Outline Background 1 The objects An enumerative problem, and two generating series The b -Conjecture 2 An algebraic generalization and the b -Conjecture A family of invariants The invariants resolve a special case Evidence that they are b -invariants The q -Conjecture 3 A remarkable identity and the q -Conjecture A refinement Future Work 4 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 10 / 27
Outline Background 1 The objects An enumerative problem, and two generating series The b -Conjecture 2 An algebraic generalization and the b -Conjecture A family of invariants The invariants resolve a special case Evidence that they are b -invariants The q -Conjecture 3 A remarkable identity and the q -Conjecture A refinement Future Work 4 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 10 / 27
Jack Symmetric Functions Definition , are a one-parameter family, denoted Jack symmetric functions, by { J θ ( α ) } θ , that generalizes both Schur functions and zonal polynomials. Proposition (Stanley) Jack symmetric functions are related to Schur functions and zonal polynomials by: � J λ , J λ � 1 = H 2 J λ (1) = H λ s λ , λ , and J λ (2) = Z λ , � J λ , J λ � 2 = H 2 λ , where 2 λ is the partition obtained from λ by multiplying each part by two. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 10 / 27
A Generalized Series b -Conjecture (Goulden and Jackson) The generalized series, � � H p ( x ) , p ( y ) , p ( z ); b �� �� := (1 + b ) t ∂ J θ ( x ; 1 + b ) J θ ( y ; 1 + b ) J θ ( z ; 1 + b ) � ∂t ln � � J θ , J θ � 1+ b � � θ ∈ P t =0 � � = c ν,ϕ,ǫ ( b ) p ν ( x ) p ϕ ( y ) p ǫ ( z ) , n ≥ 0 ν,ϕ,ǫ ⊢ n has an combinatorial interpretation involving hypermaps. In particular b β ( h ) for some invariant β of rooted hypermaps. � c ν,ϕ,ǫ ( b ) = h ∈H ν,ϕ,ǫ Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 11 / 27
b is ubiquitous The many lives of b b = 0 b = 1 Hypermaps Orientable Locally Orientable ? Symmetric Functions s θ J θ ( b ) Z θ Matrix Integrals Hermitian Real Symmetric ? Moduli Spaces over C ? over R Matching Systems Bipartite All ? Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 12 / 27
A b -Invariant The b -Conjecture assumes that c ν,ϕ,ǫ ( b ) is a polynomial, and numerical evidence suggests that its degree is the genus of the hypermaps it enumerates. A b -invariant must: 1 be zero for orientable hypermaps, 2 be positive for non-orientable hypermaps, and 3 depend on rooting. Example Rootings of precisely three maps are enumerated by c [4] , [4] , [2 2 ] ( b ) = 1 + b + 3 b 2 . Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 13 / 27
A b -Invariant The b -Conjecture assumes that c ν,ϕ,ǫ ( b ) is a polynomial, and numerical evidence suggests that its degree is the genus of the hypermaps it enumerates. A b -invariant must: 1 be zero for orientable hypermaps, 2 be positive for non-orientable hypermaps, and 3 depend on rooting. Example Rootings of precisely three maps are enumerated by c [4] , [4] , [2 2 ] ( b ) = 1 + b + 3 b 2 . Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 13 / 27
A b -Invariant The b -Conjecture assumes that c ν,ϕ,ǫ ( b ) is a polynomial, and numerical evidence suggests that its degree is the genus of the hypermaps it enumerates. A b -invariant must: 1 be zero for orientable hypermaps, 2 be positive for non-orientable hypermaps, and 3 depend on rooting. Example Rootings of precisely three maps are enumerated by c [4] , [4] , [2 2 ] ( b ) = 1 + b + 3 b 2 . Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 13 / 27
