Goulden and Jackson’s b-conjecture and Matching-Jack conjecture

Goulden and Jackson’s b-conjecture and

Matching-Jack conjecture

Houcine Ben Dali

Université de Paris, CNRS, IRIF, Paris Université de Lorraine, CNRS, IECL, Nancy

86th Séminaire Lotharingien de Combinatoire

Bad Boll, September 7, 2021

Maps

Maps

A map is a graph embedded into a surface, oriented or not.

A map is oriented if it is the case of the underlying surface.

A map is bipartite if its vertices are colored in white and black, and each white vertex has only black neighbors.

Figure 1: A non-oriented bipartite map on the Klein bottle.

Maps

A bipartite map is rooted by distinguishing an oriented white corner.

Example:

Figure 1:A rooted non-oriented bipartite map on the Klein bottle.

Maps

(λ, µ, ν ) is the profile of the bipartite map M if λ is the

partition given by the face degrees divided by 2, and µ (resp.

ν) is the partition given by the degrees of the white (resp.

black) vertices.

Figure 1: A non-oriented bipartite map on the Klein bottle with profile ([9],[4,2,2,1],[4,2,2,1]).

Generating series of

oriented bipartite maps

Oriented bipartite maps

1

For every triplet (λ, µ, ν), we have the bijection Oriented (edge-) labelled

bipartite maps of profile (λ, µ, ν)

←→ couples of permutations (σ

, σ

) such that the cyclic type of σ

, σ

and σ

σ

are respectively λ, µ and ν

2

[Representation theory of the symmetric group]

X

θ

t^|θ| |θ|!

dim(θ)sθ(p)sθ(q)sθ(r) =X

n≥0

tⁿ n!

X

σ₁,σ₂∈Sn

p_type(σ1)q_type(σ2)r_type(σ1σ₂),

sθ: the Schur function associated to the partitionθ, expressed in the power-sum basis.

p:= (p_i)i≥1; q:= (q_i)i≥1; ; r:= (r_i)i≥1.

[Classical]

t∂

∂tlog X

θ

t^|θ| |θ|!

dim(θ)s_θ(p)s_θ(q)s_θ(r)

!

= X

Mconnected rooted oriented bipartite maps

t^|M|p_Λ◦(M)q_Λ•(M)r_Λ(M).

Oriented bipartite maps

1

For every triplet (λ, µ, ν), we have the bijection Oriented (edge-) labelled

bipartite maps of profile (λ, µ, ν)

←→ couples of permutations (σ

, σ

) such that the cyclic type of σ

, σ

and σ

σ

are respectively λ, µ and ν

2

[Representation theory of the symmetric group]

X

θ

t^|θ| |θ|!

dim(θ)sθ(p)sθ(q)sθ(r) =X

n≥0

tⁿ n!

X

σ₁,σ₂∈Sn

p_type(σ1)q_type(σ2)r_type(σ1σ₂),

sθ: the Schur function associated to the partitionθ, expressed in the power-sum basis.

p:= (p_i)i≥1; q:= (q_i)i≥1; ; r:= (r_i)i≥1.

[Classical]

t∂

∂tlog X

θ

t^|θ| |θ|!

dim(θ)s_θ(p)s_θ(q)s_θ(r)

!

= X

Mconnected rooted oriented bipartite maps

t^|M|p_Λ◦(M)q_Λ•(M)r_Λ(M).

Oriented bipartite maps

1

For every triplet (λ, µ, ν), we have the bijection Oriented (edge-) labelled

bipartite maps of profile (λ, µ, ν)

←→ couples of permutations (σ

, σ

) such that the cyclic type of σ

, σ

and σ

σ

are respectively λ, µ and ν

2

[Representation theory of the symmetric group]

X

θ

t^|θ| |θ|!

dim(θ)sθ(p)sθ(q)sθ(r) =X

n≥0

tⁿ n!

X

σ₁,σ₂∈Sn

p_type(σ1)q_type(σ2)r_type(σ1σ₂),

sθ: the Schur function associated to the partitionθ, expressed in the power-sum basis.

p:= (p_i)i≥1; q:= (q_i)i≥1; ; r:= (r_i)i≥1.

[Classical]

t∂

∂tlog X

θ

t^|θ| |θ|!

dim(θ)s_θ(p)s_θ(q)s_θ(r)

!

= X

Mconnected rooted oriented bipartite maps

t^|M|p_Λ◦(M)q_Λ•(M)r_Λ(M).

Generating series of

non-oriented maps

Labelled Maps

A map is labelled if it is equipped with a bijection between its edge-sides and the set A

:= {1, 1, ..., ˆ n, ˆ n}.

