A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna...

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A new q , t -square Anna Vanden Wyngaerd Universit ´ e Libre de Bruxelles [email protected] September 11, 2017

Transcript of A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna...

Page 1: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

A new q, t-squareAnna Vanden Wyngaerd

Universite Libre de [email protected]

September 11, 2017

Page 2: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Background

The old q, t-square

A new q, t-square

Decorated Dyck paths

Page 3: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Background

Page 4: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

MacDonald PolynomialsΛC(q,t) := C(q, t)[X1, ...,XN ]SN =

⊕∞i=1 Λn

C(q,t)

I When n ≥ N, basis of ΛnC(q,t) include elementary eλ,

homogeneous hλ, power pλ and Shur sλ symmetricfunctions.

I {Hλ | λ ` n} (modified, Garsia & Haiman) MacdonaldPolynomials: basis of ΛC(q,t)

I Applications in wide variety of subjectsI ”Generalisation” of Hall-Littlewood, Jack polynomials,...I Kostka-Macdonald coefficients

Hµ =∑λ`n

Kλµ(q, t)sλ

Macdonald Positivity Conjecture

Kλµ(q, t) ∈ N[q, t ], i.e. the Macdonald polynomials are Shurpositive

Page 5: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

MacDonald PolynomialsΛC(q,t) := C(q, t)[X1, ...,XN ]SN =

⊕∞i=1 Λn

C(q,t)

I When n ≥ N, basis of ΛnC(q,t) include elementary eλ,

homogeneous hλ, power pλ and Shur sλ symmetricfunctions.

I {Hλ | λ ` n} (modified, Garsia & Haiman) MacdonaldPolynomials: basis of ΛC(q,t)

I Applications in wide variety of subjectsI ”Generalisation” of Hall-Littlewood, Jack polynomials,...I Kostka-Macdonald coefficients

Hµ =∑λ`n

Kλµ(q, t)sλ

Macdonald Positivity Conjecture

Kλµ(q, t) ∈ N[q, t ], i.e. the Macdonald polynomials are Shurpositive

Page 6: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

MacDonald PolynomialsΛC(q,t) := C(q, t)[X1, ...,XN ]SN =

⊕∞i=1 Λn

C(q,t)

I When n ≥ N, basis of ΛnC(q,t) include elementary eλ,

homogeneous hλ, power pλ and Shur sλ symmetricfunctions.

I {Hλ | λ ` n} (modified, Garsia & Haiman) MacdonaldPolynomials: basis of ΛC(q,t)

I Applications in wide variety of subjectsI ”Generalisation” of Hall-Littlewood, Jack polynomials,...I Kostka-Macdonald coefficients

Hµ =∑λ`n

Kλµ(q, t)sλ

Macdonald Positivity Conjecture

Kλµ(q, t) ∈ N[q, t ], i.e. the Macdonald polynomials are Shurpositive

Page 7: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

MacDonald PolynomialsΛC(q,t) := C(q, t)[X1, ...,XN ]SN =

⊕∞i=1 Λn

C(q,t)

I When n ≥ N, basis of ΛnC(q,t) include elementary eλ,

homogeneous hλ, power pλ and Shur sλ symmetricfunctions.

I {Hλ | λ ` n} (modified, Garsia & Haiman) MacdonaldPolynomials: basis of ΛC(q,t)

I Applications in wide variety of subjects

I ”Generalisation” of Hall-Littlewood, Jack polynomials,...I Kostka-Macdonald coefficients

Hµ =∑λ`n

Kλµ(q, t)sλ

Macdonald Positivity Conjecture

Kλµ(q, t) ∈ N[q, t ], i.e. the Macdonald polynomials are Shurpositive

Page 8: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

MacDonald PolynomialsΛC(q,t) := C(q, t)[X1, ...,XN ]SN =

⊕∞i=1 Λn

C(q,t)

I When n ≥ N, basis of ΛnC(q,t) include elementary eλ,

homogeneous hλ, power pλ and Shur sλ symmetricfunctions.

