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Jack Symmetric Functions and the Non-Orientability of Rooted Maps Michael La Croix University of Waterloo January 4, 2012 Graphs, Surfaces, and Maps Example Definition A surface is a compact 2 -manifold without boundary. Definition A graph

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SLIDE 1

Jack Symmetric Functions and the Non-Orientability of Rooted Maps

Michael La Croix

University of Waterloo

January 4, 2012

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SLIDE 2

Graphs, Surfaces, and Maps

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.

Example

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 1 / 11

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SLIDE 3

Graphs, Surfaces, and Maps

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.

Example

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 1 / 11

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SLIDE 4

Graphs, Surfaces, and Maps

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.

Example

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 1 / 11

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SLIDE 5

Ribbon Graphs and Flags

Definition

The neighbourhood of the graph determines a ribbon graph, and the boundaries of ribbons determine flags.

Definition

Automorphisms permute flags, and a rooted map is a map together with a distinguished orbit of flags.

Example

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 1 / 11

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SLIDE 6

Ribbon Graphs and Flags

Definition

The neighbourhood of the graph determines a ribbon graph, and the boundaries of ribbons determine flags.

Definition

Automorphisms permute flags, and a rooted map is a map together with a distinguished orbit of flags.

Example

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 1 / 11

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SLIDE 7

Ribbon Graphs and Flags

Definition

The neighbourhood of the graph determines a ribbon graph, and the boundaries of ribbons determine flags.

Definition

Automorphisms permute flags, and a rooted map is a map together with a distinguished orbit of flags.

Example

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 1 / 11

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SLIDE 8

Ribbon Graphs and Flags

Definition

The neighbourhood of the graph determines a ribbon graph, and the boundaries of ribbons determine flags.

Definition

Automorphisms permute flags, and a rooted map is a map together with a distinguished orbit of flags.

Example

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 1 / 11

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SLIDE 9

Rooted Maps

Definition

The neighbourhood of the graph determines a ribbon graph, and the boundaries of ribbons determine flags.

Definition

Automorphisms permute flags, and a rooted map is a map together with a distinguished orbit of flags.

Example

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 1 / 11

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SLIDE 10

Rooted Maps

Definition

The neighbourhood of the graph determines a ribbon graph, and the boundaries of ribbons determine flags.

Definition

Automorphisms permute flags, and a rooted map is a map together with a distinguished orbit of flags.

Example

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 1 / 11

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SLIDE 11

Rooted Maps

Definition

The neighbourhood of the graph determines a ribbon graph, and the boundaries of ribbons determine flags.

Definition

Automorphisms permute flags, and a rooted map is a map together with a distinguished orbit of flags.

Note

The map with no edges, , has a rooting.

Example

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 1 / 11

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SLIDE 12

A Combinatorial Encoding of Maps

Equivalence classes can be encoded by perfect matchings of flags. Start with a ribbon graph.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 2 / 11

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SLIDE 13

A Combinatorial Encoding of Maps

Equivalence classes can be encoded by perfect matchings of flags. Start with a ribbon graph.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 2 / 11

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SLIDE 14

A Combinatorial Encoding of Maps

Equivalence classes can be encoded by perfect matchings of flags. Mv Me Mf Ribbon boundaries determine 3 perfect matchings of flags.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 2 / 11

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SLIDE 15

A Combinatorial Encoding of Maps

Equivalence classes can be encoded by perfect matchings of flags. Mv Me Pairs of matchings determine, faces,

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 2 / 11

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SLIDE 16

A Combinatorial Encoding of Maps

Equivalence classes can be encoded by perfect matchings of flags. Mv Mf Pairs of matchings determine, faces, edges,

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 2 / 11

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SLIDE 17

A Combinatorial Encoding of Maps

Equivalence classes can be encoded by perfect matchings of flags. Me Mf Pairs of matchings determine, faces, edges, and vertices.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 2 / 11

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SLIDE 18

A Combinatorial Encoding of Maps

1 1’ 2 2’ 3 3’ 4 4’ 5 5’ 6 6’ 7 7’ 8 8’

Mv Me Mf Mv =

  • {1, 3}, {1′, 3′}, {2, 5}, {2′, 5′}, {4, 8′}, {4′, 8}, {6, 7}, {6′, 7′}
  • Me =
  • {1, 2′}, {1′, 4}, {2, 3′}, {3, 4′}, {5, 6′}, {5′, 8}, {6, 7′}, {7, 8′}
  • Mf =
  • {1, 1′}, {2, 2′}, {3, 3′}, {4, 4′}, {5, 5′}, {6, 6′}, {7, 7′}, {8, 8′}
  • Michael La Croix (University of Waterloo)

