A superplancherel measure associated to set partitions and its limit
A superplancherel measure associated to set partitions and its limit
Dario De Stavola
SLC, 28 March 2017
Advisor: Valentin Féray
Affiliation: University of Zürich
A superplancherel measure associated to set partitions and its limit Basic character theory
Frobenius scalar product
Letφ, ψ:G→C
hφ, ψi:= 1
|G| X
g∈G
φ(g)ψ(g)∈C
Consider the algebra of class functions of G, endowed with this product:
∃! orthonormal basis s.t. every basis element χ hasχ(1)∈N+. Call such elements irreducible characters and the basisIrr(G).
A superplancherel measure associated to set partitions and its limit Basic character theory
Frobenius scalar product
Letφ, ψ:G→C
hφ, ψi:= 1
|G| X
g∈G
φ(g)ψ(g)∈C
Consider the algebra of class functions of G, endowed with this product:
∃! orthonormal basis s.t. every basis element χ hasχ(1)∈N+. Call such elements irreducible characters and the basisIrr(G).
A superplancherel measure associated to set partitions and its limit Basic character theory
Frobenius scalar product
Letφ, ψ:G→C
hφ, ψi:= 1
|G| X
g∈G
φ(g)ψ(g)∈C
Consider the algebra of class functions of G, endowed with this product:
∃! orthonormal basis s.t. every basis element χ hasχ(1)∈N+. Call such elements irreducible characters and the basisIrr(G).
A superplancherel measure associated to set partitions and its limit Basic character theory
Frobenius scalar product
Letφ, ψ:G→C
hφ, ψi:= 1
|G| X
g∈G
φ(g)ψ(g)∈C
Consider the algebra of class functions of G, endowed with this product:
∃! orthonormal basis s.t. every basis element χ hasχ(1)∈N+.
Call such elements irreducible characters and the basisIrr(G).
A superplancherel measure associated to set partitions and its limit Basic character theory
Frobenius scalar product
Letφ, ψ:G→C
hφ, ψi:= 1
|G| X
g∈G
φ(g)ψ(g)∈C
Consider the algebra of class functions of G, endowed with this product:
∃! orthonormal basis s.t. every basis element χ hasχ(1)∈N+. Call such elements irreducible charactersand the basis Irr(G).
A superplancherel measure associated to set partitions and its limit Basic character theory
The Plancherel measure
X
χ∈Irr(G)
χ(1)2 =|G|
Measure onIrr(G)
PlG(χ) = χ(1)2
|G|
A superplancherel measure associated to set partitions and its limit Basic character theory
The Plancherel measure
X
χ∈Irr(G)
χ(1)2 =|G|
Measure onIrr(G)
PlG(χ) = χ(1)2
|G|
A superplancherel measure associated to set partitions and its limit supercharacter theory
The upper unitriangular group
Fq the finite field withq elements,q a prime power
Un(Fq) :=
1 ∗ ∗ · · · ∗ 1 ∗ · · · ∗ . .. ... . .. ... 1
, ∗ ∈Fq
Classifying the irreducible representations ofUn(Fq) is a “wild” problem
A superplancherel measure associated to set partitions and its limit supercharacter theory
The upper unitriangular group
Fq the finite field withq elements,q a prime power
Un(Fq) :=
1 ∗ ∗ · · · ∗ 1 ∗ · · · ∗ . .. ... . .. ...
1
, ∗ ∈Fq
Classifying the irreducible representations ofUn(Fq) is a “wild” problem
A superplancherel measure associated to set partitions and its limit supercharacter theory
The upper unitriangular group
Fq the finite field withq elements,q a prime power
Un(Fq) :=
1 ∗ ∗ · · · ∗ 1 ∗ · · · ∗ . .. ... . .. ...
1
, ∗ ∈Fq
Classifying the irreducible representations ofUn(Fq) is a “wild”
problem
A superplancherel measure associated to set partitions and its limit supercharacter theory
OK, no irreducible characters. Now what?