Example:

Figure 2:A labelled non-oriented bipartite map on the Klein bottle

Matchings

A matchingδonA_n={1,ˆ1, ...,n,ˆn}is a 1-regular graph.

Figure 3:An example of a matching onA₈.

Remark

A permutation of cycle typeλis associated to a matchingδsuch that Λ(ε, δ) =λ.

The profile of(δ₁, δ₂, δ₃)is the triplet of partitions (Λ(δ₁, δ₂),Λ(δ₁, δ₃),Λ(δ₂,Λ₃)).

Matchings

A matching is bipartite if each one of its edges is of the form(i,ˆj).

Figure 3:An example of a bipartite matching onA8.

Remark

A permutation of cycle typeλis associated to a matchingδsuch that Λ(ε, δ) =λ.

The profile of(δ₁, δ₂, δ₃)is the triplet of partitions (Λ(δ₁, δ₂),Λ(δ₁, δ₃),Λ(δ₂,Λ₃)).

Matchings

For everyn≥1, we denote byεthe bipartite matching onA_nformed by the pairs of the form(i,ˆi).

Figure 3:The matching onA .

Remark

A permutation of cycle typeλis associated to a matchingδsuch that Λ(ε, δ) =λ.

The profile of(δ₁, δ₂, δ₃)is the triplet of partitions (Λ(δ₁, δ₂),Λ(δ₁, δ₃),Λ(δ₂,Λ₃)).

Matchings

For two matchingsδandδ⁰ onA_n, we defineΛ(δ, δ⁰)as the partition given by half-sizes of the connected components of the graphδ∪δ⁰.

Once and for all, we fix for every partitionλa bipartite matchingδλsuch thatΛ(ε, δ_λ) =λ.

Figure 3:An example of the graph ofε∪δλforλ= [3,3,2]

Remark

A permutation of cycle typeλis associated to a matchingδsuch that Λ(ε, δ) =λ.

The profile of(δ₁, δ₂, δ₃)is the triplet of partitions (Λ(δ₁, δ₂),Λ(δ₁, δ₃),Λ(δ₂,Λ₃)).

Matchings

We have a bijection betweenS_nand bipartite matchings onA_n: σ7−→the matching formed by(i,σ(j)).ˆ

Example:

Remark

A permutation of cycle typeλis associated to a matchingδsuch that Λ(ε, δ) =λ.

The profile of(δ₁, δ₂, δ₃)is the triplet of partitions (Λ(δ₁, δ₂),Λ(δ₁, δ₃),Λ(δ₂,Λ₃)).

(1,2,3)(4,5,6)(7,8)7−→

Matchings

We have a bijection betweenS_nand bipartite matchings onA_n: σ7−→the matching formed by(i,σ(j)).ˆ

Example:

Remark

A permutation of cycle typeλis associated to a matchingδsuch that Λ(ε, δ) =λ.

The profile of(δ₁, δ₂, δ₃)is the triplet of partitions (Λ(δ₁, δ₂),Λ(δ₁, δ₃),Λ(δ₂,Λ₃)).

(1,2,3)(4,5,6)(7,8)7−→

Correspondence between bipartite maps and matchings

For a labelled bipartite map M we define three matchings;

δ

relating the labels of edge-sides forming a white corner.

δ

relating the labels of edge-sides forming a black corner.

δ

relating the labels of the two

sides of a same edge.

Correspondence between bipartite maps and matchings

For a labelled bipartite map M we define three matchings;

δ

relating the labels of edge-sides forming a white corner.

δ

relating the labels of edge-sides forming a black corner.

δ

relating the labels of the two sides of a same edge.

Λ(δ

, δ

) gives the face degrees.

Λ(δ

, δ

) gives the white vertices degrees.

Λ(δ

, δ

) gives the black vertices degrees.

Generating series of non-oriented maps [Goulden and Jackson ’96]

1

We obtain the following bijection : Labelled bipartite maps of

profile (λ, µ, ν)

←→ (δ

, δ

, δ

) of profile (λ, µ, ν)

2

[Representation Theory of the Gelfand pair (S

, B

)]

X

θ

t^|θ|dim(2θ)

|2θ|! Zθ(p)Zθ(q)Zθ(r) =X

n≥0

tⁿ (2n)!

X

δ₀,δ₁,δ₂

matchings onAn

pΛ(δ₀,δ₁)qΛ(δ₁,δ₂)rΛ(δ₁,δ₂),

Zθ: the zonal polynomial associated to the partitionθ, expressed in the power-sum basis.

p:= (pi)_i≥1;q:= (qi)_i≥1;r:= (ri)_i≥1.