The Jack Parameter and non-orientability January 4, 2012 2 / 11

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SLIDE 19

Hypermaps

Generalizing the combinatorial encoding, an arbitrary triple of perfect matchings determines a hypermap when the triple induces a connected graph, with cycles of Me ∪ Mf, Me ∪ Mv, and Mv ∪ Mf 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 non-orientability January 4, 2012 3 / 11

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SLIDE 20

Hypermaps

Generalizing the combinatorial encoding, an arbitrary triple of perfect matchings determines a hypermap when the triple induces a connected graph, with cycles of Me ∪ Mf, Me ∪ Mv, and Mv ∪ Mf determining vertices, hyperfaces, and hyperedges.

Example

Hypermaps both specialize and generalize maps.

Example

֒ →

Maps can be represented as hypermaps with ǫ = [2n].

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 3 / 11

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SLIDE 21

The Hypermap Series

Definition

The hypermap series for a set H of hypermaps is the combinatorial sum H(x, y, z) :=

  • h∈H

xν(h)yφ(h)zǫ(h) where ν(h), φ(h), and ǫ(h) are the vertex-, hyperface-, and hyperedge- degree partitions of h.

Example

Example

Rootings of contribute 12

  • x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2 x2 3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

  • (y3

y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5) z6

2 to the sum.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 4 / 11

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SLIDE 22

The Hypermap Series

Definition

The hypermap series for a set H of hypermaps is the combinatorial sum H(x, y, z) :=

  • h∈H

xν(h)yφ(h)zǫ(h) where ν(h), φ(h), and ǫ(h) are the vertex-, hyperface-, and hyperedge- degree partitions of h.

Example

Example

Rootings of contribute 12

  • x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2

x3

2 x2 3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

x2

3

  • (y3

y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y3 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y4 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5 y5) z6

2 to the sum.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 4 / 11

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SLIDE 23

Explicit Formulae

The hypermap series can be computed explicitly when H consists of

  • rientable hypermaps or all hypermaps.

sketch

Theorem (Jackson and Visentin - 1990)

When H is the set of orientable hypermaps,

encoding details

HO

  • p(x), p(y), p(z); 0
  • = t ∂

∂t ln

  • θ∈P

Hθsθ(x)sθ(y)sθ(z)

  • t=0.

Theorem (Goulden and Jackson - 1996)

When H is the set of all hypermaps (orientable and non-orientable), HA

  • p(x), p(y), p(z); 1
  • = 2t ∂

∂t ln

  • θ∈P

1 H2θ Zθ(x)Zθ(y)Zθ(z)

  • t=0.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 5 / 11

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SLIDE 24

Jack Symmetric Functions

Jack symmetric functions,

Definition , are a one-parameter family, denoted

by {Jθ(α)}θ, that generalizes both Schur functions and zonal polynomials.

Proposition (Stanley - 1989)

Jack symmetric functions are related to Schur functions and zonal polynomials by: Jλ(1) = Hλsλ, Jλ, Jλ1 = H2

λ,

Jλ(2) = Zλ, and Jλ, Jλ2 = H2λ, where 2λ is the partition obtained from λ by multiplying each part by two.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 6 / 11

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SLIDE 25

A Generalized Series

b -Conjecture (Goulden and Jackson - 1996)

The generalized series, H

  • p(x), p(y), p(z); b
  • := (1 + b)t ∂

∂t ln

  • θ∈P

Jθ(x; 1 + b)Jθ(y; 1 + b)Jθ(z; 1 + b) Jθ, Jθ1+b

  • t=0

=

  • n≥0
  • ν,φ,ǫ⊢n

cν,φ,ǫ(b)pν(x)pφ(y)pǫ(z), has an combinatorial interpretation involving hypermaps. In particular cν,φ,ǫ(b) =

  • h∈Hν,φ,ǫ

b β(h) for some invariant β of rooted hypermaps.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 7 / 11

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SLIDE 26

A b-Invariant

The b-Conjecture assumes that cν,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

  • f

precisely three maps are enumerated by c[4],[4],[22](b) = 1 + b + 3b2.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 8 / 11

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SLIDE 27

A b-Invariant

The b-Conjecture assumes that cν,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