I {Hλ | λ ` n} (modified, Garsia & Haiman) MacdonaldPolynomials: basis of ΛC(q,t)

I Applications in wide variety of subjectsI ”Generalisation” of Hall-Littlewood, Jack polynomials,...

I Kostka-Macdonald coefficients

Hµ =∑λ`n

Kλµ(q, t)sλ

Macdonald Positivity Conjecture

Kλµ(q, t) ∈ N[q, t ], i.e. the Macdonald polynomials are Shurpositive

Page 9: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

MacDonald PolynomialsΛC(q,t) := C(q, t)[X1, ...,XN ]SN =

⊕∞i=1 Λn

C(q,t)

I When n ≥ N, basis of ΛnC(q,t) include elementary eλ,

homogeneous hλ, power pλ and Shur sλ symmetricfunctions.

I {Hλ | λ ` n} (modified, Garsia & Haiman) MacdonaldPolynomials: basis of ΛC(q,t)

I Applications in wide variety of subjectsI ”Generalisation” of Hall-Littlewood, Jack polynomials,...I Kostka-Macdonald coefficients

Hµ =∑λ`n

Kλµ(q, t)sλ

Macdonald Positivity Conjecture

Kλµ(q, t) ∈ N[q, t ], i.e. the Macdonald polynomials are Shurpositive

Page 10: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

MacDonald PolynomialsΛC(q,t) := C(q, t)[X1, ...,XN ]SN =

⊕∞i=1 Λn

C(q,t)

I When n ≥ N, basis of ΛnC(q,t) include elementary eλ,

homogeneous hλ, power pλ and Shur sλ symmetricfunctions.

I {Hλ | λ ` n} (modified, Garsia & Haiman) MacdonaldPolynomials: basis of ΛC(q,t)

I Applications in wide variety of subjectsI ”Generalisation” of Hall-Littlewood, Jack polynomials,...I Kostka-Macdonald coefficients

Hµ =∑λ`n

Kλµ(q, t)sλ

Macdonald Positivity Conjecture

Kλµ(q, t) ∈ N[q, t ], i.e. the Macdonald polynomials are Shurpositive

Page 11: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

n! conjecture

Strategy to prove Shur positivity of Macdonald Polynomials

I Construction, for each µ, a bi-graded module Mµ (GarsiaHaiman module), affording regular representation of Sn

I Hµ is image of the bi-graded character of this module byFrobenius characteristic map

I Garsia and Haiman reduced this to the problem of showingthat Dim(Mµ) = n!

I Proved by Haiman in 2001, using tools from AlgebraicGeometry

Page 12: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

n! conjecture

Strategy to prove Shur positivity of Macdonald PolynomialsI Construction, for each µ, a bi-graded module Mµ (Garsia

Haiman module), affording regular representation of Sn

I Hµ is image of the bi-graded character of this module byFrobenius characteristic map

I Garsia and Haiman reduced this to the problem of showingthat Dim(Mµ) = n!

I Proved by Haiman in 2001, using tools from AlgebraicGeometry

Page 13: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

n! conjecture

Strategy to prove Shur positivity of Macdonald PolynomialsI Construction, for each µ, a bi-graded module Mµ (Garsia

Haiman module), affording regular representation of Sn

I Hµ is image of the bi-graded character of this module byFrobenius characteristic map

I Garsia and Haiman reduced this to the problem of showingthat Dim(Mµ) = n!

I Proved by Haiman in 2001, using tools from AlgebraicGeometry

Page 14: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

n! conjecture

Strategy to prove Shur positivity of Macdonald PolynomialsI Construction, for each µ, a bi-graded module Mµ (Garsia

Haiman module), affording regular representation of Sn

I Hµ is image of the bi-graded character of this module byFrobenius characteristic map

I Garsia and Haiman reduced this to the problem of showingthat Dim(Mµ) = n!

I Proved by Haiman in 2001, using tools from AlgebraicGeometry

Page 15: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

n! conjecture

Strategy to prove Shur positivity of Macdonald PolynomialsI Construction, for each µ, a bi-graded module Mµ (Garsia

Haiman module), affording regular representation of Sn

I Hµ is image of the bi-graded character of this module byFrobenius characteristic map

I Garsia and Haiman reduced this to the problem of showingthat Dim(Mµ) = n!

