ( u , z ) ( t ) := τ KP ( u , z ) ( 0 , t ) = � u | λ | h − 1 λ e z cont ( λ ) S λ ( t ) λ ∞ ∞ z d � � � u n H d ( µ ) P µ ( t ) = d ! n = 0 d = 0 µ, | µ | = n τ 2 D ( u , z ) ( t , s ) := τ 2 DToda ( u , z ) ( 0 , t , s ) = � u | λ | e z cont ( λ ) S λ ( t ) S λ ( s ) λ ∞ ∞ z d � � � u n H d = exp ( µ, ν ) P µ ( t ) P ν ( s ) d ! n = 0 d = 0 µ,ν, | µ | = ν | = n These are therefore generating functions for the single and double Hurwitz numbers . Harnad (CRM and Concordia) Weighted Hurwitz numbers and hypergeometric tau - functions May 11 - 15 , 2015 12 / 39
KP and 2D Toda τ -functions as generating functions Fermionic representation Fermionic representation of KP and 2D Toda τ -functions F 1 e z ˆ ˆ τ mKP ( u , z ) ( N , t ) = � N | ˆ F 2 ˆ γ + ( t ) u γ − ( 1 , 0 , 0 . . . ) | N � F 1 e z ˆ ˆ τ 2 DToda ( u , z ) ( N , t , s ) = � N | ˆ F 2 ˆ γ + ( t ) u γ − ( s ) | N � where the fermionic creation and annihiliation operators { ψ i , ψ † i } i ∈ Z satisfy the usual anticommutation relations and vacuum state | 0 � vanishing conditions [ ψ i , ψ † ψ † j ] + = δ ij ψ i | 0 � = 0 , for i < 0 , i | 0 � = 0 , for i ≥ 0 , F k := 1 j k : ψ j ψ † ˆ � j k j ∈ Z � ∞ � ∞ � ψ k ψ † i = 1 t i J i , i = 1 s i J i , ˆ γ + ( t ) = e γ − ( s ) = e ˆ J i = k + i , i ∈ Z . k ∈ Z Harnad (CRM and Concordia) Weighted Hurwitz numbers and hypergeometric tau - functions May 11 - 15 , 2015 13 / 39
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