A b -Invariant The b -Conjecture assumes that c ν,ϕ,ǫ ( b ) is a polynomial, and numerical evidence suggests that its degree is the genus of the hypermaps it enumerates. A b -invariant must: 1 be zero for orientable hypermaps, 2 be positive for non-orientable hypermaps, and 3 depend on rooting. Example Rootings of precisely three maps are enumerated by c [4] , [4] , [2 2 ] ( b ) = 1 + b + 3 b 2 . Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 13 / 27
A b -Invariant The b -Conjecture assumes that c ν,ϕ,ǫ ( b ) is a polynomial, and numerical evidence suggests that its degree is the genus of the hypermaps it enumerates. A b -invariant must: 1 be zero for orientable hypermaps, 2 be positive for non-orientable hypermaps, and 3 depend on rooting. Example Rootings of precisely three maps are enumerated by c [4] , [4] , [2 2 ] ( b ) = 1 + b + 3 b 2 . 2 b 2 1 + b 1 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 13 / 27
Outline Background 1 The objects An enumerative problem, and two generating series The b -Conjecture 2 An algebraic generalization and the b -Conjecture A family of invariants The invariants resolve a special case Evidence that they are b -invariants The q -Conjecture 3 A remarkable identity and the q -Conjecture A refinement Future Work 4 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 14 / 27
A root-edge classification There are four possible types of root edges in a map. Example Borders Bridges A handle Handles Cross-Borders Example A cross-border Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 14 / 27
A root-edge classification There are four possible types of root edges in a map. Example Borders Bridges A handle Handles Cross-Borders Example A cross-border Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 14 / 27
A root-edge classification Handles occur in pairs e v 4 v 4 v 3 v 3 v 5 v 5 f 2 f 2 v 2 v 2 v k v k v 1 v 1 v j v j u 4 u i u 4 u i f 1 f 1 u 3 u 1 u 3 u 1 e ′ u 2 u 2 Untwisted Twisted Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 14 / 27
A family of invariants The invariant η Iteratively deleting the root edge assigns a type to each edge in a map. An invariant, η , is given by η ( m ) := ( # of cross-borders ) + ( # of twisted handles ) . Different handle twisting determines a different invariant. Example Example Handle Cross-Border Border Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 15 / 27
A family of invariants The invariant η Iteratively deleting the root edge assigns a type to each edge in a map. An invariant, η , is given by η ( m ) := ( # of cross-borders ) + ( # of twisted handles ) . Different handle twisting determines a different invariant. Example Example Handle Cross-Border Border Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 15 / 27
A family of invariants The invariant η Iteratively deleting the root edge assigns a type to each edge in a map. An invariant, η , is given by η ( m ) := ( # of cross-borders ) + ( # of twisted handles ) . Different handle twisting determines a different invariant. Example Example Handle Cross-Border Border 1 or 2 1 0 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 15 / 27
A family of invariants The invariant η Iteratively deleting the root edge assigns a type to each edge in a map. An invariant, η , is given by η ( m ) := ( # of cross-borders ) + ( # of twisted handles ) . Different handle twisting determines a different invariant. Example Example Handle Cross-Border Border 1 or 2 1 0 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 15 / 27