  • f

precisely three maps are enumerated by c[4],[4],[22](b) = 1 + b + 3b2.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 8 / 11

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SLIDE 28

A b-Invariant

The b-Conjecture assumes that cν,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

  • f

precisely three maps are enumerated by c[4],[4],[22](b) = 1 + b + 3b2.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 8 / 11

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SLIDE 29

A b-Invariant

The b-Conjecture assumes that cν,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

There are precisely eight rooted maps enumerated by c[4,4],[3,5],[24](b) = 8b2.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 8 / 11

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SLIDE 30

A b-Invariant

The b-Conjecture assumes that cν,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

There are precisely eight rooted maps enumerated by c[4,4],[3,5],[24](b) = 8b2.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 8 / 11

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SLIDE 31

A root-edge classification

There are four possible types of root edges in a map.

Borders Bridges Handles Cross-Borders Example

A handle

Example

A cross-border

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 9 / 11

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SLIDE 32

A root-edge classification

Handles occur in pairs

e e’

Untwisted Twisted

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 9 / 11

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SLIDE 33

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

η(m) = 1 or 2 η(m) = 2 or 1 η(m) = 1 η(m) = 0

Handle Cross-Border Border

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 10 / 11

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SLIDE 34

Main result (marginal b-invariants exist)

Theorem (La Croix)

If φ partitions 2n and η is a member of the family of invariants then, dv,φ(b) :=

  • ℓ(ν)=v

cν,φ,[2n](b) =

  • m∈Mv,φ

bη(m).

Proof (sketch).

A

generating series for maps with respect η satisfies a PDE with a

unique solution. The corresponding specialization of H has an analytic presentation.

Details

An algebraic refinement to distinguish between root and non-root faces in the generating series satisfies the same PDE.

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 11 / 11

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SLIDE 35

Main result (marginal b-invariants exist)

Theorem (La Croix)

If φ partitions 2n and η is a member of the family of invariants then, dv,φ(b) :=

  • ℓ(ν)=v

cν,φ,[2n](b) =

  • m∈Mv,φ

bη(m).

Implications of the proof

dv,φ(b) is of the form

  • 0≤i≤g/2

hv,φ,ibg−2i(1 + b)i. The degree of dv,φ(b) is the genus of the maps it enumerates. The top coefficient, hv,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 non-orientability January 4, 2012 11 / 11

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SLIDE 36

The End

Thank You

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 11 / 11

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SLIDE 37

Finding a partial differential equation

Root-edge type Schematic Contribution to M Cross-border z

  • i≥0

(i + 1)bri+2 ∂ ∂ri M Border z

  • i≥0

i+1

  • j=1

rjyi−j+2 ∂ ∂ri M Handle z

  • i,j≥0

(1 + b)jri+j+2 ∂2 ∂ri∂yj M Bridge z

  • i,j≥0

ri+j+2 ∂ ∂ri M ∂ ∂rj M

  • Return

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 12 / 11

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SLIDE 38

Example

Mf Mv Me ν = [23] ǫ = [32] φ = [6]

Return Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 13 / 11

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SLIDE 39

Example

is enumerated by

  • x3

2 x2 3

  • (y3 y4 y5)
  • z6

2

  • .

ν = [23, 32] φ = [3, 4, 5] ǫ = [26]

Return Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 14 / 11

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SLIDE 40

Jack Symmetric Functions

With respect to the inner product defined by pλ(x), pµ(x)α = δλ,µ |λ|! |Cλ|αℓ(λ), Jack symmetric functions are the unique family satisfying: (P1) (Orthogonality) If λ = µ, then Jλ, Jµα = 0. (P2) (Triangularity) Jλ =

  • µλ

vλµ(α)mµ, where vλµ(α) is a rational function in α, and ‘’ denotes the natural order on partitions. (P3) (Normalization) If |λ| = n, then vλ,[1n](α) = n!.

Return Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 15 / 11

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SLIDE 41

Encoding Orientable Maps

1 Orient and label the edges. 2 This induces labels on flags. 3 Clockwise circulations at each

vertex determine ν.

4 Face circulations are the cycles

  • f ǫν.

1 2 3 4 5 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 non-orientability January 4, 2012 16 / 11

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SLIDE 42

Encoding Orientable Maps

1 Orient and label the edges. 2 This induces labels on flags. 3 Clockwise circulations at each

vertex determine ν.

4 Face circulations are the cycles

  • f ǫν.