G = F
g∈G/∼
[g], G is an union of conjugacy classes Irr(G) ={χ:G/∼→C orthonormal w.r.t. h·,·i}
idea (André and Yan):
G = F
K∈K
K,
H={ψ:K →Corthogonal w.r.t. h·,·i}
A superplancherel measure associated to set partitions and its limit supercharacter theory
OK, no irreducible characters. Now what?
G = F
g∈G/∼
[g], G is an union of conjugacy classes Irr(G) ={χ:G/∼→C orthonormal w.r.t. h·,·i}
idea (André and Yan):
G = F
K∈K
K,
H={ψ:K →Corthogonal w.r.t. h·,·i}
A superplancherel measure associated to set partitions and its limit supercharacter theory
OK, no irreducible characters. Now what?
G = F
g∈G/∼
[g], G is an union of conjugacy classes Irr(G) ={χ:G/∼→C orthonormal w.r.t. h·,·i}
idea (André and Yan):
G = F
K∈K
K,
H={ψ:K →Corthogonal w.r.t. h·,·i}
A superplancherel measure associated to set partitions and its limit supercharacter theory
OK, no irreducible characters. Now what?
G = F
K∈K
K,
H={ψ:K →Corthogonal w.r.t. h·,·i}
Each K is a union of conjugacy classes;
thus,ψ are class functions (constant on conjugacy classes); Irr(G) is a basis for the algebra of class functions, so eachψ must be a linear combination of irreducible characters;
A superplancherel measure associated to set partitions and its limit supercharacter theory
OK, no irreducible characters. Now what?
G = F
K∈K
K,
H={ψ:K →Corthogonal w.r.t. h·,·i}
Each K is a union of conjugacy classes;
thus,ψ are class functions (constant on conjugacy classes); Irr(G) is a basis for the algebra of class functions, so eachψ must be a linear combination of irreducible characters;
A superplancherel measure associated to set partitions and its limit supercharacter theory
OK, no irreducible characters. Now what?
G = F
K∈K
K,
H={ψ:K →Corthogonal w.r.t. h·,·i}
Each K is a union of conjugacy classes;
thus,ψ are class functions (constant on conjugacy classes);
Irr(G) is a basis for the algebra of class functions, so eachψ must be a linear combination of irreducible characters;
A superplancherel measure associated to set partitions and its limit supercharacter theory
OK, no irreducible characters. Now what?
G = F
K∈K
K,
H={ψ:K →Corthogonal w.r.t. h·,·i}
Each K is a union of conjugacy classes;
thus,ψ are class functions (constant on conjugacy classes);
Irr(G) is a basis for the algebra of class functions, so eachψ must be a linear combination of irreducible characters;
A superplancherel measure associated to set partitions and its limit supercharacter theory
Supercharacter theory (Diaconis and Isaacs)
Asupercharacter theoryis a pair (K,H) whereK is a set partition ofG andH is an orthogonal set of characters such that
1 |K|=|H|;
2 if ψ∈ H then ψ constant onK,∀K ∈ K;
3 if χ∈Irr(G) then ∃!ψ such thathχ, ψi 6= 0. Given a suitableK then His fixed
(and viceversa).
A superplancherel measure associated to set partitions and its limit supercharacter theory
Supercharacter theory (Diaconis and Isaacs)
Asupercharacter theoryis a pair (K,H) whereK is a set partition ofG andH is an orthogonal set of characters such that
1 |K|=|H|;
2 if ψ∈ H then ψ constant onK,∀K ∈ K;
3 if χ∈Irr(G) then ∃!ψ such thathχ, ψi 6= 0. Given a suitableK then His fixed
(and viceversa).
A superplancherel measure associated to set partitions and its limit supercharacter theory
Supercharacter theory (Diaconis and Isaacs)
Asupercharacter theoryis a pair (K,H) whereK is a set partition ofG andH is an orthogonal set of characters such that
1 |K|=|H|;
2 if ψ∈ H then ψ constant onK,∀K ∈ K;
3 if χ∈Irr(G) then ∃!ψ such thathχ, ψi 6= 0. Given a suitableK then His fixed
(and viceversa).