Outline Background 1 The objects An enumerative problem, and two generating series The b -Conjecture 2 An algebraic generalization and the b -Conjecture A family of invariants The invariants resolve a special case Evidence that they are b -invariants The q -Conjecture 3 A remarkable identity and the q -Conjecture A refinement Future Work 4 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 16 / 27
Main result (marginal b -invariants exist) Theorem (La Croix) If ϕ partitions 2 n and η is a member of the family of invariants then, � � b η ( m ) . d v,ϕ ( b ) := c ν,ϕ, [2 n ] ( b ) = ℓ ( ν )= v m ∈M v,ϕ Corollary x | V ( m ) | y ϕ ( m ) z | E ( m ) | is an element of Z + [ x, y , b ] � z � . � M ( x, y , z ; b ) = m ∈M Corollary There is an uncountably infinite family of marginal b -invariants. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 16 / 27
Main result (marginal b -invariants exist) Theorem (La Croix) If ϕ partitions 2 n and η is a member of the family of invariants then, � � b η ( m ) . d v,ϕ ( b ) := c ν,ϕ, [2 n ] ( b ) = ℓ ( ν )= v m ∈M v,ϕ Corollary x | V ( m ) | y ϕ ( m ) z | E ( m ) | is an element of Z + [ x, y , b ] � z � . � M ( x, y , z ; b ) = m ∈M Corollary There is an uncountably infinite family of marginal b -invariants. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 16 / 27
Main result (marginal b -invariants exist) Theorem (La Croix) If ϕ partitions 2 n and η is a member of the family of invariants then, � � b η ( m ) . d v,ϕ ( b ) := c ν,ϕ, [2 n ] ( b ) = ℓ ( ν )= v m ∈M v,ϕ Corollary x | V ( m ) | y ϕ ( m ) z | E ( m ) | is an element of Z + [ x, y , b ] � z � . � M ( x, y , z ; b ) = m ∈M Corollary There is an uncountably infinite family of marginal b -invariants. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 16 / 27
Main result (marginal b -invariants exist) Theorem (La Croix) If ϕ partitions 2 n and η is a member of the family of invariants then, � � b η ( m ) . d v,ϕ ( b ) := c ν,ϕ, [2 n ] ( b ) = ℓ ( ν )= v m ∈M v,ϕ Proof (sketch). Distinguish between root and non-root faces in the generating series. Show that this series satisfies a PDE with a unique solution. Predict an expression for the corresponding algebraic refinement. Show that the refined series satisfies the same PDE. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 16 / 27
Main result (marginal b -invariants exist) Theorem (La Croix) If ϕ partitions 2 n and η is a member of the family of invariants then, � � b η ( m ) . d v,ϕ ( b ) := c ν,ϕ, [2 n ] ( b ) = ℓ ( ν )= v m ∈M v,ϕ Proof (sketch). Distinguish between root and non-root faces in the generating series. Show that this series satisfies a PDE with a unique solution. Predict an expression for the corresponding algebraic refinement. Show that the refined series satisfies the same PDE. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 16 / 27
Main result (marginal b -invariants exist) Theorem (La Croix) If ϕ partitions 2 n and η is a member of the family of invariants then, � � b η ( m ) . d v,ϕ ( b ) := c ν,ϕ, [2 n ] ( b ) = ℓ ( ν )= v m ∈M v,ϕ Proof (sketch). Distinguish between root and non-root faces in the generating series. Show that this series satisfies a PDE with a unique solution. Predict an expression for the corresponding algebraic refinement. Show that the refined series satisfies the same PDE. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 16 / 27
Main result (marginal b -invariants exist) Theorem (La Croix) If ϕ partitions 2 n and η is a member of the family of invariants then, � � b η ( m ) . d v,ϕ ( b ) := c ν,ϕ, [2 n ] ( b ) = ℓ ( ν )= v m ∈M v,ϕ Proof (sketch). Distinguish between root and non-root faces in the generating series. Show that this series satisfies a PDE with a unique solution. Predict an expression for the corresponding algebraic refinement. Show that the refined series satisfies the same PDE. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 16 / 27