I Proved by Haiman in 2001, using tools from AlgebraicGeometry

Page 16: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

The nabla operator

I Working on the Macdonald positivity conjecture, Garsiaand Haiman introduced the SN -module DHn of diagonalharmonics

I Conjecture:F(DHn; q, t) = ∇en

Where ∇ is a linear operator defined by

∇Hµ = TµHµ Tλ := qn(λ)tn(λ′)

(now proved using n! conjecture)I Conjecture: for each partition λ, ∇sλ is Shur positive.I Shuffle conjecture (now theorem Carlsson & Mellit 2015)

gives a combinatorial interpretation of ∇en

Page 17: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

The nabla operator

I Working on the Macdonald positivity conjecture, Garsiaand Haiman introduced the SN -module DHn of diagonalharmonics

I Conjecture:F(DHn; q, t) = ∇en

Where ∇ is a linear operator defined by

∇Hµ = TµHµ Tλ := qn(λ)tn(λ′)

(now proved using n! conjecture)

I Conjecture: for each partition λ, ∇sλ is Shur positive.I Shuffle conjecture (now theorem Carlsson & Mellit 2015)

gives a combinatorial interpretation of ∇en

Page 18: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

The nabla operator

I Working on the Macdonald positivity conjecture, Garsiaand Haiman introduced the SN -module DHn of diagonalharmonics

I Conjecture:F(DHn; q, t) = ∇en

Where ∇ is a linear operator defined by

∇Hµ = TµHµ Tλ := qn(λ)tn(λ′)

(now proved using n! conjecture)I Conjecture: for each partition λ, ∇sλ is Shur positive.

I Shuffle conjecture (now theorem Carlsson & Mellit 2015)gives a combinatorial interpretation of ∇en

Page 19: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

The nabla operator

I Working on the Macdonald positivity conjecture, Garsiaand Haiman introduced the SN -module DHn of diagonalharmonics

I Conjecture:F(DHn; q, t) = ∇en

Where ∇ is a linear operator defined by

∇Hµ = TµHµ Tλ := qn(λ)tn(λ′)

(now proved using n! conjecture)I Conjecture: for each partition λ, ∇sλ is Shur positive.I Shuffle conjecture (now theorem Carlsson & Mellit 2015)

gives a combinatorial interpretation of ∇en

Page 20: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

The delta operator and conjecture

The linear operator ∆f

I Eigenoperator of Macdonald polynomials

∆f Hµ = f [Bµ]Hµ

I Generalisation of ∇, on ΛnC(q,t) :

∆en = ∇.

I Delta conjecture (generalisation of the Shufle Conjecture)gives a combinatorial interpretation of ∆ek en and is stillopen (Haglund, Remmel & Wilson 2016)

Page 21: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

The delta operator and conjecture

The linear operator ∆f

I Eigenoperator of Macdonald polynomials

∆f Hµ = f [Bµ]Hµ

I Generalisation of ∇, on ΛnC(q,t) :

∆en = ∇.

I Delta conjecture (generalisation of the Shufle Conjecture)gives a combinatorial interpretation of ∆ek en and is stillopen (Haglund, Remmel & Wilson 2016)

Page 22: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

The delta operator and conjecture

The linear operator ∆f

I Eigenoperator of Macdonald polynomials

∆f Hµ = f [Bµ]Hµ

I Generalisation of ∇, on ΛnC(q,t) :

∆en = ∇.