A superplancherel measure associated to set partitions and its limit supercharacter theory
Supercharacter theory (Diaconis and Isaacs)
Asupercharacter theoryis a pair (K,H) whereK is a set partition ofG andH is an orthogonal set of characters such that
1 |K|=|H|;
2 if ψ∈ H then ψ constant onK,∀K ∈ K;
3 if χ∈Irr(G) then∃!ψ such thathχ, ψi 6= 0.
Given a suitableK then His fixed (and viceversa).
A superplancherel measure associated to set partitions and its limit supercharacter theory
Supercharacter theory (Diaconis and Isaacs)
Asupercharacter theoryis a pair (K,H) whereK is a set partition ofG andH is an orthogonal set of characters such that
1 |K|=|H|;
2 if ψ∈ H then ψ constant onK,∀K ∈ K;
3 if χ∈Irr(G) then∃!ψ such thathχ, ψi 6= 0.
Given a suitableK then His fixed (and viceversa).
A superplancherel measure associated to set partitions and its limit supercharacter theory
(Bergeron and Thiem) A supercharacter theory for U
n(F
q)
h=
1 0 1 0 1 4 0 3
1 5 2 0 3 6 0
1 0 0 0 4 3
1 0 0 3 0
1 0 4 0
1 0 0 1 0 1
,
1 0 · 0 ? · · ·
1 ? · · · · ·
1 0 0 0 · ?
1 0 0 · 0
1 0 ? · 1 0 0 1 0 1
π=
1 2 3 4 5 6 7 8
A superplancherel measure associated to set partitions and its limit supercharacter theory
(Bergeron and Thiem) A supercharacter theory for U
n(F
q)
h=
1 0 1 0 1 4 0 3
1 5 2 0 3 6 0
1 0 0 0 4 3
1 0 0 3 0
1 0 4 0
1 0 0 1 0 1
,
1 0 · 0 ? · · ·
1 ? · · · · ·
1 0 0 0 · ?
1 0 0 · 0
1 0 ? · 1 0 0 1 0 1
π=
1 2 3 4 5 6 7 8
A superplancherel measure associated to set partitions and its limit supercharacter theory
(Bergeron and Thiem) A supercharacter theory for U
n(F
q)
h=
1 0 1 0 1 4 0 3
1 5 2 0 3 6 0
1 0 0 0 4 3
1 0 0 3 0
1 0 4 0
1 0 0 1 0 1
,
1 0 · 0 ? · · ·
1 ? · · · · ·
1 0 0 0 · ?
1 0 0 · 0
1 0 ? · 1 0 0 1 0 1
π=
1 2 3 4 5 6 7 8
A superplancherel measure associated to set partitions and its limit supercharacter theory
Why this supercharacter theory?
1 Superclasses (and supercharacters) are indexed by nice combinatorial objects;
2 the supercharacters have an explicit formula;
3 the supercharacters have rational values;
4 the algebra of superclass functions is isomorphic to the algebra of symmetric functions in noncommutative variables;
5 Nice decomposition of the supercharacter table (Bergeron and Thiem).
A superplancherel measure associated to set partitions and its limit supercharacter theory
Why this supercharacter theory?
1 Superclasses (and supercharacters) are indexed by nice combinatorial objects;
2 the supercharacters have an explicit formula;
3 the supercharacters have rational values;
4 the algebra of superclass functions is isomorphic to the algebra of symmetric functions in noncommutative variables;
5 Nice decomposition of the supercharacter table (Bergeron and Thiem).
A superplancherel measure associated to set partitions and its limit supercharacter theory
Why this supercharacter theory?
1 Superclasses (and supercharacters) are indexed by nice combinatorial objects;
2 the supercharacters have an explicit formula;
3 the supercharacters have rational values;
4 the algebra of superclass functions is isomorphic to the algebra of symmetric functions in noncommutative variables;
5 Nice decomposition of the supercharacter table (Bergeron and Thiem).
A superplancherel measure associated to set partitions and its limit supercharacter theory
Why this supercharacter theory?