Main result (marginal b -invariants exist) Theorem (La Croix) If ϕ partitions 2 n and η is a member of the family of invariants then, � � b η ( m ) . d v,ϕ ( b ) := c ν,ϕ, [2 n ] ( b ) = ℓ ( ν )= v m ∈M v,ϕ Proof (sketch). Distinguish between root and non-root faces in the generating series. Show that this series satisfies a PDE with a unique solution. Predict an expression for the corresponding algebraic refinement. Show that the refined series satisfies the same PDE. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 16 / 27
Main result (marginal b -invariants exist) Theorem (La Croix) If ϕ partitions 2 n and η is a member of the family of invariants then, � � b η ( m ) . d v,ϕ ( b ) := c ν,ϕ, [2 n ] ( b ) = ℓ ( ν )= v m ∈M v,ϕ Implications of the proof h v,ϕ,i b g − 2 i (1 + b ) i is an element of span Z + ( B g ) . � d v,ϕ ( b ) = 0 ≤ i ≤ g/ 2 The degree of d v,ϕ ( b ) is the genus of the maps it enumerates. The top coefficient, h v,ϕ, 0 , enumerates unhandled maps. η and root-face degree are independent among maps with given ϕ . Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 16 / 27
Finding a partial differential equation Root-edge type Schematic Contribution to M ∂ � Cross-border z ( i + 1) br i +2 M ∂r i i ≥ 0 i +1 ∂ � � Border z r j y i − j +2 M ∂r i i ≥ 0 j =1 ∂ 2 � Handle z (1 + b ) jr i + j +2 M ∂r i ∂y j i,j ≥ 0 � ∂ � � ∂ � � Bridge z r i + j +2 M M ∂r i ∂r j i,j ≥ 0 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 17 / 27
Finding a partial differential equation Root-edge type Schematic Contribution to M ∂ � Cross-border z ( i + 1) br i +2 M ∂r i i ≥ 0 i +1 ∂ � � Border z r j y i − j +2 M ∂r i i ≥ 0 j =1 ∂ 2 � Handle z (1 + b ) jr i + j +2 M ∂r i ∂y j i,j ≥ 0 � ∂ � � ∂ � � Bridge z r i + j +2 M M ∂r i ∂r j i,j ≥ 0 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 17 / 27
An integral expression for M ( N, y , z ; b ) Define the expectation operator �·� by � 2 � � 1+ b f ( λ ) exp 1 � � � f � := � V ( λ ) − 2(1+ b ) p 2 ( λ ) d λ . � R N Theorem (Goulden, Jackson, Okounkov) � �� k y k p k ( λ ) √ z k M ( N, y , z ; b ) = (1 + b )2 z ∂ � � 1 1 ∂z ln exp 1+ b k ≥ 1 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 18 / 27
An integral expression for M ( N, y , z ; b ) Define the expectation operator �·� by � 2 � � 1+ b f ( λ ) exp 1 � � � f � := � V ( λ ) − 2(1+ b ) p 2 ( λ ) d λ . � R N Theorem (Goulden, Jackson, Okounkov) � �� k y k p k ( λ ) √ z k M ( N, y , z ; b ) = (1 + b )2 z ∂ � � 1 1 ∂z ln exp 1+ b k ≥ 1 Predict that replacing 2 z ∂ ∂ � ∂z with jr j gives the refinement. ∂y j j ≥ 1 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 18 / 27
An integral expression for M ( N, y , z ; b ) Define the expectation operator �·� by � 2 � � 1+ b f ( λ ) exp 1 � � � f � := � V ( λ ) − 2(1+ b ) p 2 ( λ ) d λ . � R N Theorem (Goulden, Jackson, Okounkov) � �� k y k p k ( λ ) √ z k M ( N, y , z ; b ) = (1 + b )2 z ∂ � � 1 1 ∂z ln exp 1+ b k ≥ 1 Verify the guess using the following lemma. Lemma (La Croix) If N is a fixed positive integer, then j � � � p j +2 p θ � = ( j +1) b � p j p θ � +(1+ b ) i m i ( θ ) � p j + i p θ � i � + � p i p j − i p θ � . i ∈ θ i =0 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 18 / 27
Outline Background 1 The objects An enumerative problem, and two generating series The b -Conjecture 2 An algebraic generalization and the b -Conjecture A family of invariants The invariants resolve a special case Evidence that they are b -invariants The q -Conjecture 3 A remarkable identity and the q -Conjecture A refinement Future Work 4 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 19 / 27