I Delta conjecture (generalisation of the Shufle Conjecture)gives a combinatorial interpretation of ∆ek en and is stillopen (Haglund, Remmel & Wilson 2016)

Page 23: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

q, t-Catalan Numbers: Garsia & Haglund 2002

Combinatorial interpretation of 〈∇en,en〉 = 〈∇en, s1n〉:∑D∈Dyckn

qarea(D)tbounce(D) =∑

D∈Dyckn

qdinv(D)tarea(D)

0000122233

Page 24: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

q, t-Catalan Numbers: Garsia & Haglund 2002

Combinatorial interpretation of 〈∇en,en〉 = 〈∇en, s1n〉:∑D∈Dyckn

qarea(D)tbounce(D) =∑

D∈Dyckn

qdinv(D)tarea(D)

0000122233

Page 25: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

q, t-Catalan Numbers: Garsia & Haglund 2002

Combinatorial interpretation of 〈∇en,en〉 = 〈∇en, s1n〉:∑D∈Dyckn

qarea(D)tbounce(D) =∑

D∈Dyckn

qdinv(D)tarea(D)

0000122233

Page 26: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

q, t-Catalan Numbers: Garsia & Haglund 2002

Combinatorial interpretation of 〈∇en,en〉 = 〈∇en, s1n〉:∑D∈Dyckn

qarea(D)tbounce(D) =∑

D∈Dyckn

qdinv(D)tarea(D)

0000122233

Page 27: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

q, t-Catalan Numbers: Garsia & Haglund 2002

Combinatorial interpretation of 〈∇en,en〉 = 〈∇en, s1n〉:∑D∈Dyckn

qarea(D)tbounce(D) =∑

D∈Dyckn

qdinv(D)tarea(D)

0000122233

Page 28: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

q, t-Catalan Numbers: Garsia & Haglund 2002

Combinatorial interpretation of 〈∇en,en〉 = 〈∇en, s1n〉:∑D∈Dyckn

qarea(D)tbounce(D) =∑

D∈Dyckn

qdinv(D)tarea(D)

0000122233

Page 29: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

q, t-Catalan Numbers: Garsia & Haglund 2002

Combinatorial interpretation of 〈∇en,en〉 = 〈∇en, s1n〉:∑D∈Dyckn

qarea(D)tbounce(D) =∑

D∈Dyckn

qdinv(D)tarea(D)

0000122233

Page 30: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

The old q, t-square

Page 31: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Loehr and Warrington 2006Combinatorial interpretation of 〈(−1)n−1∇pn,en〉∑

P∈SQEn

qarea1(P)tbounce1(P) =∑

P∈SQNn

qarea1(P)tbounce1(P)

1

0

3

3

2

2

Page 32: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Loehr and Warrington 2006Combinatorial interpretation of 〈(−1)n−1∇pn,en〉∑

P∈SQEn

qarea1(P)tbounce1(P) =∑

P∈SQNn

qarea1(P)tbounce1(P)

1

0

3

3

2

2

Page 33: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Loehr and Warrington 2006Combinatorial interpretation of 〈(−1)n−1∇pn,en〉∑

P∈SQEn

qarea1(P)tbounce1(P) =∑

P∈SQNn

qarea1(P)tbounce1(P)

1

0

3

3

2

2

Page 34: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Loehr and Warrington 2006Combinatorial interpretation of 〈(−1)n−1∇pn,en〉∑

P∈SQEn

qarea1(P)tbounce1(P) =∑

P∈SQNn

qarea1(P)tbounce1(P)

1

0

3

3

2

2

Page 35: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Loehr and Warrington 2006Combinatorial interpretation of 〈(−1)n−1∇pn,en〉∑

P∈SQEn

qarea1(P)tbounce1(P) =∑

P∈SQNn

qarea1(P)tbounce1(P)

1

0

3

3

2

2

Page 36: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Loehr and Warrington 2006Combinatorial interpretation of 〈(−1)n−1∇pn,en〉∑

P∈SQEn

qarea1(P)tbounce1(P) =∑

P∈SQNn

qarea1(P)tbounce1(P)

1

0

3

3

2

2

Page 37: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Loehr and Warrington 2006Combinatorial interpretation of 〈(−1)n−1∇pn,en〉∑

P∈SQEn

qarea1(P)tbounce1(P) =∑

P∈SQNn

qarea1(P)tbounce1(P)

1

0

3

3

2

2

Page 38: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Loehr and Warrington 2006Combinatorial interpretation of 〈(−1)n−1∇pn,en〉∑

P∈SQEn

qarea1(P)tbounce1(P) =∑

P∈SQNn

qarea1(P)tbounce1(P)

1

0

3

3

2

2

Page 39: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Context

I Statistics generalize Catalan statistics

I Link with Delta conjecture

Theorem (M. D’Adderio, A.V.W.)

