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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

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シェア "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"

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(1)

The Compositional Delta Conjecture

Alessandro Iraci

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

(2)

Diagonal Harmonics

LetR:=C[x1, . . . , xn, y1, . . . , yn]on whichSn acts diagonally:

σf(x1, . . . , xn, y1, . . . , yn) :=f(xσ1, . . . , xσn, yσ1, . . . , yσn)

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

DHn:=R/I

R is bi-graded,I is homogeneous and invariant so DHn is a bi-graded Sn-module.

(3)

Diagonal Harmonics

LetR:=C[x1, . . . , xn, y1, . . . , yn]on whichSn acts diagonally:

σf(x1, . . . , xn, y1, . . . , yn) :=f(xσ1, . . . , xσn, yσ1, . . . , yσn) LetI be the ideal generated by the constant free invariants of this action.

DHn:=R/I

R is bi-graded,I is homogeneous and invariant so DHn is a bi-graded Sn-module.

(4)

Diagonal Harmonics

LetR:=C[x1, . . . , xn, y1, . . . , yn]on whichSn acts diagonally:

σf(x1, . . . , xn, y1, . . . , yn) :=f(xσ1, . . . , xσn, yσ1, . . . , yσn) LetI be the ideal generated by the constant free invariants of this action.

DHn:=R/I

R is bi-graded,I is homogeneous and invariant so DHn is a bi-graded Sn-module.

(5)

Diagonal Harmonics

LetR:=C[x1, . . . , xn, y1, . . . , yn]on whichSn acts diagonally:

σf(x1, . . . , xn, y1, . . . , yn) :=f(xσ1, . . . , xσn, yσ1, . . . , yσn) LetI be the ideal generated by the constant free invariants of this action.

DHn:=R/I

R is bi-graded,I is homogeneous and invariant so DHn is a bi-graded Sn-module.

We have

(6)

Diagonal Harmonics

LetR:=C[x1, . . . , xn, y1, . . . , yn]on whichSn acts diagonally:

σf(x1, . . . , xn, y1, . . . , yn) :=f(xσ1, . . . , xσn, yσ1, . . . , yσn) LetI be the ideal generated by the constant free invariants of this action.

DHn:=R/I

R is bi-graded,I is homogeneous and invariant so DHn is a bi-graded Sn-module.

We dene its bi-graded Frobenius characteristic Fq,t(DHn) := X

V⊆DHn irreducible

qdegx(V)tdegy(V)sλ(V) ∈ΛQ(q,t)

(7)

The nabla operator

TheMacdonald polynomials,{Hµ}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 ofDHn (Garsia-Haiman).

Denition

Thenabla operator is the linear operator dened by

∇: ΛQ(q,t)→ΛQ(q,t) Hµ7→e|µ|[Bµ]Hµ Theorem (Haiman)

The bi-graded Frobenius characteristic ofDHn is ∇en.

(8)

The nabla operator

TheMacdonald polynomials,{Hµ}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 ofDHn (Garsia-Haiman).

Denition

Thenabla operator is the linear operator dened by

∇: ΛQ(q,t)→ΛQ(q,t) Hµ7→e|µ|[Bµ]Hµ Theorem (Haiman)

The bi-graded Frobenius characteristic ofDHn is ∇en.

(9)

The nabla operator

TheMacdonald polynomials,{Hµ}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 ofDHn (Garsia-Haiman).

Denition

Thenabla operator is the linear operator dened by

∇: ΛQ(q,t)→ΛQ(q,t) Hµ7→e|µ|[Bµ]Hµ Theorem (Haiman)

The bi-graded Frobenius characteristic ofDHn is ∇en.

(10)

The nabla operator

TheMacdonald polynomials,{Hµ}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 ofDHn (Garsia-Haiman).

Denition

Thenabla operator is the linear operator dened by

∇: ΛQ(q,t)→ΛQ(q,t) Hµ7→e|µ|[Bµ]Hµ Theorem (Haiman)

The bi-graded Frobenius characteristic ofDHn is ∇en.

(11)

The nabla operator

TheMacdonald polynomials,{Hµ}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 ofDHn (Garsia-Haiman).

Denition

Thenabla operator is the linear operator dened by

∇: ΛQ(q,t)→ΛQ(q,t) Hµ7→e|µ|[Bµ]Hµ

Theorem (Haiman)

The bi-graded Frobenius characteristic ofDHn is ∇en.