The basis B g Is c ν,ϕ,ǫ ( b ) in span Z + ( B g ) � The sum c ν,ϕ, [2 n ] ( b ) is. ℓ ( ν )= v If so, then c ν,ϕ,ǫ ( b ) satisfies a functional equation. This has been verified. For polynomials Ξ g equals span Z ( B g ) . b g − 2 i (1 + b ) i : 0 ≤ i ≤ g/ 2 � � B g := p : p ( b − 1) = ( − b ) g p � 1 � � � Ξ g := b − 1 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 19 / 27
The basis B g Is c ν,ϕ,ǫ ( b ) in span Z + ( B g ) � The sum c ν,ϕ, [2 n ] ( b ) is. ℓ ( ν )= v If so, then c ν,ϕ,ǫ ( b ) satisfies a functional equation. This has been verified. For polynomials Ξ g equals span Z ( B g ) . b g − 2 i (1 + b ) i : 0 ≤ i ≤ g/ 2 � � B g := p : p ( b − 1) = ( − b ) g p � 1 � � � Ξ g := b − 1 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 19 / 27
The basis B g c ν,ϕ,ǫ ( b ) ∈ span Z + ( B g ) c ν,ϕ,ǫ ( b ) ∈ Z + [ b ] c ν,ϕ,ǫ ( b ) ∈ span Z ( B g ) c ν,ϕ,ǫ ( b ) ∈ Q + � b � c ν,ϕ,ǫ ( b ) ∈ span Q ( B g ) c ν,ϕ,ǫ ( b ) ∈ Q [ b ] b g − 2 i (1 + b ) i : 0 ≤ i ≤ g/ 2 � � B g := p : p ( b − 1) = ( − b ) g p � 1 � � � Ξ g := b − 1 c ν,ϕ,ǫ ( b ) ∈ Ξ g Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 19 / 27
Low genus coefficients can be verified Each dot represents a coefficient of c ν,ϕ, [2 n ] ( b ) with respect to B g . 0 0 Class of maps b 1 0 Cross-caps 2 1 Handles 1 Orientable 0 3 2 Unhandled − 1 Genus 4 3 2 All 1 5 4 6 5 3 7 6 8 7 4 8 Shaded sums can be obtained by evaluating M at special values of b . Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 20 / 27
Possible extensions Genus Edges Vertices What is needed? ❶ ≤ 1 any number any number ≤ 2 any number ≤ 3 ❶ ❶ and ❸ ≤ 2 and number any number ≤ 4 any number ≤ 2 ❶ and ❷ ❶ , ❷ , and ❸ ≤ 4 any number any number any genus ≤ 4 any number Verified ❸ or ❹ any genus ≤ 5 any number ❶ and ❹ any genus ≤ 6 any number any genus any number 1 Verified ❶ c ν,ϕ, [2 n ] ( b ) is a polynomial ❷ M ( − x , − y , − z ; − 1) enumerates unhandled maps ❸ Combinatorial sums are in span( B g ) ❹ An analogue of duality Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 21 / 27
Outline Background 1 The objects An enumerative problem, and two generating series The b -Conjecture 2 An algebraic generalization and the b -Conjecture A family of invariants The invariants resolve a special case Evidence that they are b -invariants The q -Conjecture 3 A remarkable identity and the q -Conjecture A refinement Future Work 4 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 22 / 27
Outline Background 1 The objects An enumerative problem, and two generating series The b -Conjecture 2 An algebraic generalization and the b -Conjecture A family of invariants The invariants resolve a special case Evidence that they are b -invariants The q -Conjecture 3 A remarkable identity and the q -Conjecture A refinement Future Work 4 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 22 / 27
A remarkable identity Theorem (Jackson and Visentin) Q ( u 2 , x, y, z ) = 1 2 M (4 u 2 , y + u, y, xz 2 ) + 1 2 M (4 u 2 , y − u, y, xz 2 ) = bis even u M (4 u 2 , y + u, y, xz 2 ) M is the genus series for rooted orientable maps, and Q is the corresponding series for 4 -regular maps. M ( u 2 , x, y, z ) := � u 2 g ( m ) x v ( m ) y f ( m ) z e ( m ) m ∈M Q ( u 2 , x, y, z ) := � u 2 g ( m ) x v ( m ) y f ( m ) z e ( m ) m ∈ Q g ( m ) , v ( m ) , f ( m ) , and e ( m ) are genus, # vertices, # faces, and # edges Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 22 / 27