∆en−1en∣∣t=1/q

= (−1)n−1∇pn∣∣t=1/q

→ Our new q, t-square is also about ∆en−1en

Page 40: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Context

I Statistics generalize Catalan statisticsI Link with Delta conjecture

Theorem (M. D’Adderio, A.V.W.)

∆en−1en∣∣t=1/q

= (−1)n−1∇pn∣∣t=1/q

→ Our new q, t-square is also about ∆en−1en

Page 41: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Context

I Statistics generalize Catalan statisticsI Link with Delta conjecture

Theorem (M. D’Adderio, A.V.W.)

∆en−1en∣∣t=1/q

= (−1)n−1∇pn∣∣t=1/q

→ Our new q, t-square is also about ∆en−1en

Page 42: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

A new q, t-square

Page 43: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D’Adderio & A.V.W, 2017Combinatorial interpretation of 〈∆en−1en,en〉∑

P∈SQEn

qarea2(P)tbounce2(P) =∑

P∈SQNn

qarea2(P)tbounce2(P)

c

j22300111122

Page 44: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D’Adderio & A.V.W, 2017Combinatorial interpretation of 〈∆en−1en,en〉∑

P∈SQEn

qarea2(P)tbounce2(P) =∑

P∈SQNn

qarea2(P)tbounce2(P)

c

j

22300111122

Page 45: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D’Adderio & A.V.W, 2017Combinatorial interpretation of 〈∆en−1en,en〉∑

P∈SQEn

qarea2(P)tbounce2(P) =∑

P∈SQNn

qarea2(P)tbounce2(P)

c

j22300111122

Page 46: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D’Adderio & A.V.W, 2017Combinatorial interpretation of 〈∆en−1en,en〉∑

P∈SQEn

qarea2(P)tbounce2(P) =∑

P∈SQNn

qarea2(P)tbounce2(P)

c

j22300111122

Page 47: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D’Adderio & A.V.W, 2017Combinatorial interpretation of 〈∆en−1en,en〉∑

P∈SQEn

qarea2(P)tbounce2(P) =∑

P∈SQNn

qarea2(P)tbounce2(P)

c

j22300111122

Page 48: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D’Adderio & A.V.W, 2017Combinatorial interpretation of 〈∆en−1en,en〉∑

P∈SQEn

qarea2(P)tbounce2(P) =∑

P∈SQNn

qarea2(P)tbounce2(P)

c

j

22300111122

Page 49: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Combinatorial interpretation of 〈∆en−1en,en〉

j22300111222

Page 50: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Combinatorial interpretation of 〈∆en−1en,en〉

j

22300111222

Page 51: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Combinatorial interpretation of 〈∆en−1en,en〉

j22300111222

Page 52: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Combinatorial interpretation of 〈∆en−1en,en〉

j22300111222

Page 53: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Combinatorial interpretation of 〈∆en−1en,en〉

j22300111222

Page 54: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Combinatorial interpretation of 〈∆en−1en,en〉

j

22300111222

Page 55: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

The old and the new

I area1, area2 generalise area and bounce1, bounce2generalise bounce

I We have

〈∆en−1en,en〉∣∣t=1/q= 〈(−1)n−1∇pn,en〉∣∣t=1/q

=

[2nn

]q

11 + qn

∑P∈SQE

n

qarea1(P)−bounce1(P) =∑

P∈SQEn

qarea2(P)−bounce2(P)

Page 56: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

The old and the new

I area1, area2 generalise area and bounce1, bounce2generalise bounce

I We have

〈∆en−1en,en〉∣∣t=1/q= 〈(−1)n−1∇pn,en〉∣∣t=1/q

=

[2nn

]q

11 + qn

∑P∈SQE

n

qarea1(P)−bounce1(P) =∑

P∈SQEn

qarea2(P)−bounce2(P)

