S -module. isbi-graded, I ishomogeneousandinvariantso DH isabi-graded DH R/IR := Let I betheidealgeneratedbytheconstantfreeinvariantsofthisaction. σf ( x ,...,x ,y ,...,y ):= f ( x ,...,x ,y ,...,y ) Let R C [ x ,...,x ,y ,...,y ] onwhich S acts diagonall

∗

The Compositional Delta Conjecture

Alessandro Iraci

joint work with Michele D’Adderio and Anna Vanden Wyngaerd 2 september, 2019

Diagonal Harmonics

LetR:=C[x₁, . . . , x_n, y₁, . . . , y_n]on whichS_n acts diagonally:

σf(x₁, . . . , x_n, y₁, . . . , y_n) :=f(x_σ1, . . . , x_σn, y_σ1, . . . , y_σn)

LetI be the ideal generated by the constant free invariants of this action.

DH_n:=R/I

R is bi-graded,I is homogeneous and invariant so DH_n is a bi-graded S_n-module.

Diagonal Harmonics

LetR:=C[x₁, . . . , x_n, y₁, . . . , y_n]on whichS_n acts diagonally:

σf(x₁, . . . , x_n, y₁, . . . , y_n) :=f(x_σ1, . . . , x_σn, y_σ1, . . . , y_σn) LetI be the ideal generated by the constant free invariants of this action.

DH_n:=R/I

R is bi-graded,I is homogeneous and invariant so DH_n is a bi-graded S_n-module.

Diagonal Harmonics

LetR:=C[x₁, . . . , x_n, y₁, . . . , y_n]on whichS_n acts diagonally:

σf(x₁, . . . , x_n, y₁, . . . , y_n) :=f(x_σ1, . . . , x_σn, y_σ1, . . . , y_σn) LetI be the ideal generated by the constant free invariants of this action.

DH_n:=R/I

R is bi-graded,I is homogeneous and invariant so DH_n is a bi-graded S_n-module.

Diagonal Harmonics

LetR:=C[x1, . . . , x_n, y₁, . . . , y_n]on whichS_n acts diagonally:

σf(x₁, . . . , x_n, y₁, . . . , y_n) :=f(x_σ1, . . . , x_σn, y_σ1, . . . , y_σn) LetI be the ideal generated by the constant free invariants of this action.

DH_n:=R/I

R is bi-graded,I is homogeneous and invariant so DH_n is a bi-graded S_n-module.

We have

Diagonal Harmonics

LetR:=C[x1, . . . , x_n, y₁, . . . , y_n]on whichS_n acts diagonally:

σf(x₁, . . . , x_n, y₁, . . . , y_n) :=f(x_σ1, . . . , x_σn, y_σ1, . . . , y_σn) LetI be the ideal generated by the constant free invariants of this action.

DH_n:=R/I

R is bi-graded,I is homogeneous and invariant so DH_n is a bi-graded S_n-module.

We dene its bi-graded Frobenius characteristic F_q,t(DH_n) := X

V⊆DH_n irreducible

q^degx(V)t^degy(V)s_λ(V) ∈Λ_Q(q,t)

The nabla operator

TheMacdonald polynomials,{H_µ}_µ`d form a basis of the ringΛ^(d)_Q(q,t)

I Generalisation of Schur, Hall-Littlewood, Jack . . . I Schur positive.

I Bi-graded Frobenius characteristic of submodules ofDH_n (Garsia-Haiman).

Denition

Thenabla operator is the linear operator dened by

∇: Λ_Q(q,t)→Λ_Q(q,t) H_µ7→e_|µ|[B_µ]H_µ Theorem (Haiman)

The bi-graded Frobenius characteristic ofDH_n is ∇e_n.

The nabla operator

TheMacdonald polynomials,{H_µ}_µ`d form a basis of the ringΛ^(d)_Q(q,t) I Generalisation of Schur, Hall-Littlewood, Jack . . .

I Schur positive.

I Bi-graded Frobenius characteristic of submodules ofDH_n (Garsia-Haiman).

Denition

Thenabla operator is the linear operator dened by

∇: Λ_Q(q,t)→Λ_Q(q,t) H_µ7→e_|µ|[B_µ]H_µ Theorem (Haiman)

The bi-graded Frobenius characteristic ofDH_n is ∇e_n.