A remarkable identity Theorem (Jackson and Visentin) Q ( u 2 , x, y, z ) = 1 2 M (4 u 2 , y + u, y, xz 2 ) + 1 2 M (4 u 2 , y − u, y, xz 2 ) = bis even u M (4 u 2 , y + u, y, xz 2 ) The right hand side is a generating series for a set M consisting of elements of M with each handle decorated independently in one of 4 ways, and an even subset of vertices marked. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 22 / 27
A remarkable identity Theorem (Jackson and Visentin) Q ( u 2 , x, y, z ) = 1 2 M (4 u 2 , y + u, y, xz 2 ) + 1 2 M (4 u 2 , y − u, y, xz 2 ) q -Conjecture (Jackson and Visentin) The identity is explained by a natural bijection ϕ from M to Q . A decorated map with A 4-regular map with v vertices e vertices ϕ → 2 k marked vertices 2 e edges e edges f + v − 2 k faces f faces genus g + k genus g Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 22 / 27
Products of rooted maps Two special cases suggest comparing products on M and Q . Details Products acting on Q π 1 ( ) π 2 , π 3 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 23 / 27
Products of rooted maps Two special cases suggest comparing products on M and Q . Details Products acting on M ( m 1 , m 2 ) ρ 1 ( m 1 , m 2 ) ρ 2 ( m 1 , m 2 ) ρ 3 ( m 1 , m 2 ) ( ) , g 1 + g 2 g 1 + g 2 g 1 + g 2 + 1 ( , ) g 1 + g 2 g 1 + g 2 g 1 + g 2 ( ) , g 1 + g 2 g 1 + g 2 g 1 + g 2 ( ) , g 1 + g 2 g 1 + g 2 − 1 g 1 + g 2 − 1 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 23 / 27
Outline Background 1 The objects An enumerative problem, and two generating series The b -Conjecture 2 An algebraic generalization and the b -Conjecture A family of invariants The invariants resolve a special case Evidence that they are b -invariants The q -Conjecture 3 A remarkable identity and the q -Conjecture A refinement Future Work 4 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 24 / 27
A refined q -Conjecture Conjecture (La Croix) There is a natural bijection ϕ from M to Q such that: A decorated map with A 4-regular map with v vertices e vertices ϕ → 2 k marked vertices 2 e edges e edges f + v − 2 k faces f faces genus g + k genus g and the root edge of ϕ ( m ) the root vertex of m if and only if is face-separating is not decorated. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 24 / 27
Root vertices in M are related to root edges in Q Example (planar maps with 2 edges and 2 decorated vertices) Nine of eleven rooted maps have a decorated root vertex. Example ( 4 -regular maps on the torus with two vertices) Nine of fifteen rooted maps have face-non-separating root edges. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 25 / 27
Testing the refined conjecture The refined conjecture has been tested numerically for images of maps with at most 20 edges by expressing the relevant generating series as linear combination of Q and the generating series for (3 , 1) -pseudo- 4 -regular maps. An analytic reformulation The existence of an appropriate bijection, modulo the definition of ‘natural’, is equivalent to the following conjectured identity: � ( p 4 + p 1 p 3 )e p 4 x � ( N ) � e p 4 x � ( N +1) = − m [1 , 3] e p 4 x � ( N +1) � e p 4 x � ( N ) . � for every positive integer N . Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 26 / 27
Outline Background 1 The objects An enumerative problem, and two generating series The b -Conjecture 2 An algebraic generalization and the b -Conjecture A family of invariants The invariants resolve a special case Evidence that they are b -invariants The q -Conjecture 3 A remarkable identity and the q -Conjecture A refinement Future Work 4 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 27 / 27
