Post on 16-Mar-2020
A new q, t-squareAnna Vanden Wyngaerd
Universite Libre de Bruxellesanvdwyng@ulb.ac.be
September 11, 2017
Background
The old q, t-square
A new q, t-square
Decorated Dyck paths
Background
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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)
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)
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
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
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
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
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
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
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
The old q, t-square
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
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
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
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
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
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
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
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
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
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
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
A new q, t-square
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
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
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
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
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
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
Combinatorial interpretation of 〈∆en−1en,en〉
j22300111222
Combinatorial interpretation of 〈∆en−1en,en〉
j
22300111222
Combinatorial interpretation of 〈∆en−1en,en〉
j22300111222
Combinatorial interpretation of 〈∆en−1en,en〉
j22300111222
Combinatorial interpretation of 〈∆en−1en,en〉
j22300111222
Combinatorial interpretation of 〈∆en−1en,en〉
j
22300111222
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)
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)
Decorated Dyck paths
D(a,b)n : definitions
∗∗
∗
00111111122
I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3I Top peak is not decorated
D(a,b)n : definitions
∗∗
∗
00111111122
I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3I Top peak is not decorated
D(a,b)n : definitions
∗∗
∗
00111111122
I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3I Top peak is not decorated
D(a,b)n : definitions
∗∗
∗
00111111122
I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3I Top peak is not decorated
D(a,b)n : definitions
∗∗
∗
00111111122
I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3I Top peak is not decorated
D(a,b)n : definitions
∗∗
∗
00111111122
I a := decorated peaks, b := decorated double falls
I Statistics area3 and bounce3I Top peak is not decorated
D(a,b)n : definitions
∗∗
∗
00111111122
I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3
I Top peak is not decorated
D(a,b)n : definitions
∗∗
∗
00111111122
I a := decorated peaks, b := decorated double fallsI Statistics area3 and bounce3I Top peak is not decorated
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)
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)
Next...
The unifying language of decorated Dyck paths has many linksto existing work.
→ More results on symmetric functions related to the Deltaconjecture
Next...
The unifying language of decorated Dyck paths has many linksto existing work.
→ More results on symmetric functions related to the Deltaconjecture
A new q, t-squareAnna Vanden Wyngaerd
Universite Libre de Bruxellesanvdwyng@ulb.ac.be
September 11, 2017
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.
D(a,b)n : definitions
◦
∗
0
0
0
0
0
1
I Statistics area3 and bounce3
I Top peak is not decorated
D(a,b)n : definitions
◦
∗
0
0
0
0
0
1
I Statistics area3 and bounce3
I Top peak is not decorated
D(a,b)n : definitions
◦
∗
0
0
0
0
0
1
I Statistics area3 and bounce3
I Top peak is not decorated
D(a,b)n : definitions
◦
∗
0
0
0
0
0
1
I Statistics area3 and bounce3
I Top peak is not decorated
D(a,b)n : definitions
◦
∗
0
0
0
0
0
1
I Statistics area3 and bounce3
I Top peak is not decorated
D(a,b)n : definitions
◦
∗
0
0
0
0
0
1
I Statistics area3 and bounce3
I Top peak is not decorated
D(a,b)n : definitions
◦
∗
0
0
0
0
0
1
I Statistics area3 and bounce3
I Top peak is not decorated
D(a,b)n : definitions
◦
∗
0
0
0
0
0
1
I Statistics area3 and bounce3
I Top peak is not decorated
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 )
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
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
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
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
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
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