Page 57: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Decorated Dyck paths

Page 58: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

∗∗

00111111122

I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3I Top peak is not decorated

Page 59: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

∗∗

00111111122

I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3I Top peak is not decorated

Page 60: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

∗∗

00111111122

I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3I Top peak is not decorated

Page 61: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

∗∗

00111111122

I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3I Top peak is not decorated

Page 62: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

∗∗

00111111122

I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3I Top peak is not decorated

Page 63: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

∗∗

00111111122

I a := decorated peaks, b := decorated double falls

I Statistics area3 and bounce3I Top peak is not decorated

Page 64: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

∗∗

00111111122

I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3

I Top peak is not decorated

Page 65: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

∗∗

00111111122

I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3I Top peak is not decorated

Page 66: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between our q, t-square and DecoratedDyck paths

∑P∈SQE

n

qarea2(P)tbounce2(P) =∑

P∈D(0,0)n

qarea3(P)tbounce3(P)

+∑

P∈D(0,1)n

qarea3(P)tbounce3(P)

∑P∈SQN

n

qarea2(P)tbounce2(P) =∑

P∈D(0,0)n

qarea3(P)tbounce3(P)

+∑

P∈D(1,0)n

qarea3(P)tbounce3(P)

Page 67: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between our q, t-square and DecoratedDyck paths

∑P∈SQE

n

qarea2(P)tbounce2(P) =∑

P∈D(0,0)n

qarea3(P)tbounce3(P)

+∑

P∈D(0,1)n

qarea3(P)tbounce3(P)

∑P∈SQN

n

qarea2(P)tbounce2(P) =∑

P∈D(0,0)n

qarea3(P)tbounce3(P)

+∑

P∈D(1,0)n

qarea3(P)tbounce3(P)

Page 68: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Next...

The unifying language of decorated Dyck paths has many linksto existing work.

→ More results on symmetric functions related to the Deltaconjecture

Page 69: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Next...

The unifying language of decorated Dyck paths has many linksto existing work.

→ More results on symmetric functions related to the Deltaconjecture

Page 70: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

A new q, t-squareAnna Vanden Wyngaerd

Universite Libre de [email protected]

September 11, 2017

Page 71: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Garsia and Remmel, Breakthroughs in the theory ofmacdonald polynomials, PNAS, 2005.

J. Haglund, The q,t-catalan numbers and the space ofdiagonal harmonics, Mathematics subject classification,1991.

Page 72: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

0

0

0

0

0

1

I Statistics area3 and bounce3

I Top peak is not decorated

Page 73: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

0

0

0

0

0

1

I Statistics area3 and bounce3

I Top peak is not decorated

Page 74: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

0

0

0

0

0

1

I Statistics area3 and bounce3

I Top peak is not decorated

Page 75: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

0

0

0

0

0

1

I Statistics area3 and bounce3

I Top peak is not decorated

Page 76: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

0

0

0

0

0

1

I Statistics area3 and bounce3

I Top peak is not decorated

Page 77: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

0

0

0

0

0

1

I Statistics area3 and bounce3

I Top peak is not decorated

Page 78: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

0

0

0

0

0

1

I Statistics area3 and bounce3

I Top peak is not decorated

Page 79: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

D(a,b)n : definitions

0

0

0

0

0

1

I Statistics area3 and bounce3

I Top peak is not decorated

Page 80: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(0,1)n and SQE

n

∑P∈SQE

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(0,1)n

qarea3(P)tbounce3(P)

j

c

1

2

0

0

0

c

j

0

0

0

1

2

Page 81: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(0,1)n and SQE

n

∑P∈SQE

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(0,1)n

qarea3(P)tbounce3(P)

j

c

1

2

0

0

0

c

j

0

0

0

1

2

Page 82: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(0,1)n and SQE

n

∑P∈SQE

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(0,1)n

qarea3(P)tbounce3(P)

j

c

1

2

0

0

0

c

j

0

0

0

1

2

Page 83: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(0,1)n and SQE

n

∑P∈SQE

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(0,1)n

qarea3(P)tbounce3(P)