Future Work On the b -Conjecture Show that c ν,ϕ,ǫ ( b ) is a polynomial for every ν , ϕ , and ǫ . Show that the generating series for maps is an element of span( B g ) . Explicitly compute the generating series for unhandled maps. Extend the analysis to hypermaps. On the q -Conjecture Verify one of the algebraic or analytic properties that characterizes the refinement. Use the refinement to determine additional structure of the bijection. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 27 / 27
Future Work On the b -Conjecture Show that c ν,ϕ,ǫ ( b ) is a polynomial for every ν , ϕ , and ǫ . Show that the generating series for maps is an element of span( B g ) . Explicitly compute the generating series for unhandled maps. Extend the analysis to hypermaps. On the q -Conjecture Verify one of the algebraic or analytic properties that characterizes the refinement. Use the refinement to determine additional structure of the bijection. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 27 / 27
Future Work On the b -Conjecture Show that c ν,ϕ,ǫ ( b ) is a polynomial for every ν , ϕ , and ǫ . Show that the generating series for maps is an element of span( B g ) . Explicitly compute the generating series for unhandled maps. Extend the analysis to hypermaps. On the q -Conjecture Verify one of the algebraic or analytic properties that characterizes the refinement. Use the refinement to determine additional structure of the bijection. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 27 / 27
Future Work On the b -Conjecture Show that c ν,ϕ,ǫ ( b ) is a polynomial for every ν , ϕ , and ǫ . Show that the generating series for maps is an element of span( B g ) . Explicitly compute the generating series for unhandled maps. Extend the analysis to hypermaps. On the q -Conjecture Verify one of the algebraic or analytic properties that characterizes the refinement. Use the refinement to determine additional structure of the bijection. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 27 / 27
Future Work On the b -Conjecture Show that c ν,ϕ,ǫ ( b ) is a polynomial for every ν , ϕ , and ǫ . Show that the generating series for maps is an element of span( B g ) . Explicitly compute the generating series for unhandled maps. Extend the analysis to hypermaps. On the q -Conjecture Verify one of the algebraic or analytic properties that characterizes the refinement. Use the refinement to determine additional structure of the bijection. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 27 / 27
Future Work On the b -Conjecture Show that c ν,ϕ,ǫ ( b ) is a polynomial for every ν , ϕ , and ǫ . Show that the generating series for maps is an element of span( B g ) . Explicitly compute the generating series for unhandled maps. Extend the analysis to hypermaps. On the q -Conjecture Verify one of the algebraic or analytic properties that characterizes the refinement. Use the refinement to determine additional structure of the bijection. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 27 / 27
Future Work On the b -Conjecture Show that c ν,ϕ,ǫ ( b ) is a polynomial for every ν , ϕ , and ǫ . Show that the generating series for maps is an element of span( B g ) . Explicitly compute the generating series for unhandled maps. Extend the analysis to hypermaps. On the q -Conjecture Verify one of the algebraic or analytic properties that characterizes the refinement. Use the refinement to determine additional structure of the bijection. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 27 / 27