The nabla operator

TheMacdonald polynomials,{H_µ}_µ`d form a basis of the ringΛ^(d)_Q(q,t) I Generalisation of Schur, Hall-Littlewood, Jack . . .

I Schur positive.

I Bi-graded Frobenius characteristic of submodules ofDH_n (Garsia-Haiman).

Denition

Thenabla operator is the linear operator dened by

∇: Λ_Q(q,t)→Λ_Q(q,t) H_µ7→e_|µ|[B_µ]H_µ Theorem (Haiman)

The bi-graded Frobenius characteristic ofDH_n is ∇e_n.

The nabla operator

TheMacdonald polynomials,{H_µ}_µ`d form a basis of the ringΛ^(d)_Q(q,t) I Generalisation of Schur, Hall-Littlewood, Jack . . .

I Schur positive.

I Bi-graded Frobenius characteristic of submodules ofDH_n (Garsia-Haiman).

Denition

Thenabla operator is the linear operator dened by

∇: Λ_Q(q,t)→Λ_Q(q,t) H_µ7→e_|µ|[B_µ]H_µ Theorem (Haiman)

The bi-graded Frobenius characteristic ofDH_n is ∇e_n.

The nabla operator

TheMacdonald polynomials,{H_µ}_µ`d form a basis of the ringΛ^(d)_Q(q,t) I Generalisation of Schur, Hall-Littlewood, Jack . . .

I Schur positive.

I Bi-graded Frobenius characteristic of submodules ofDH_n (Garsia-Haiman).

Denition

Thenabla operator is the linear operator dened by

∇: Λ_Q(q,t)→Λ_Q(q,t) H_µ7→e_|µ|[B_µ]H_µ

Theorem (Haiman)

The bi-graded Frobenius characteristic ofDH_n is ∇e_n.

The nabla operator

TheMacdonald polynomials,{H_µ}_µ`d form a basis of the ringΛ^(d)_Q(q,t) I Generalisation of Schur, Hall-Littlewood, Jack . . .

I Schur positive.

I Bi-graded Frobenius characteristic of submodules ofDH_n (Garsia-Haiman).

Denition

Thenabla operator is the linear operator dened by

∇: Λ_Q(q,t)→Λ_Q(q,t) H_µ7→e_|µ|[B_µ]H_µ Theorem (Haiman)

The bi-graded Frobenius characteristic ofDH_n is∇e_n.

The Shue Theorem

Theorem (Carlsson-Mellit)

∇e_n= X

D∈LD(n)

q^dinv(D)t^area(D)x^D

2 4 5

1 3

3 6

1

The Shue Theorem

Theorem (Carlsson-Mellit)

∇e_n= X

D∈LD(n)

q^dinv(D)t^area(D)x^D

2 4 5

1 3

3 6

1

LD(n): labelled Dyck paths of size n

The Shue Theorem

Theorem (Carlsson-Mellit)

∇e_n= X

D∈LD(n)

q^dinv(D)t^area(D)x^D

5 1 3

3 6

1

Area: number of whole squares between the path andy =x.

The Shue Theorem

Theorem (Carlsson-Mellit)

∇e_n= X

D∈LD(n)

q^dinv(D)t^area(D)x^D

2 4 5

1 3

3 6

1

2

3

Dinv: count the number of pairs I same diagonal,

lower label < upper label (primary dinv)

I lower step one diagonal above upper steplower label > upper label

(secondary dinv)

The Shue Theorem

Theorem (Carlsson-Mellit)

∇e_n= X

D∈LD(n)

q^dinv(D)t^area(D)x^D

5 1 3

3 6

1

5 1

Dinv: count the number of pairs I same diagonal,

lower label < upper label (primary dinv)

I lower step one diagonal above upper step

The Shue Theorem

Theorem (Carlsson-Mellit)

∇e_n= X

D∈LD(n)

q^dinv(D)t^area(D)x^D

2 4 5

1 3

3 6

1

x^D :=

n

Y

i=1

x_li(D)

wherel_i(D) is the label of thei-th vertical step ofD.

x²₁x₂x²₃x₄x₅x₆

The Compositional Shue Theorem

Theorem (Carlsson-Mellit)

∇C_α= X

D∈LD(α)

q^dinv(D)t^area(D)x^D

5 1 3

3 6

1 We have

X

αn

Cα =en

LD(α): labelled Dyck paths withdiagonal compositionα.