3

2t∂

∂t log X

θ

t^|θ|dim(2θ)

|2θ|! Zθ(p)Zθ(q)Zθ(r)

!

= X

Mconnected rooted

bipartite maps

t^|M|pΛ^◦(M)qΛ^•(M)rΛ(M)

Generating series of non-oriented maps [Goulden and Jackson ’96]

1

We obtain the following bijection : Labelled bipartite maps of

profile (λ, µ, ν)

←→ (δ

, δ

, δ

) of profile (λ, µ, ν)

2

[Representation Theory of the Gelfand pair (S

, B

)]

X

θ

t^|θ|dim(2θ)

|2θ|! Zθ(p)Zθ(q)Zθ(r) =X

n≥0

tⁿ (2n)!

X

δ₀,δ₁,δ₂

matchings onAn

pΛ(δ₀,δ₁)qΛ(δ₁,δ₂)rΛ(δ₁,δ₂),

Zθ: the zonal polynomial associated to the partitionθ, expressed in the power-sum basis.

p:= (pi)_i≥1;q:= (qi)_i≥1;r:= (ri)_i≥1.

3

2t∂

∂t log X

θ

t^|θ|dim(2θ)

|2θ|! Zθ(p)Zθ(q)Zθ(r)

!

= X

Mconnected rooted

bipartite maps

t^|M|pΛ^◦(M)qΛ^•(M)rΛ(M)

Jack polynomials and a one parameter deformation of

the generating series of

bipartite maps

Jack polynomials

We consider the following deformation of the Hall scalar producth., .i_bdefined on symmetric functions by

hpλ,pµi_b=δλµzλ(1+b)^`(λ).

Definition

Jack polynomials of parameter 1+b, denotedJ_λ^(b)are defined as follows :

1 Triangularity and normalisation: ifλ`n, then

J_λ^(b)= X

µ`n,µ≤λ

uλµmµ,

such thatu_λ[1ⁿ]=n!.

(predominance orderµ≤λ:µ1+µ2+...+µi≤λ1+λ2...+λi∀i) 2 Orthogonality: ifλ6=µthenhJ_λ^(b),Jµ^(b)i_b=0.

Jack polynomials

We consider the following deformation of the Hall scalar producth., .i_bdefined on symmetric functions by

hpλ,pµi_b=δλµzλ(1+b)^`(λ).

Definition

Jack polynomials of parameter 1+b, denotedJ_λ^(b)are defined as follows :

1 Triangularity and normalisation: ifλ`n, then

J_λ^(b)= X

µ`n,µ≤λ

uλµmµ,

such thatu_λ[1ⁿ]=n!.

(predominance orderµ≤λ:µ1+µ2+...+µi≤λ1+λ2...+λi∀i)

Orthogonality: ifλ6=µthenhJ^(b),J^(b)i =0.

Jack polynomials

For b = 0 −→ Schur functions J

=

s

.

For b = 1 −→ Zonal polynomials J

= Z

. We define

τ

(t, p, q, r) := X

θ

t

j

J

(p)J

(q)J

(r),

where j

= hJ

, J

i

.

b=0

τ₀(t,p,q,r) =X

n≥0

X

λ`n

tⁿ zλ

X

δbipartite

matching onAn

pλqΛ(ε,δ)rΛ(δλ,δ).

t∂

∂t log (τ₀(t,p,q,r)) = X

Moriented rooted

connected bipartite map

t^|M|pΛ^◦(M)qΛ^•(M)rΛ(M).

b=1

τ₁(t,p,q,r) =X

n≥0

X

λ`n

tⁿ zλ2^`(λ)

X

δmatching onAn

pλqΛ(ε,δ)rΛ(δλ,δ).

2t∂

∂tlog (τ₁(t,p,q,r)) = X

Mrooted

connected bipartite map

t^|M|p_Λ^◦_(M)q_Λ^•_(M)r_Λ(M).

b=0

τ₀(t,p,q,r) =X

n≥0

X

λ`n

tⁿ zλ

X

δbipartite

matching onAn

pλqΛ(ε,δ)rΛ(δλ,δ).

t∂

∂t log (τ₀(t,p,q,r)) = X

Moriented rooted

connected bipartite map

t^|M|pΛ^◦(M)qΛ^•(M)rΛ(M).

b=1

τ₁(t,p,q,r) =X

n≥0

X

λ`n

tⁿ zλ2^`(λ)

X

δmatching onAn

pλqΛ(ε,δ)rΛ(δλ,δ).

2t∂

∂tlog (τ₁(t,p,q,r)) = X

Mrooted

connected bipartite map

t^|M|p_Λ^◦_(M)q_Λ^•_(M)r_Λ(M).