j

c

1

2

0

0

0

c

j

0

0

0

1

2

Page 84: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(0,1)n and SQE

n

∑P∈SQE

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(0,1)n

qarea3(P)tbounce3(P)

j

c

1

2

0

0

0

c

j

0

0

0

1

2

Page 85: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(0,1)n and SQE

n

∑P∈SQE

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(0,1)n

qarea3(P)tbounce3(P)

j

c

1

2

0

0

0

c

j

0

0

0

1

2

Page 86: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(0,1)n and SQE

n

∑P∈SQE

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(0,1)n

qarea3(P)tbounce3(P)

j

c

1

2

0

0

0

c

j

0

0

0

1

2

Page 87: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(0,1)n and SQE

n

∑P∈SQE

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(0,1)n

qarea3(P)tbounce3(P)

j

c

1

2

0

0

0

c

j

0

0

0

1

2

Page 88: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(0,1)n and SQE

n

∑P∈SQE

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(0,1)n

qarea3(P)tbounce3(P)

j

c

1

2

0

0

0

c

j

0

0

0

1

2

Page 89: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(1,0)n and SQN

n

∑P∈SQN

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(1,0)n

qarea3(P)tbounce3(P)

j1

2

0

1

1

j

0

1

1

1

2

Page 90: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(1,0)n and SQN

n

∑P∈SQN

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(1,0)n

qarea3(P)tbounce3(P)

j1

2

0

1

1

j

0

1

1

1

2

Page 91: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(1,0)n and SQN

n

∑P∈SQN

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(1,0)n

qarea3(P)tbounce3(P)

j1

2

0

1

1

j

0

1

1

1

2

Page 92: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(1,0)n and SQN

n

∑P∈SQN

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(1,0)n

qarea3(P)tbounce3(P)

j1

2

0

1

1

j

0

1

1

1

2

Page 93: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(1,0)n and SQN

n

∑P∈SQN

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(1,0)n

qarea3(P)tbounce3(P)

j1

2

0

1

1

j

0

1

1

1

2

Page 94: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(1,0)n and SQN

n

∑P∈SQN

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(1,0)n

qarea3(P)tbounce3(P)

j1

2

0

1

1

j

0

1

1

1

2

Page 95: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(1,0)n and SQN

n

∑P∈SQN

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(1,0)n

qarea3(P)tbounce3(P)

j1

2

0

1

1

j

0

1

1

1

2

Page 96: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(1,0)n and SQN

n

∑P∈SQN

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(1,0)n

qarea3(P)tbounce3(P)

j1

2

0

1

1

j

0

1

1

1

2

Page 97: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Link between D(1,0)n and SQN

n

∑P∈SQN

n

qareaE2 (P)tbounce2(P) =

∑D∈Dyckn

qarea(D)tbounce(D)

+∑

P∈D(1,0)n

qarea3(P)tbounce3(P)

j1

2

0

1

1

j

0

1

1

1

2

Page 98: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Representation theory

ρ : Sn −→ GL

⊕(i,j)∈N×N

V (j,j)

I V (i,j) are ρ invariantI Character

χρ = tr ◦ ρ : Sn → C

I We can decompose χρ =∑

(i,j) χ(i,j)ρ and χ(i,j)

ρ =∑

cλχλwhere cλ ∈ N (multiplicity) and χλ are the irreduciblecharacters of (ρ|V (i,j) ,V (i,j)) (one per conjugacy class

Page 99: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Frobenius Characteristic map

F : Class(Sn)→ ΛnC

f 7→ 1n!

∑σ∈Sn

f (σ)pλ(σ)

I Irreducible characters get sent to Shur functionsI If a symmetric function is the image of the character of a

representation by the Frobenius map then is must be Shurpositive because F is linear

I Bi-graded Frobenius characteristic map

F : χρ 7→∑(i,j)

qi t jF(χ(i,j)ρ )

Page 100: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Symmetric functionsΛK := K [X1, ....,XN ]SN space of symmetric functions.