1 Superclasses (and supercharacters) are indexed by nice combinatorial objects;
2 the supercharacters have an explicit formula;
3 the supercharacters have rational values;
4 the algebra of superclass functions is isomorphic to the algebra of symmetric functions in noncommutative variables;
5 Nice decomposition of the supercharacter table (Bergeron and Thiem).
A superplancherel measure associated to set partitions and its limit supercharacter theory
Why this supercharacter theory?
1 Superclasses (and supercharacters) are indexed by nice combinatorial objects;
2 the supercharacters have an explicit formula;
3 the supercharacters have rational values;
4 the algebra of superclass functions is isomorphic to the algebra of symmetric functions in noncommutative variables;
5 Nice decomposition of the supercharacter table (Bergeron and Thiem).
A superplancherel measure associated to set partitions and its limit supercharacter theory
Set partitions notation
π=
1 2 3 4 5 6 7 8 Arcs(π) ={(1,5),(2,3),(3,8),(5,7)};
a(π) =|Arcs(π)|= 4;
dim(π) = P
(i,j)∈Arcs(π)
j −i = 12; crs(π) =]crossings ofπ = 1;
A superplancherel measure associated to set partitions and its limit supercharacter theory
Set partitions notation
π=
1 2 3 4 5 6 7 8 Arcs(π) ={(1,5),(2,3),(3,8),(5,7)};
a(π) =|Arcs(π)|= 4;
dim(π) = P
(i,j)∈Arcs(π)
j −i = 12; crs(π) =]crossings ofπ = 1;
A superplancherel measure associated to set partitions and its limit supercharacter theory
Set partitions notation
π=
1 2 3 4 5 6 7 8 Arcs(π) ={(1,5),(2,3),(3,8),(5,7)};
a(π) =|Arcs(π)|= 4;
dim(π) = P
(i,j)∈Arcs(π)
j −i = 12;
crs(π) =]crossings ofπ = 1;
A superplancherel measure associated to set partitions and its limit supercharacter theory
Set partitions notation
π=
1 2 3 4 5 6 7 8 Arcs(π) ={(1,5),(2,3),(3,8),(5,7)};
a(π) =|Arcs(π)|= 4;
dim(π) = P
(i,j)∈Arcs(π)
j −i = 12;
crs(π) =]crossings ofπ = 1;
A superplancherel measure associated to set partitions and its limit supercharacter theory
The dimension of a supercharacter is χπ(1) = (q−1)a(π)·qdim(π)−a(π);
hχπ, χπi= (q−1)a(π)·qcrs(π);
The superplancherel measure
SPlG(χ) = 1
|G| χ(1)2
hχ, χi = 1 qn(n−1)2
(q−1)a(π)·q2 dim(π)−2a(π)
qcrs(π)
A superplancherel measure associated to set partitions and its limit supercharacter theory
The dimension of a supercharacter is χπ(1) = (q−1)a(π)·qdim(π)−a(π);
hχπ, χπi= (q−1)a(π)·qcrs(π);
The superplancherel measure
SPlG(χ) = 1
|G| χ(1)2
hχ, χi = 1 qn(n−1)2
(q−1)a(π)·q2 dim(π)−2a(π)
qcrs(π)
A superplancherel measure associated to set partitions and its limit supercharacter theory
The dimension of a supercharacter is χπ(1) = (q−1)a(π)·qdim(π)−a(π);
hχπ, χπi= (q−1)a(π)·qcrs(π);
The superplancherel measure
SPlG(χ) = 1
|G| χ(1)2
hχ, χi = 1 qn(n−1)2
(q−1)a(π)·q2 dim(π)−2a(π)
qcrs(π)
A superplancherel measure associated to set partitions and its limit supercharacter theory
The dimension of a supercharacter is χπ(1) = (q−1)a(π)·qdim(π)−a(π);
hχπ, χπi= (q−1)a(π)·qcrs(π);