Goulden and Jackson’s conjectures ’96 Matching-Jack conjecture

τ_b(t,p,q,r) =X

n≥0

X

λ`n

tⁿ zλ(1+b)^`(λ)

X

δmatching onAn

b^ϑλ(δ)pλqΛ(ε,δ)rΛ(δ_λ,δ),

where for every partitionλ`n,ϑλa function on the matchings ofA_nwith non-negative integer values, such thatϑλ(δ) =0iffδis a bipartite matching.

b-conjecture (Hypermap-Jack conjecture)

(1+b)t∂

∂t log (τ_b(t,p,q,r)) = X

M rooted connected

bipartite map

t^|M|b^ϑ(M)pΛ^◦(M)qΛ^•(M)rΛ(M)

whereϑis a function on connected rooted maps with non-negative integer value, such thatϑ(M) =0iff M is oriented.

Goulden and Jackson’s conjectures ’96 Matching-Jack conjecture

τ_b(t,p,q,r) =X

n≥0

X

λ`n

tⁿ zλ(1+b)^`(λ)

X

δmatching onAn

b^ϑλ(δ)pλqΛ(ε,δ)rΛ(δ_λ,δ),

where for every partitionλ`n,ϑλa function on the matchings ofA_nwith non-negative integer values, such thatϑλ(δ) =0iffδis a bipartite matching.

b-conjecture (Hypermap-Jack conjecture)

(1+b)t∂

∂t log (τ_b(t,p,q,r)) = X

M rooted connected

bipartite map

t^|M|b^ϑ(M)pΛ^◦(M)qΛ^•(M)rΛ(M)

whereϑis a function on connected rooted maps with non-negative integer value, such thatϑ(M) =0iff M is oriented.

Some partial results

Theorem (Doł ˛ega-Féray ’15)

The coefficient of p

q

r

in the function τ

(t , p, q, r) multiplied by z

(1 + b)

is a polynomial in b with rational coefficients.

Theorem (Doł ˛ega-Féray ’17)

The coefficient of p

q

r

in the function (1+b)

log (τ

(t , p, q, r))

is a polynomial in b with rational coefficients.

Some partial results

Theorem (Chapuy-Doł ˛ega ’20)

(1+b)t∂

∂t log (τ_b(t,p,q,u)) = X

M rooted connected

bipartite map

t^|M|b^ϑ(M)pΛqΛ^◦(M)u^`(Λ•(M))

whereϑis a function on connected rooted maps with non-negative integer value, such thatϑ(M) =0iff M is oriented.

p := (p

, p

, p

, ...),

q := (q

, q

, q

, ...),

u := (u, u, u...).

Some partial results

Theorem (B.D. ’21, arXiv:2106.15414)

τ_b(t,p,q,u) =X

n≥0

X

λ`n

tⁿ zλ(1+b)^`(λ)

X

δmatching onAn

b^ϑλ(δ)pλqΛ(ε,δ)u^{`(Λ(δλ,δ))},

where for every partitionλ`n,ϑλ a function on the matchings ofA_n with non-negative integer values, such thatϑλ(δ) =0iffδis a bipartite matching.

p := (p

, p

, p

, ...), q := (q

, q

, q

, ...), u := (u, u, u...).

Remark

All the precedent results can be generalized to the case ofk-constellations.

Some partial results

Theorem (B.D. ’21, arXiv:2106.15414)

τ_b(t,p,q,u) =X

n≥0

X

λ`n

tⁿ zλ(1+b)^`(λ)

X

δmatching onAn

b^ϑλ(δ)pλqΛ(ε,δ)u^{`(Λ(δλ,δ))},

where for every partitionλ`n,ϑλ a function on the matchings ofA_n with non-negative integer values, such thatϑλ(δ) =0iffδis a bipartite matching.

p := (p

, p

, p

, ...), q := (q

, q

, q

, ...), u := (u, u, u...).

Remark

All the precedent results can be generalized to the case ofk-constellations.

Recall:

Theorem (Doł ˛ega-Féray ’15)

The coefficient of p

q

r

in the function τ

(t , p, q, r) multiplied by

z

(1 + b)

is a polynomial in b with rational coefficients.

Recall:

Theorem (Doł ˛ega-Féray ’15)

The coefficient of p

q

r

in the function τ

(t , p, q, r) multiplied by z

(1 + b)

is a polynomial in b with rational coefficients.

Upcoming result (joint work with Chapuy and Doł ˛ega):

Theorem

The coefficient of p

q

r

in the function τ

(t , p, q, r) multiplied by z

(1 + b)

is a polynomial in b with integer coefficients.

Proof: The integrality of the coefficients τ

(t , p, q, u) +

Farahat-Higman Algebra.