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All work and no play makes Jack enumerate maps Michael La Croix PhD University of Waterloo February 15, 2011 Outline Combinatorial Enumeration 1 Graphs, Maps, and Surfaces 2 Rooted Maps and Flags 3 Quantum gravity and the q -Conjecture 4

  1. Three Utilities A classical challenge is how to place conduits so that three utilities can be connected to three houses. This should be done so that no conduits cross. GAS SEWERS WATER Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 9 / 38

  2. Three Utilities Together, the houses, utilities, and conduits define a graph. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 9 / 38

  3. Three Utilities We want to embed the graph. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 9 / 38

  4. Three Utilities After a lot of effort, we con- clude that the problem cannot be solved as it appears to be stated. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 9 / 38

  5. Three Utilities After a lot of effort, we con- clude that the problem cannot be solved as it appears to be stated. This is actually a classic result (Kuratowski’s Theorem). The given graph is one of two ob- stacles to being able to draw a graph on the plane. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 9 / 38

  6. Three Utilities Solving the problems relies on finding a loop-hole in its state- ment. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 9 / 38

  7. Three Utilities Solving the problems relies on finding a loop-hole in its state- ment. One solution is that, as stated, the problem does not say that the houses are on a plane. We can draw them on a torus. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 9 / 38

  8. Representing Surfaces The torus (or any surface) can be represented schematically in terms of the surgery required to stitch it together from a rubber sheet. = Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 10 / 38

  9. Representing Surfaces The torus (or any surface) can be represented schematically in terms of the surgery required to stitch it together from a rubber sheet. Example = Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 10 / 38

  10. Other Surfaces are Also Obtained by Surgery = = Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 11 / 38

  11. Graphs, Surfaces, and Maps Example Definition A surface is a compact 2-manifold without boundary. Definition A graph is a finite set of vertices together with a finite set of edges , such that each edge is associated with either one or two vertices. Definition A map is a 2-cell embedding of a graph in a surface. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 12 / 38

  12. Graphs, Surfaces, and Maps Example Definition A surface is a compact 2-manifold without boundary. Definition A graph is a finite set of vertices together with a finite set of edges , such that each edge is associated with either one or two vertices. Definition A map is a 2-cell embedding of a graph in a surface. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 12 / 38

  13. Graphs, Surfaces, and Maps Example Definition A surface is a compact 2-manifold without boundary. Definition A graph is a finite set of vertices together with a finite set of edges , such that each edge is associated with either one or two vertices. Definition A map is a 2-cell embedding of a graph in a surface. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 12 / 38

  14. Maps and Faces Once a graph is drawn, the unused portion of the paper is split into faces. f3 f1 f2 A map is a graph together with an embedding in a surfaces. It is defined by its vertices, edges, and faces. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 13 / 38

  15. Maps and Faces Once a graph is drawn, the unused portion of the paper is split into faces. For symmetry, the outer face is thought of as part of a sphere. A map is a graph together with an embedding in a surfaces. It is defined by its vertices, edges, and faces. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 13 / 38

  16. Tiling the Representation The faces of a map can be made more evident by tessellating the tile that represents the surface. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 14 / 38

  17. Tiling the Representation Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 14 / 38

  18. Outline Combinatorial Enumeration 1 Graphs, Maps, and Surfaces 2 Rooted Maps and Flags 3 Quantum gravity and the q -Conjecture 4 Map Enumeration 5 Orientable Maps Non-Orientable Maps Hypermaps Generating Series What does Jack have to do with it? 6 The invariants resolve a special case Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 14 / 38

  19. Ribbon Graphs Example The homeomorphism class of an embedding is determined by a neighbourhood of the graph. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 15 / 38

  20. Ribbon Graphs Example Neighbourhoods of vertices and edges can be replaced by discs and ribbons to form a ribbon graph. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 15 / 38

  21. Flags Example The boundaries of ribbons determine flags. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 15 / 38

  22. Flags Example The boundaries of ribbons determine flags, and these can be associated with quarter edges. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 15 / 38

  23. Rooted Maps Definition A rooted map is a map together with a distinguished orbit of flags under the action of its automorphism group. Example Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 16 / 38

  24. Rooted Maps Definition A rooted map is a map together with a distinguished orbit of flags under the action of its automorphism group. Example Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 16 / 38