ΛK =∞⊕

i=1

ΛnK

where ΛnK is the space of homogeneous symmetric functions of

degree n.I A lot of different basis for Λn

K , indexed by partitions of n:elementary eλ, homogeneous hλ, power symmetric pλ.

I Link with representation theory of Sn: the Frobeniuscharacteristic map:

F : Class(Sn)→ ΛnK

I Shur functions sλ form another basis and are the image ofthe irreducible characters by the Frobenius map.

I Scalar product 〈, 〉 on ΛnK such that sλ are orthonormal→

F is an isometry

Page 101: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Symmetric functionsΛK := K [X1, ....,XN ]SN space of symmetric functions.

ΛK =∞⊕

i=1

ΛnK

where ΛnK is the space of homogeneous symmetric functions of

degree n.

I A lot of different basis for ΛnK , indexed by partitions of n:

elementary eλ, homogeneous hλ, power symmetric pλ.I Link with representation theory of Sn: the Frobenius

characteristic map:

F : Class(Sn)→ ΛnK

I Shur functions sλ form another basis and are the image ofthe irreducible characters by the Frobenius map.

I Scalar product 〈, 〉 on ΛnK such that sλ are orthonormal→

F is an isometry

Page 102: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Symmetric functionsΛK := K [X1, ....,XN ]SN space of symmetric functions.

ΛK =∞⊕

i=1

ΛnK

where ΛnK is the space of homogeneous symmetric functions of

degree n.I A lot of different basis for Λn

K , indexed by partitions of n:elementary eλ, homogeneous hλ, power symmetric pλ.

I Link with representation theory of Sn: the Frobeniuscharacteristic map:

F : Class(Sn)→ ΛnK

I Shur functions sλ form another basis and are the image ofthe irreducible characters by the Frobenius map.

I Scalar product 〈, 〉 on ΛnK such that sλ are orthonormal→

F is an isometry

Page 103: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Symmetric functionsΛK := K [X1, ....,XN ]SN space of symmetric functions.

ΛK =∞⊕

i=1

ΛnK

where ΛnK is the space of homogeneous symmetric functions of

degree n.I A lot of different basis for Λn

K , indexed by partitions of n:elementary eλ, homogeneous hλ, power symmetric pλ.

I Link with representation theory of Sn: the Frobeniuscharacteristic map:

F : Class(Sn)→ ΛnK

I Shur functions sλ form another basis and are the image ofthe irreducible characters by the Frobenius map.

I Scalar product 〈, 〉 on ΛnK such that sλ are orthonormal→

F is an isometry

Page 104: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Symmetric functionsΛK := K [X1, ....,XN ]SN space of symmetric functions.

ΛK =∞⊕

i=1

ΛnK

where ΛnK is the space of homogeneous symmetric functions of

degree n.I A lot of different basis for Λn

K , indexed by partitions of n:elementary eλ, homogeneous hλ, power symmetric pλ.

I Link with representation theory of Sn: the Frobeniuscharacteristic map:

F : Class(Sn)→ ΛnK

I Shur functions sλ form another basis and are the image ofthe irreducible characters by the Frobenius map.

I Scalar product 〈, 〉 on ΛnK such that sλ are orthonormal→

F is an isometry

Page 105: A new q,t-square - mat.univie.ac.atslc/wpapers/s79vortrag/wyngaerd.pdf · A new q ;t-square Anna Vanden Wyngaerd Universit e Libre de Bruxelles´ anvdwyng@ulb.ac.be September 11,

Symmetric functionsΛK := K [X1, ....,XN ]SN space of symmetric functions.

ΛK =∞⊕

i=1

ΛnK

where ΛnK is the space of homogeneous symmetric functions of

degree n.I A lot of different basis for Λn

K , indexed by partitions of n:elementary eλ, homogeneous hλ, power symmetric pλ.

I Link with representation theory of Sn: the Frobeniuscharacteristic map:

F : Class(Sn)→ ΛnK

I Shur functions sλ form another basis and are the image ofthe irreducible characters by the Frobenius map.

I Scalar product 〈, 〉 on ΛnK such that sλ are orthonormal→

F is an isometry