1 2 3 4 5 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 non-orientability January 4, 2012 16 / 11

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SLIDE 43

Encoding Orientable Maps

1 Orient and label the edges. 2 This induces labels on flags. 3 Clockwise circulations at each

vertex determine ν.

4 Face circulations are the cycles

  • f ǫν.

1’ 1 2’ 2 3’ 3 4’ 4 5’ 5 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′)

Return Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 16 / 11

slide-44
SLIDE 44

Encoding Orientable Maps

1 Orient and label the edges. 2 This induces labels on flags. 3 Clockwise circulations at each

vertex determine ν.

4 Face circulations are the cycles

  • f ǫν.

1’ 1 2’ 2 3’ 3 4’ 4 5’ 5 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′)

Return Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 16 / 11

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SLIDE 45

The Map Series

An enumerative problem associated with maps is to determine the number

  • f rooted maps with specified vertex- and face- degree partitions.

Definition

The map series for a set M of rooted maps is the combinatorial sum M = M(x, y, z, r; b) :=

  • m∈M

x|V (m)|yφ(m)ρ(m)z|E(m)|rρ(m)bη(m), where the sum is taken over all rooted maps, including the map with no edges, V (m) is the vertex set of m, φ(m) is the face-degree partition of m, ρ(m) is the degree of the root face of m, and E(m) is the edge set of m.

Return Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 17 / 11

slide-46
SLIDE 46

How to enumerate maps with symmetric functions

Instead of counting rooted maps, we can count labelled hypermaps. This adds easily computable multiplicities. 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. The logarithms restrict to connected maps, and the differential

  • perators remove the decoration.

Return Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 18 / 11

slide-47
SLIDE 47

How to enumerate maps with symmetric functions

Instead of counting rooted maps, we can count labelled hypermaps. This adds easily computable multiplicities. 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. The logarithms restrict to connected maps, and the differential

  • perators remove the decoration.

Return Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 18 / 11

slide-48
SLIDE 48

How to enumerate maps with symmetric functions

Instead of counting rooted maps, we can count labelled hypermaps. This adds easily computable multiplicities. 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. The logarithms restrict to connected maps, and the differential

  • perators remove the decoration.

Return Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 18 / 11

slide-49
SLIDE 49

How to enumerate maps with symmetric functions

Instead of counting rooted maps, we can count labelled hypermaps. This adds easily computable multiplicities. 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. The logarithms restrict to connected maps, and the differential

  • perators remove the decoration.

Return Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 18 / 11

slide-50
SLIDE 50

How to enumerate maps with symmetric functions

Instead of counting rooted maps, we can count labelled hypermaps. This adds easily computable multiplicities. 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. The logarithms restrict to connected maps, and the differential

  • perators remove the decoration.

Return Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 18 / 11

slide-51
SLIDE 51

A Specialization

Definition

For a function f : RN → R, define an expectation operator · by f1+b := c1+b

  • RN |V (λ)|

2 1+b f(λ)e− 1 2(1+b) p2(λ) dλ,

with c1+b chosen such that 11+b = 1.

Theorem (Okounkov - 1997)

If N is a positive integer, 1 + b is a positive real number, and θ is an integer partition of 2n, then Jθ(λ, 1 + b)1+b = Jθ(1N, 1 + b)[p[2n]]Jθ, where 1N = (1, . . . , 1, 0, 0, . . . ) consists of N leading 1’s followed by 0’s.

Return Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 19 / 11

slide-52
SLIDE 52

A Specialization

Definition

For a function f : RN → R, define an expectation operator · by f1+b := c1+b

  • RN |V (λ)|

2 1+b f(λ)e− 1 2(1+b) p2(λ) dλ,

with c1+b chosen such that 11+b = 1. M(N, y, z, r; b) = r0N + (1 + b)

  • j≥1

rj ∂ ∂yj ln

  • e

1 1+b

  • k≥1

1 k ykpk(λ)√zk

1+b

Return Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 19 / 11

slide-53
SLIDE 53

b is ubiquitous

The many lives of b

b = 0 b = 1 Hypermaps Orientable ? Locally Orientable Symmetric Functions sθ Jθ(b) Zθ Matrix Integrals Hermitian ? Real Symmetric Moduli Spaces

  • ver C

?

  • ver R

Matching Systems Bipartite ? All

Michael La Croix (University of Waterloo) The Jack Parameter and non-orientability January 4, 2012 20 / 11