  25. Rooted Maps Definition A rooted map is a map together with a distinguished orbit of flags under the action of its automorphism group. Example Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 16 / 38

  26. Outline Combinatorial Enumeration 1 Graphs, Maps, and Surfaces 2 Rooted Maps and Flags 3 Quantum gravity and the q -Conjecture 4 Map Enumeration 5 Orientable Maps Non-Orientable Maps Hypermaps Generating Series What does Jack have to do with it? 6 The invariants resolve a special case Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 16 / 38

  27. 2-dimensional Quantum Gravity Two models of 2-dimensional quantum gravity are analyzed by enumerating rooted orientable maps. The Penner Model involves all smooth maps. φ − 4 model involves only 4-regular maps. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 17 / 38

  28. 2-dimensional Quantum Gravity Two models of 2-dimensional quantum gravity are analyzed by enumerating rooted orientable maps. The Penner Model involves all smooth maps. φ − 4 model involves only 4-regular maps. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 17 / 38

  29. 2-dimensional Quantum Gravity Two models of 2-dimensional quantum gravity are analyzed by enumerating rooted orientable maps. The Penner Model involves all smooth maps. φ − 4 model involves only 4-regular maps. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 17 / 38

  30. 2-dimensional Quantum Gravity Two models of 2-dimensional quantum gravity are analyzed by enumerating rooted orientable maps. The Penner Model involves all smooth maps. φ − 4 model involves only 4-regular maps. The models have the same behaviour. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 17 / 38

  31. An algebraic explanation - 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 Map Enumeration February 15, 2011 18 / 38

  32. An algebraic explanation - 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 Map Enumeration February 15, 2011 18 / 38

  33. An algebraic explanation - 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 Map Enumeration February 15, 2011 18 / 38

  34. Two Clues The radial construction for undecorated maps One extra image of φ Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 19 / 38

  35. Two Clues The radial construction for undecorated maps One extra image of φ Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 19 / 38

  36. Two Clues One extra image of φ [ ] , = [ ] , = Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 19 / 38

  37. 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 Map Enumeration February 15, 2011 20 / 38

  38. ¯ 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 Map Enumeration February 15, 2011 21 / 38

  39. Outline Combinatorial Enumeration 1 Graphs, Maps, and Surfaces 2 Rooted Maps and Flags 3 Quantum gravity and the q -Conjecture 4 Map Enumeration 5 Orientable Maps Non-Orientable Maps Hypermaps Generating Series What does Jack have to do with it? 6 The invariants resolve a special case Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 21 / 38

  40. The Map Series Extra symmetry makes it easier to work with a more refined series. An enumerative problem associated with maps is to determine the number of rooted maps with specified vertex- and face- degree partitions. Definition The map series for a set M of rooted maps is the combinatorial sum � x ν ( m ) y φ ( m ) z | E ( m ) | M ( x , y , z ) := m ∈M where ν ( m ) and φ ( m ) are the the vertex- and face-degree partitions of m . Example x 3 x 3 x 3 x 3 x 3 2 x 2 x 2 x 2 x 2 x 2 y 5 ) z 6 . � � Rootings of are enumerated by ( y 3 y 3 y 3 y 3 y 3 y 4 y 4 y 4 y 5 y 4 y 4 y 5 y 5 y 5 2 2 2 2 3 3 3 3 3 Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 22 / 38

  41. The Map Series Extra symmetry makes it easier to work with a more refined series. An enumerative problem associated with maps is to determine the number of rooted maps with specified vertex- and face- degree partitions. Definition The map series for a set M of rooted maps is the combinatorial sum � x ν ( m ) y φ ( m ) z | E ( m ) | M ( x , y , z ) := m ∈M where ν ( m ) and φ ( m ) are the the vertex- and face-degree partitions of m . Example x 3 x 3 x 3 x 3 x 3 2 x 2 x 2 x 2 x 2 x 2 y 5 ) z 6 . � � Rootings of are enumerated by ( y 3 y 3 y 3 y 3 y 3 y 4 y 4 y 4 y 5 y 4 y 4 y 5 y 5 y 5 2 2 2 2 3 3 3 3 3 Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 22 / 38