Combinatorial recursion

Theorem (Zabrocki) SetD_q,t(α) = P

D∈D(α)

q^dinv(D)t^area(D). We have the recursion

Dq,t(a, α) =t^a−1 X

βa−1

q^`(α)Dq,t(α, β)

for a >0, with initial conditionsDq,t(∅) = 1.

Combinatorial recursion

Theorem (Zabrocki) SetD_q,t(α) = P

D∈D(α)

q^dinv(D)t^area(D). We have the recursion

Dq,t(a, α) =t^a−1 X

βa−1

q^`(α)Dq,t(α, β)

for a >0, with initial conditionsDq,t(∅) = 1.

Combinatorial recursion

Theorem (Zabrocki) SetD_q,t(α) = P

D∈D(α)

q^dinv(D)t^area(D). We have the recursion

Dq,t(a, α) =t^a−1 X

βa−1

q^`(α)Dq,t(α, β)

for a >0, with initial conditionsDq,t(∅) = 1.

Combinatorial recursion

Theorem (Zabrocki) SetD_q,t(α) = P

D∈D(α)

q^dinv(D)t^area(D). We have the recursion

Dq,t(a, α) =t^a−1 X

βa−1

q^`(α)Dq,t(α, β)

for a >0, with initial conditionsDq,t(∅) = 1.

Combinatorial recursion

Theorem (Zabrocki) SetD_q,t(α) = P

D∈D(α)

q^dinv(D)t^area(D). We have the recursion

Dq,t(a, α) =t^a−1 X

βa−1

q^`(α)Dq,t(α, β)

for a >0, with initial conditionsDq,t(∅) = 1.

Combinatorial recursion

Theorem (Zabrocki) SetD_q,t(α) = P

D∈D(α)

q^dinv(D)t^area(D). We have the recursion

Dq,t(a, α) =t^a−1 X

βa−1

q^`(α)Dq,t(α, β)

for a >0, with initial conditionsDq,t(∅) = 1.

Super-diagonal coinvariants and ∆

LetR:=C[x₁, . . . , x_n, y₁, . . . , y_n, θ₁, . . . , θ_n]whereθ_iθ_j =−θ_jθ_i.

Consider again the diagonal action of Sn onR and letI be the ideal generated by the constant free invariants of this action.

M_n:=R/I

R is bi-graded,I is homogeneous and invariant so M_nis a bi-graded S_n-module.

Denition

Forf ∈Λ_Q(q,t) we dene theDelta operatorsas

∆f: Λ_Q(q,t)→Λ_Q(q,t):Hµ7→f[Bµ]Hµ

∆⁰_f: Λ_Q(q,t)→Λ_Q(q,t):H_µ7→f[B_µ−1]H_µ Conjecture (Zabrocki)

The bi-graded Frobenius characteristic of the submodule of M_n in θ-degreek is∆⁰_en−k−1e_n.

Super-diagonal coinvariants and ∆

LetR:=C[x₁, . . . , x_n, y₁, . . . , y_n, θ₁, . . . , θ_n]whereθ_iθ_j =−θ_jθ_i. Consider again the diagonal action of Sn onR and letI be the ideal generated by the constant free invariants of this action.

M_n:=R/I

R is bi-graded,I is homogeneous and invariant so M_nis a bi-graded S_n-module.

Denition

Forf ∈Λ_Q(q,t) we dene theDelta operatorsas

∆f: Λ_Q(q,t)→Λ_Q(q,t):Hµ7→f[Bµ]Hµ

∆⁰_f: Λ_Q(q,t)→Λ_Q(q,t):H_µ7→f[B_µ−1]H_µ Conjecture (Zabrocki)

The bi-graded Frobenius characteristic of the submodule of M_n in θ-degreek is∆⁰_en−k−1e_n.

Super-diagonal coinvariants and ∆

LetR:=C[x₁, . . . , x_n, y₁, . . . , y_n, θ₁, . . . , θ_n]whereθ_iθ_j =−θ_jθ_i. Consider again the diagonal action of Sn onR and letI be the ideal generated by the constant free invariants of this action.

M_n:=R/I

R is bi-graded,I is homogeneous and invariant so M_nis a bi-graded S_n-module.

Denition

Forf ∈Λ_Q(q,t) we dene theDelta operatorsas

∆_f: Λ_Q(q,t)→Λ_Q(q,t):Hµ7→f[Bµ]Hµ

∆⁰_f: Λ_Q(q,t)→Λ_Q(q,t):H_µ7→f[B_µ−1]H_µ

Conjecture (Zabrocki)

The bi-graded Frobenius characteristic of the submodule of M_n in θ-degreek is∆⁰_en−k−1e_n.