The superplancherel measure
SPlG(χ) = 1
|G| χ(1)2 hχ, χi
= 1
qn(n−1)2
(q−1)a(π)·q2 dim(π)−2a(π)
qcrs(π)
A superplancherel measure associated to set partitions and its limit supercharacter theory
The dimension of a supercharacter is χπ(1) = (q−1)a(π)·qdim(π)−a(π);
hχπ, χπi= (q−1)a(π)·qcrs(π);
The superplancherel measure
SPlG(χ) = 1
|G| χ(1)2
hχ, χi = 1 qn(n−1)2
(q−1)a(π)·q2 dim(π)−2a(π)
qcrs(π)
A superplancherel measure associated to set partitions and its limit supercharacter theory
Plan
1 See set partitions as objects of the same space
(some renormalization happens);
2 interpret our statistics w.r.t. this new setting;
3 let n→ ∞;
A superplancherel measure associated to set partitions and its limit supercharacter theory
Plan
1 See set partitions as objects of the same space (some renormalization happens);
2 interpret our statistics w.r.t. this new setting;
3 let n→ ∞;
A superplancherel measure associated to set partitions and its limit supercharacter theory
Plan
1 See set partitions as objects of the same space (some renormalization happens);
2 interpret our statistics w.r.t. this new setting;
3 let n→ ∞;
A superplancherel measure associated to set partitions and its limit supercharacter theory
Plan
1 See set partitions as objects of the same space (some renormalization happens);
2 interpret our statistics w.r.t. this new setting;
3 let n→ ∞;
A superplancherel measure associated to set partitions and its limit supercharacter theory
First step to look for a limit
→ → →
1 0 · 0 ? · · · 1 ? · · · · · 1 0 0 0 · ? 1 0 0 · 0 1 0 ? · 1 0 0 1 0 1
→
10·0?··· 1?····· 1000·? 100·0 10?· 100 10 1
→
1
0 1
→ ?
A superplancherel measure associated to set partitions and its limit supercharacter theory
First step to look for a limit
→
→ →
1 0 · 0 ? · · · 1 ? · · · · · 1 0 0 0 · ? 1 0 0 · 0 1 0 ? · 1 0 0 1 0 1
→
10·0?··· 1?····· 1000·? 100·0 10?· 100 10 1
→
1
0 1
→ ?
A superplancherel measure associated to set partitions and its limit supercharacter theory
First step to look for a limit
→ →
→
1 0 · 0 ? · · · 1 ? · · · · · 1 0 0 0 · ? 1 0 0 · 0 1 0 ? · 1 0 0 1 0 1
→
10·0?··· 1?····· 1000·? 100·0 10?· 100 10 1
→
1
0 1
→ ?
A superplancherel measure associated to set partitions and its limit supercharacter theory
First step to look for a limit
→ → →
1 0 · 0 ? · · · 1 ? · · · · · 1 0 0 0 · ? 1 0 0 · 0 1 0 ? · 1 0 0 1 0 1
→
10·0?··· 1?····· 1000·? 100·0 10?· 100 10 1
→
1
0 1
→ ?
A superplancherel measure associated to set partitions and its limit supercharacter theory
First step to look for a limit
→ → →
1 0 · 0 ? · · · 1 ? · · · · · 1 0 0 0 · ? 1 0 0 · 0 1 0 ? · 1 0 0 1 0 1
→
10·0?··· 1?····· 1000·? 100·0 10?· 100 10 1
→
1
0 1
→ ?
A superplancherel measure associated to set partitions and its limit supercharacter theory
First step to look for a limit
→ → →
1 0 · 0 ? · · · 1 ? · · · · · 1 0 0 0 · ? 1 0 0 · 0 1 0 ? · 1 0 0 1 0 1
→
10·0?··· 1?····· 1000·? 100·0 10?· 100 10 1
→
1
0 1
→ ?