(12)

The nabla operator

TheMacdonald polynomials,{Hµ}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 ofDHn (Garsia-Haiman).

Denition

Thenabla operator is the linear operator dened by

∇: ΛQ(q,t)→ΛQ(q,t) Hµ7→e|µ|[Bµ]Hµ Theorem (Haiman)

The bi-graded Frobenius characteristic ofDHn is∇en.

(13)

The Shue Theorem

Theorem (Carlsson-Mellit)

∇en= X

D∈LD(n)

qdinv(D)tarea(D)xD

2 4 5

1 3

3 6

1

(14)

The Shue Theorem

Theorem (Carlsson-Mellit)

∇en= X

D∈LD(n)

qdinv(D)tarea(D)xD

2 4 5

1 3

3 6

1

LD(n): labelled Dyck paths of size n

(15)

The Shue Theorem

Theorem (Carlsson-Mellit)

∇en= X

D∈LD(n)

qdinv(D)tarea(D)xD

5 1 3

3 6

1

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

(16)

The Shue Theorem

Theorem (Carlsson-Mellit)

∇en= X

D∈LD(n)

qdinv(D)tarea(D)xD

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)

(17)

The Shue Theorem

Theorem (Carlsson-Mellit)

∇en= X

D∈LD(n)

qdinv(D)tarea(D)xD

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

(18)

The Shue Theorem

Theorem (Carlsson-Mellit)

∇en= X

D∈LD(n)

qdinv(D)tarea(D)xD

2 4 5

1 3

3 6

1

xD :=

n

Y

i=1

xli(D)

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

x21x2x23x4x5x6

(19)

The Compositional Shue Theorem

Theorem (Carlsson-Mellit)

∇Cα= X

D∈LD(α)

qdinv(D)tarea(D)xD

5 1 3

3 6

1 We have

X

αn

Cα =en

LD(α): labelled Dyck paths withdiagonal compositionα.

(20)

Combinatorial recursion

Theorem (Zabrocki) SetDq,t(α) = P

D∈D(α)

qdinv(D)tarea(D). We have the recursion

Dq,t(a, α) =ta−1 X

βa−1

q`(α)Dq,t(α, β)

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

(21)

Combinatorial recursion

Theorem (Zabrocki) SetDq,t(α) = P

D∈D(α)

qdinv(D)tarea(D). We have the recursion

Dq,t(a, α) =ta−1 X

βa−1

q`(α)Dq,t(α, β)

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

(22)

Combinatorial recursion

Theorem (Zabrocki) SetDq,t(α) = P

D∈D(α)

qdinv(D)tarea(D). We have the recursion

Dq,t(a, α) =ta−1 X

βa−1

q`(α)Dq,t(α, β)

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

(23)

Combinatorial recursion

Theorem (Zabrocki) SetDq,t(α) = P

D∈D(α)

qdinv(D)tarea(D). We have the recursion

Dq,t(a, α) =ta−1 X

βa−1

q`(α)Dq,t(α, β)

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

(24)

Combinatorial recursion

Theorem (Zabrocki) SetDq,t(α) = P

D∈D(α)

qdinv(D)tarea(D). We have the recursion

Dq,t(a, α) =ta−1 X

βa−1

q`(α)Dq,t(α, β)

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

(25)

Combinatorial recursion

Theorem (Zabrocki) SetDq,t(α) = P

D∈D(α)

qdinv(D)tarea(D). We have the recursion

Dq,t(a, α) =ta−1 X

βa−1

q`(α)Dq,t(α, β)

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

(26)

Super-diagonal coinvariants and ∆

f

LetR:=C[x1, . . . , xn, y1, . . . , yn, θ1, . . . , θn]whereθiθj =−θjθi.

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

Mn:=R/I

R is bi-graded,I is homogeneous and invariant so Mnis a bi-graded Sn-module.

Denition

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

f: ΛQ(q,t)→ΛQ(q,t):Hµ7→f[Bµ]Hµ

0f: ΛQ(q,t)→ΛQ(q,t):Hµ7→f[Bµ−1]Hµ Conjecture (Zabrocki)

The bi-graded Frobenius characteristic of the submodule of Mn in θ-degreek is∆0en−k−1en.

(27)

Super-diagonal coinvariants and ∆

f

LetR:=C[x1, . . . , xn, y1, . . . , yn, θ1, . . . , θn]whereθiθj =−θjθi. Consider again the diagonal action of Sn onR and letI be the ideal generated by the constant free invariants of this action.