Super-diagonal coinvariants and ∆

LetR:=C[x₁, . . . , x_n, y₁, . . . , y_n, θ₁, . . . , θ_n]whereθ_iθ_j =−θ_jθ_i. Consider again the diagonal action of Sn onR and letI be the ideal generated by the constant free invariants of this action.

M_n:=R/I

R is bi-graded,I is homogeneous and invariant so M_nis a bi-graded S_n-module.

Denition

Forf ∈Λ_Q(q,t) we dene theDelta operatorsas

∆_f: Λ_Q(q,t)→Λ_Q(q,t):Hµ7→f[Bµ]Hµ

∆⁰_f: Λ_Q(q,t)→Λ_Q(q,t):H_µ7→f[B_µ−1]H_µ

The Delta Conjecture

Conjecture (Haglund-Remmel-Wilson)

∆⁰_en−k−1e_n= X

D∈LD(n)^∗k

q^dinv(D)t^area(D)x^D

The Delta Conjecture

Conjecture (Haglund-Remmel-Wilson)

∆⁰_en−k−1en= X

D∈LD(n)^∗k

q^dinv(D)t^area(D)x^D

∗

∗

4 5

1 3

3 6

1

LD(n)^∗k: labelled Dyck paths of size n withkdecoratedrises

The Delta Conjecture

Conjecture (Haglund-Remmel-Wilson)

∆⁰_en−k−1en= X

D∈LD(n)^∗k

q^dinv(D)t^area(D)x^D

∗

∗

2 4 5

1 3

3 6

1

Area: number of whole squares between the path andy =x and in rows not containing decorated rises.

The Delta Conjecture

Conjecture (Haglund-Remmel-Wilson)

∆⁰_en−k−1en= X

D∈LD(n)^∗k

q^dinv(D)t^area(D)x^D

∗

∗

4 5

1 3

3 6

1 3

5 1

Dinv,x^D: same as for the undecorated case

The Delta Conjecture

Conjecture (Haglund-Remmel-Wilson)

∆⁰_en−k−1e_n= X

D∈LD(n)^∗k

q^dinv(D)t^area(D)x^D

Denition

Forf ∈Λ_Q(q,t) we dene the following operators onΛ_Q(q,t)

Π:=X

i∈N

(−1)ⁱ∆_ei Θ_f :=Πf

X (1−q)(1−t)

Π⁻¹

Theorem (D'Adderio-Iraci-Vanden Wyngaerd) Θek∇e_n−k = ∆⁰_en−k−1en

The Compositional Delta Conjecture

Conjecture (D'Adderio-Iraci-Vanden Wyngaerd) Θek∇C_α = X

D∈LD(α)^∗k

q^dinv(D)t^area(D)x^D

∗

∗

4 5

1 3

3 6

1 LD(α)^∗k: labelled decorated Dyck paths withdiagonal compositionα. Rows containing decorated rises do not count.

α = (4,2)

Combinatorial recursion

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∗

∗

∗

∗

∗

Combinatorial recursion

Theorem (D'Adderio-Iraci-Vanden Wyngaerd) We have the combinatorial recursion, fora, k >0,

Dq,t(a, α)^∗k =t^a−1 X

βa−1

q^`(α)Dq,t(α, β)^∗k

+t^a−1X

βa

q^`(α)D_q,t(α, β)^∗k−1

with initial conditions D_q,t(∅)^∗k=δ_k,0 and D_q,t(α)^∗0 =D_q,t(α).

Towards a Theta Conjecture

I Algebra side: diagonal coinvariants of

R:=C[x1, . . . , x_n, y₁, . . . , y_n, θ₁, . . . , θ_n, η₁, . . . , η_n]

whereθ_iθ_j =−θ_jθ_i andη_iη_j =−η_jη_i inθ-degreek andη-degreel. I Symmetric function side: Θ_ekΘ_el∇e_n−k−l.

I Combinatorial side: labelled Dyck paths with decorated rises and valleys. Statistics?

Thank you for your attention!

t³ t²

t qt q²t 1 q q² q³

Forf ∈Λ_Q(q,t) we setf[Bµ]to be f evaluated in the content of this picture