A superplancherel measure associated to set partitions and its limit supercharacter theory
First step to look for a limit
→ → →
1 0 · 0 ? · · · 1 ? · · · · · 1 0 0 0 · ? 1 0 0 · 0 1 0 ? · 1 0 0 1 0 1
→
10·0?··· 1?····· 1000·? 100·0 10?· 100 10 1
→
1
0 1
→ ?
A superplancherel measure associated to set partitions and its limit supercharacter theory
First step to look for a limit
→ → →
1 0 · 0 ? · · · 1 ? · · · · · 1 0 0 0 · ? 1 0 0 · 0 1 0 ? · 1 0 0 1 0 1
→
10·0?··· 1?····· 1000·? 100·0 10?· 100 10 1
→
1
0 1
→ ?
A superplancherel measure associated to set partitions and its limit supercharacter theory
Our setting
∆ =
1
0
1
Γ =
measures µon ∆ s.t. R
∆µ≤1 (subprobability) µhassub-uniform marginals
1
0
a b 1 c
d
Z x=1 x=0
Z y=d y=c
dµ≤d−c SUB-UNIFORM
A superplancherel measure associated to set partitions and its limit supercharacter theory
Our setting
∆ =
1
0
1
Γ =
measures µon ∆ s.t.
R
∆µ≤1 (subprobability) µhassub-uniform marginals
1
0
a b 1 c
d
Z x=1 x=0
Z y=d y=c
dµ≤d−c SUB-UNIFORM
A superplancherel measure associated to set partitions and its limit supercharacter theory
Our setting
∆ =
1
0
1
Γ =
measures µon ∆ s.t.
R
∆µ≤1 (subprobability) µhassub-uniform marginals
1
0
a b 1 c
d
Z x=1 x=0
Z y=d y=c
dµ≤d−c SUB-UNIFORM
A superplancherel measure associated to set partitions and its limit supercharacter theory
Our setting
∆ =
1
0
1
Γ =
measures µon ∆ s.t.
R
∆µ≤1 (subprobability) µhassub-uniform marginals
1
0
a b 1 c
d
Z x=1 x=0
Z y=d y=c
dµ≤d−c SUB-UNIFORM
A superplancherel measure associated to set partitions and its limit supercharacter theory
Theorem (DDS)
There exists a measure Ω∈Γ such thatµπ →Ω almost surely.
1 2 3 n
A superplancherel measure associated to set partitions and its limit supercharacter theory
Theorem (DDS)
There exists a measure Ω∈Γ such thatµπ →Ω almost surely.
1 2 3 n
A superplancherel measure associated to set partitions and its limit supercharacter theory
Theorem (DDS)
There exists a measure Ω∈Γ such thatµπ →Ω almost surely.
1 2 3 n
A superplancherel measure associated to set partitions and its limit supercharacter theory
SPln(χπ) = 1 qn(n−1)2
q2 dim(π)−2a(π)
(q−1)a(π)qcrs(π) =
exp
−n2logq 1
2 −2Idim(µπ) +Icrs(µπ)
+O(n)
dim(π)→Idim(µπ) := Z
∆
(y−x)dµ crs(π)→
Z
∆2
1[x1 <x2<y1 <y2]dµπ(x1,y1)dµπ(x2,y2)
H(µ) := 1
2−2Idim(µ) +Icrs(µ)
A superplancherel measure associated to set partitions and its limit supercharacter theory
SPln(χπ) = 1 qn(n−1)2
q2 dim(π)−2a(π)
(q−1)a(π)qcrs(π) =
exp
−n2logq 1
2−2Idim(µπ) +Icrs(µπ)
+O(n)
dim(π)→Idim(µπ) := Z
∆
(y−x)dµ crs(π)→
Z
∆2
1[x1 <x2<y1 <y2]dµπ(x1,y1)dµπ(x2,y2)
H(µ) := 1
2−2Idim(µ) +Icrs(µ)
A superplancherel measure associated to set partitions and its limit supercharacter theory
SPln(χπ) = 1 qn(n−1)2