Mn:=R/I

R is bi-graded,I is homogeneous and invariant so Mnis a bi-graded Sn-module.

Denition

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

f: ΛQ(q,t)→ΛQ(q,t):Hµ7→f[Bµ]Hµ

0f: ΛQ(q,t)→ΛQ(q,t):Hµ7→f[Bµ−1]Hµ Conjecture (Zabrocki)

The bi-graded Frobenius characteristic of the submodule of Mn in θ-degreek is∆0en−k−1en.

(28)

Super-diagonal coinvariants and ∆

f

LetR:=C[x1, . . . , xn, y1, . . . , yn, θ1, . . . , θn]whereθiθj =−θjθi. Consider again the diagonal action of Sn onR and letI be the ideal generated by the constant free invariants of this action.

Mn:=R/I

R is bi-graded,I is homogeneous and invariant so Mnis a bi-graded Sn-module.

Denition

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

f: ΛQ(q,t)→ΛQ(q,t):Hµ7→f[Bµ]Hµ

0f: ΛQ(q,t)→ΛQ(q,t):Hµ7→f[Bµ−1]Hµ

Conjecture (Zabrocki)

The bi-graded Frobenius characteristic of the submodule of Mn in θ-degreek is∆0en−k−1en.

(29)

Super-diagonal coinvariants and ∆

f

LetR:=C[x1, . . . , xn, y1, . . . , yn, θ1, . . . , θn]whereθiθj =−θjθi. Consider again the diagonal action of Sn onR and letI be the ideal generated by the constant free invariants of this action.

Mn:=R/I

R is bi-graded,I is homogeneous and invariant so Mnis a bi-graded Sn-module.

Denition

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

f: ΛQ(q,t)→ΛQ(q,t):Hµ7→f[Bµ]Hµ

0f: ΛQ(q,t)→ΛQ(q,t):Hµ7→f[Bµ−1]Hµ

(30)

The Delta Conjecture

Conjecture (Haglund-Remmel-Wilson)

0en−k−1en= X

D∈LD(n)∗k

qdinv(D)tarea(D)xD

(31)

The Delta Conjecture

Conjecture (Haglund-Remmel-Wilson)

0en−k−1en= X

D∈LD(n)∗k

qdinv(D)tarea(D)xD

4 5

1 3

3 6

1

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

(32)

The Delta Conjecture

Conjecture (Haglund-Remmel-Wilson)

0en−k−1en= X

D∈LD(n)∗k

qdinv(D)tarea(D)xD

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.

(33)

The Delta Conjecture

Conjecture (Haglund-Remmel-Wilson)

0en−k−1en= X

D∈LD(n)∗k

qdinv(D)tarea(D)xD

4 5

1 3

3 6

1 3

5 1

Dinv,xD: same as for the undecorated case

(34)

The Delta Conjecture

Conjecture (Haglund-Remmel-Wilson)

0en−k−1en= X

D∈LD(n)∗k

qdinv(D)tarea(D)xD

Denition

Forf ∈ΛQ(q,t) we dene the following operators onΛQ(q,t)

Π:=X

i∈N

(−1)iei Θf :=Πf

X (1−q)(1−t)

Π−1

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

(35)

The Compositional Delta Conjecture

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

D∈LD(α)∗k

qdinv(D)tarea(D)xD

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)

(36)

(37)

Combinatorial recursion

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

Dq,t(a, α)∗k =ta−1 X

βa−1

q`(α)Dq,t(α, β)∗k

+ta−1X

βa

q`(α)Dq,t(α, β)∗k−1

with initial conditions Dq,t(∅)∗kk,0 and Dq,t(α)∗0 =Dq,t(α).

(38)

Towards a Theta Conjecture

I Algebra side: diagonal coinvariants of

R:=C[x1, . . . , xn, y1, . . . , yn, θ1, . . . , θn, η1, . . . , ηn]

whereθiθj =−θjθi andηiηj =−ηjηi inθ-degreek andη-degreel. I Symmetric function side: ΘekΘel∇en−k−l.

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

(39)

(40)

t3 t2

t qt q2t 1 q q2 q3

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

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(The Elliott-Halberstam conjecture does allow one to take B = 2 in (1.39), and therefore leads to small improve- ments in Huxley’s results, which for r ≥ 2 are weaker than the result

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のようにすべきだと考えていますか。 やっと開通します。長野、太田地区方面

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