Future Work On the b -Conjecture Show that c ν,ϕ,ǫ ( b ) is a polynomial for every ν , ϕ , and ǫ . Show that the generating series for maps is an element of span( B g ) . Explicitly compute the generating series for unhandled maps. Extend the analysis to hypermaps. On the q -Conjecture Verify one of the algebraic or analytic properties that characterizes the refinement. Use the refinement to determine additional structure of the bijection. Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 27 / 27
The End Thank You Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 27 / 27
Appendices Symmetric Functions 5 Computing η 6 Encodings 7 Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 28 / 27
Jack Symmetric Functions With respect to the inner product defined by | λ | ! | C λ | α ℓ ( λ ) , � p λ ( x ) , p µ ( x ) � = δ λ,µ Jack symmetric functions are the unique family satisfying: (P1) (Orthogonality) If λ � = µ , then � J λ , J µ � α = 0 . (P2) (Triangularity) J λ = � µ � λ v λµ ( α ) m µ , where v λµ ( α ) is a rational function in α , and ‘ � ’ denotes the natural order on partitions. (P3) (Normalization) If | λ | = n , then v λ, [1 n ] ( α ) = n ! . Return Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 29 / 27
Computing η m 1 m 2 Return Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 30 / 27
Computing η m 1 m 3 m 2 Return Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 30 / 27
Computing η m 1 m 3 m 4 m 2 Return Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 30 / 27
Computing η m 1 m 3 m 4 η ( m 4 ) = 0 η ( m 3 ) = η ( m 4 ) + 1 = 1 m 2 η ( m 2 ) = η ( m 3 ) + 1 = 2 η ( m 1 ) = η ( m 3 ) = 1 Return Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 30 / 27
Computing η m 1 m 3 m 4 η ( m 4 ) = 0 η ( m 3 ) = η ( m 4 ) + 1 = 1 m 2 η ( m 2 ) = η ( m 3 ) + 1 = 2 η ( m 1 ) = η ( m 3 ) = 1 Return Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 30 / 27
Computing η m 1 m 3 m 4 η ( m 4 ) = 0 η ( m 3 ) = η ( m 4 ) + 1 = 1 m 2 η ( m 2 ) = η ( m 3 ) + 1 = 2 η ( m 1 ) = η ( m 3 ) = 1 Return Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 30 / 27
Computing η m 5 m 6 Return Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 30 / 27
Computing η m 5 m 7 m 6 Return Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 30 / 27
Computing η m 5 m 7 m 8 m 6 Return Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 30 / 27
Computing η m 5 m 7 m 8 η ( m 8 ) = 0 η ( m 7 ) = η ( m 8 ) + 1 = 1 m 6 η ( m 6 ) = η ( m 7 ) or η ( m 7 ) + 1 η ( m 5 ) = η ( m 7 ) + 1 or η ( m 7 ) Return { η ( m 5 ) , η ( m 6 ) } = { 1 , 2 } Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 30 / 27
Computing η m 5 m 7 m 8 η ( m 8 ) = 0 η ( m 7 ) = η ( m 8 ) + 1 = 1 m 6 η ( m 6 ) = η ( m 7 ) or η ( m 7 ) + 1 η ( m 5 ) = η ( m 7 ) + 1 or η ( m 7 ) Return { η ( m 5 ) , η ( m 6 ) } = { 1 , 2 } Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 30 / 27
Computing η m 5 m 7 m 8 η ( m 8 ) = 0 η ( m 7 ) = η ( m 8 ) + 1 = 1 m 6 η ( m 6 ) = η ( m 7 ) or η ( m 7 ) + 1 η ( m 5 ) = η ( m 7 ) + 1 or η ( m 7 ) Return { η ( m 5 ) , η ( m 6 ) } = { 1 , 2 } Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 30 / 27
Encoding Orientable Maps 1 Orient and label the edges. 1 2 This induces labels on flags. 2 3 Clockwise circulations at each 4 vertex determine ν . 4 Face circulations are the cycles 5 3 of ǫν . 6 ǫ = (1 1 ′ )(2 2 ′ )(3 3 ′ )(4 4 ′ )(5 5 ′ )(6 6 ′ ) ν = (1 2 3)(1 ′ 4)(2 ′ 5)(3 ′ 5 ′ 6)(4 ′ 6 ′ ) ǫν = ϕ = (1 4 6 ′ 3 ′ )(1 ′ 2 5 6 4 ′ )(2 ′ 3 5 ′ ) Return Michael La Croix (University of Waterloo) The Jack parameter and maps July 29, 2009 31 / 27
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