  42. Encoding Orientable Maps 1 1 Orient and label the edges. 2 This induces labels on flags. 2 4 3 Clockwise circulations at each vertex determine ν . 3 5 4 Face circulations are the cycles 6 of ǫν . 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 ′ ) Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 23 / 38

  43. Encoding Orientable Maps 1 Orient and label the edges. 1 2 This induces labels on flags. 4 2 3 Clockwise circulations at each vertex determine ν . 3 5 4 Face circulations are the cycles 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 ′ ) Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 23 / 38

  44. Encoding Orientable Maps 1’ 1 Orient and label the edges. 4 2 This induces labels on flags. 2 1 3 Clockwise circulations at each 2’ 3 vertex determine ν . 5 4 Face circulations are the cycles 4’ 5’ 3’ 6’ 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 ′ ) Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 23 / 38

  45. Encoding Orientable Maps 1’ 1 Orient and label the edges. 4 2 This induces labels on flags. 3 Clockwise circulations at each 2 1 2’ vertex determine ν . 3 5 4 Face circulations are the cycles 4’ 5’ 3’ of ǫν . 6’ 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 ′ ) Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 23 / 38

  46. A M¨ obius Strip Maps can also be drawn in surfaces that contain M¨ obius strips. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 24 / 38

  47. Encoding Locally Orientable Maps A new encoding is needed to record twisting. Start with a ribbon graph. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 25 / 38

  48. Encoding Locally Orientable Maps A new encoding is needed to record twisting. Start with a ribbon graph. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 25 / 38

  49. Encoding Locally Orientable Maps A new encoding is needed to record twisting. M v M e M f Ribbon boundaries determine 3 perfect matchings of flags. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 25 / 38

  50. Encoding Locally Orientable Maps A new encoding is needed to record twisting. M v M e Pairs of matchings determine, faces, Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 25 / 38

  51. Encoding Locally Orientable Maps A new encoding is needed to record twisting. M v M f Pairs of matchings determine, faces, edges, Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 25 / 38

  52. Encoding Locally Orientable Maps A new encoding is needed to record twisting. M e M f Pairs of matchings determine, faces, edges, and vertices. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 25 / 38

  53. Encoding Locally Orientable Maps M v 5 6’ 5’ 6 7’ 2 7 2’ 8 8’ 3’ M e 1 3 1’ 4’ 4 M f { 1 , 3 } , { 1 ′ , 3 ′ } , { 2 , 5 } , { 2 ′ , 5 ′ } , { 4 , 8 ′ } , { 4 ′ , 8 } , { 6 , 7 } , { 6 ′ , 7 ′ } � � M v = { 1 , 2 ′ } , { 1 ′ , 4 } , { 2 , 3 ′ } , { 3 , 4 ′ } , { 5 , 6 ′ } , { 5 ′ , 8 } , { 6 , 7 ′ } , { 7 , 8 ′ } � � M e = { 1 , 1 ′ } , { 2 , 2 ′ } , { 3 , 3 ′ } , { 4 , 4 ′ } , { 5 , 5 ′ } , { 6 , 6 ′ } , { 7 , 7 ′ } , { 8 , 8 ′ } � � M f = Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 25 / 38

  54. Hypermaps Generalizing the combinatorial encoding, an arbitrary triple of perfect matchings determines a hypermap when the triple induces a connected graph, with cycles of M e ∪ M f , M e ∪ M v , and M v ∪ M f determining vertices, hyperfaces, and hyperedges. Example Hypermaps both specialize and generalize maps. Example → ֒ Hypermaps can be represented as face-bipartite maps. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 26 / 38

  55. Hypermaps Generalizing the combinatorial encoding, an arbitrary triple of perfect matchings determines a hypermap when the triple induces a connected graph, with cycles of M e ∪ M f , M e ∪ M v , and M v ∪ M f determining vertices, hyperfaces, and hyperedges. Example Hypermaps both specialize and generalize maps. Example → ֒ Maps can be represented as hypermaps with ǫ = [2 n ]. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 26 / 38

  56. The Hypermap Series Definition The hypermap series for a set H of hypermaps is the combinatorial sum � x ν ( h ) y φ ( h ) z ǫ ( h ) H ( x , y , z ) := h ∈H where ν ( h ), φ ( h ), and ǫ ( h ) are the vertex-, hyperface-, and hyperedge- degree partitions of h . Example Note � M ( x , y , z ) = H ( x , y , z ) � � z i = zδ i, 2 Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 27 / 38