q2 dim(π)−2a(π)
(q−1)a(π)qcrs(π) =
exp
−n2logq 1
2−2Idim(µπ) +Icrs(µπ)
+O(n)
dim(π)→Idim(µπ) :=
Z
∆
(y−x)dµ
crs(π)→ Z
∆2
1[x1 <x2<y1 <y2]dµπ(x1,y1)dµπ(x2,y2)
H(µ) := 1
2−2Idim(µ) +Icrs(µ)
A superplancherel measure associated to set partitions and its limit supercharacter theory
SPln(χπ) = 1 qn(n−1)2
q2 dim(π)−2a(π)
(q−1)a(π)qcrs(π) =
exp
−n2logq 1
2−2Idim(µπ) +Icrs(µπ)
+O(n)
dim(π)→Idim(µπ) :=
Z
∆
(y−x)dµ crs(π)→
Z
∆2
1[x1 <x2<y1 <y2]dµπ(x1,y1)dµπ(x2,y2)
H(µ) := 1
2−2Idim(µ) +Icrs(µ)
A superplancherel measure associated to set partitions and its limit supercharacter theory
SPln(χπ) = 1 qn(n−1)2
q2 dim(π)−2a(π)
(q−1)a(π)qcrs(π) =
exp
−n2logq 1
2−2Idim(µπ) +Icrs(µπ)
+O(n)
dim(π)→Idim(µπ) :=
Z
∆
(y−x)dµ crs(π)→
Z
∆2
1[x1 <x2<y1 <y2]dµπ(x1,y1)dµπ(x2,y2)
H(µ) := 1
2−2Idim(µ) +Icrs(µ)
A superplancherel measure associated to set partitions and its limit supercharacter theory
Playing with I
dim(µ) =
R∆(y − x ) d µ
7→
µ φ(µ)
A superplancherel measure associated to set partitions and its limit supercharacter theory
Playing with crossings
Icrs(µπ) = Z
∆2
1[x1<x2 <y1<y2]dµπ(x1,y1)dµπ(x2,y2)
Proposition
Ifµhas mass in andµ has uniform marginals then Icrs(µ) = 0⇔µ= = Ω
A superplancherel measure associated to set partitions and its limit supercharacter theory
Playing with crossings
Icrs(µπ) = Z
∆2
1[x1<x2 <y1<y2]dµπ(x1,y1)dµπ(x2,y2)
Proposition
Ifµhas mass in andµ has uniform marginals then Icrs(µ) = 0⇔µ= = Ω
A superplancherel measure associated to set partitions and its limit supercharacter theory
Summary
H(µ) = 0⇔
µhas uniform marginals
µinside the top left square Icrs(µ) = 0
⇔µ= Ω
The last step (technical) is to prove thatµπ(n) →Ω iff H(µπ(n))→ H(Ω) = 0.
A superplancherel measure associated to set partitions and its limit supercharacter theory
Summary
H(µ) = 0⇔
µhas uniform marginals µinside the top left square
Icrs(µ) = 0
⇔µ= Ω
The last step (technical) is to prove thatµπ(n) →Ω iff H(µπ(n))→ H(Ω) = 0.
A superplancherel measure associated to set partitions and its limit supercharacter theory
Summary
H(µ) = 0⇔
µhas uniform marginals µinside the top left square Icrs(µ) = 0
⇔µ= Ω
The last step (technical) is to prove thatµπ(n) →Ω iff H(µπ(n))→ H(Ω) = 0.
A superplancherel measure associated to set partitions and its limit supercharacter theory
Summary
H(µ) = 0⇔
µhas uniform marginals µinside the top left square Icrs(µ) = 0
⇔µ= Ω
The last step (technical) is to prove thatµπ(n) →Ω iff H(µπ(n))→ H(Ω) = 0.
A superplancherel measure associated to set partitions and its limit supercharacter theory
Summary
H(µ) = 0⇔
µhas uniform marginals µinside the top left square Icrs(µ) = 0
⇔µ= Ω
The last step (technical) is to prove thatµπ(n) →Ω iff H(µπ(n))→ H(Ω) = 0.