  57. The Hypermap Series Definition The hypermap series for a set H of hypermaps is the combinatorial sum � x ν ( h ) y φ ( h ) z ǫ ( h ) H ( x , y , z ) := h ∈H where ν ( h ), φ ( h ), and ǫ ( h ) are the vertex-, hyperface-, and hyperedge- degree partitions of h . Example Note � M ( x , y , z ) = H ( x , y , z ) � � z i = zδ i, 2 Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 27 / 38

  58. How does this help? Instead of counting rooted maps, we can count labelled hypermaps. The numbers are different, but the correction factor is easy. Labelled counting problems are turned into problems involving counting factorizations. These can be answered via character theory. Appropriate characters appear as coefficients of symmetric functions. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 28 / 38

  59. How does this help? Instead of counting rooted maps, we can count labelled hypermaps. The numbers are different, but the correction factor is easy. Labelled counting problems are turned into problems involving counting factorizations. These can be answered via character theory. Appropriate characters appear as coefficients of symmetric functions. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 28 / 38

  60. How does this help? Instead of counting rooted maps, we can count labelled hypermaps. The numbers are different, but the correction factor is easy. Labelled counting problems are turned into problems involving counting factorizations. These can be answered via character theory. Appropriate characters appear as coefficients of symmetric functions. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 28 / 38

  61. How does this help? Instead of counting rooted maps, we can count labelled hypermaps. The numbers are different, but the correction factor is easy. Labelled counting problems are turned into problems involving counting factorizations. These can be answered via character theory. Appropriate characters appear as coefficients of symmetric functions. Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 28 / 38

  62. Explicit Formulae The hypermap series can be computed explicitly when H consists of all orientable or locally orientable hypermaps. Theorem (Jackson and Visentin) When H is the set of orientable hypermaps, �� �� = t ∂ � � � H p ( x ) , p ( y ) , p ( z ); 0 ∂t ln H θ s θ ( x ) s θ ( y ) s θ ( z ) � � θ ∈ P � t =0 . Theorem (Goulden and Jackson) When H is the set of locally orientable hypermaps, �� �� = 2 t ∂ 1 � � � H p ( x ) , p ( y ) , p ( z ); 1 ∂t ln Z θ ( x ) Z θ ( y ) Z θ ( z ) � H 2 θ � � θ ∈ P t =0 . Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 29 / 38

  63. Outline Combinatorial Enumeration 1 Graphs, Maps, and Surfaces 2 Rooted Maps and Flags 3 Quantum gravity and the q -Conjecture 4 Map Enumeration 5 Orientable Maps Non-Orientable Maps Hypermaps Generating Series What does Jack have to do with it? 6 The invariants resolve a special case Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 29 / 38

  64. Parallel Problems We started with two similar problems, applied similar techniques, and found similar looking solutions. The natural question is, “Could we have solved both problems at once?” Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 30 / 38

  65. 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 Map Enumeration February 15, 2011 31 / 38

  66. 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 Map Enumeration February 15, 2011 32 / 38

  67. 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 Map Enumeration February 15, 2011 33 / 38

  68. 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 Map Enumeration February 15, 2011 34 / 38

  69. 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 Map Enumeration February 15, 2011 34 / 38

  70. 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 Map Enumeration February 15, 2011 34 / 38

  71. 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 Map Enumeration February 15, 2011 34 / 38

  72. 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 Map Enumeration February 15, 2011 34 / 38

  73. A root-edge classification There are four possible types of root edges in a map. Example Bridges Borders A handle Handles Cross-Borders Example A cross-border Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 35 / 38

  74. A root-edge classification There are four possible types of root edges in a map. Example Bridges Borders A handle Handles Cross-Borders Example A cross-border Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 35 / 38

  75. A root-edge classification Handles occur in pairs e e’ Untwisted Twisted Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 35 / 38

  76. 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 Handle Cross-Border Border Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 36 / 38

  77. 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 Handle Cross-Border Border Michael La Croix (University of Waterloo) The Jack parameter and Map Enumeration February 15, 2011 36 / 38

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