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

Let*φ, 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* has*χ(1)*∈**N**_{+}.
Call such elements *irreducible characters* and the basis*Irr*(G).

A superplancherel measure associated to set partitions and its limit Basic character theory

## Frobenius scalar product

Let*φ, 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* has*χ(1)*∈**N**_{+}.
Call such elements *irreducible characters* and the basis*Irr*(G).

A superplancherel measure associated to set partitions and its limit Basic character theory

## Frobenius scalar product

Let*φ, 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* has*χ(1)*∈**N**_{+}.
Call such elements *irreducible characters* and the basis*Irr*(G).

A superplancherel measure associated to set partitions and its limit Basic character theory

## Frobenius scalar product

Let*φ, 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* has*χ(1)*∈**N**_{+}.

Call such elements *irreducible characters* and the basis*Irr*(G).

A superplancherel measure associated to set partitions and its limit Basic character theory

## Frobenius scalar product

Let*φ, 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:

*χ* has*χ(1)*∈**N**_{+}.
Call such elements *irreducible characters*and 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 on*Irr*(G)

Pl* _{G}*(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 on*Irr*(G)

Pl* _{G}*(χ) =

*χ(1)*

^{2}

|G|

A superplancherel measure associated to set partitions and its limit supercharacter theory

## The upper unitriangular group

**F*** _{q}* the finite field with

*q*elements,

*q*a prime power

*U** _{n}*(F

*) :=*

_{q}

1 ∗ ∗ · · · ∗ 1 ∗ · · · ∗ . .. ... . .. ... 1

*,* ∗ ∈**F**_{q}

Classifying the irreducible representations of*U** _{n}*(F

*) is a “wild” problem*

_{q}A superplancherel measure associated to set partitions and its limit supercharacter theory

## The upper unitriangular group

**F*** _{q}* the finite field with

*q*elements,

*q*a prime power

*U** _{n}*(F

*) :=*

_{q}

1 ∗ ∗ · · · ∗ 1 ∗ · · · ∗ . .. ... . .. ...

1

*,* ∗ ∈**F**_{q}

Classifying the irreducible representations of*U** _{n}*(F

*) is a “wild” problem*

_{q}A superplancherel measure associated to set partitions and its limit supercharacter theory

## The upper unitriangular group

**F*** _{q}* the finite field with

*q*elements,

*q*a prime power

*U** _{n}*(F

*) :=*

_{q}

1 ∗ ∗ · · · ∗ 1 ∗ · · · ∗ . .. ... . .. ...

1

*,* ∗ ∈**F**_{q}

Classifying the irreducible representations of*U** _{n}*(F

*) is a “wild”*

_{q}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 →**C**orthogonal 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 →**C**orthogonal 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 →**C**orthogonal 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 →**C**orthogonal 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 each*ψ*
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 →**C**orthogonal 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 each*ψ*
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 →**C**orthogonal 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 each*ψ*
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 →**C**orthogonal 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 each*ψ*
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)

A*supercharacter theory*is a pair (K,H) whereK is a set partition
of*G* andH is an orthogonal set of characters such that

1 |K|=|H|;

2 if *ψ*∈ H then *ψ* constant on*K*,∀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)

A*supercharacter theory*is a pair (K,H) whereK is a set partition
of*G* andH is an orthogonal set of characters such that

1 |K|=|H|;

2 if *ψ*∈ H then *ψ* constant on*K*,∀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)

A*supercharacter theory*is a pair (K,H) whereK is a set partition
of*G* andH is an orthogonal set of characters such that

1 |K|=|H|;

2 if *ψ*∈ H then *ψ* constant on*K*,∀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)

*supercharacter theory*is a pair (K,H) whereK is a set partition
of*G* andH is an orthogonal set of characters such that

1 |K|=|H|;

2 if *ψ*∈ H then *ψ* constant on*K*,∀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)

*supercharacter theory*is a pair (K,H) whereK is a set partition
of*G* andH is an orthogonal set of characters such that

1 |K|=|H|;

2 if *ψ*∈ H then *ψ* constant on*K*,∀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;

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;

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
*χ** ^{π}*(1) = (q−1)

*·*

^{a(π)}*q*dim(π)−a(π);

hχ^{π}*, χ** ^{π}*i= (q−1)

*·*

^{a(π)}*q*

*;*

^{crs(π)}The superplancherel measure

SPl*G*(χ) = 1

|G|
*χ(1)*^{2}

hχ, χi = 1
*q*^{n(n−1)}^{2}

(q−1)* ^{a(π)}*·

*q*2 dim(π)−2a(π)

*q*^{crs(π)}

A superplancherel measure associated to set partitions and its limit supercharacter theory

The dimension of a supercharacter is
*χ** ^{π}*(1) = (q−1)

*·*

^{a(π)}*q*dim(π)−a(π);

hχ^{π}*, χ** ^{π}*i= (q−1)

*·*

^{a(π)}*q*

*;*

^{crs(π)}The superplancherel measure

SPl*G*(χ) = 1

|G|
*χ(1)*^{2}

hχ, χi = 1
*q*^{n(n−1)}^{2}

(q−1)* ^{a(π)}*·

*q*2 dim(π)−2a(π)

*q*^{crs(π)}

A superplancherel measure associated to set partitions and its limit supercharacter theory

The dimension of a supercharacter is
*χ** ^{π}*(1) = (q−1)

*·*

^{a(π)}*q*dim(π)−a(π);

hχ^{π}*, χ** ^{π}*i= (q−1)

*·*

^{a(π)}*q*

*;*

^{crs(π)}The superplancherel measure

SPl*G*(χ) = 1

|G|
*χ(1)*^{2}

hχ, χi = 1
*q*^{n(n−1)}^{2}

(q−1)* ^{a(π)}*·

*q*2 dim(π)−2a(π)

*q*^{crs(π)}

A superplancherel measure associated to set partitions and its limit supercharacter theory

The dimension of a supercharacter is
*χ** ^{π}*(1) = (q−1)

*·*

^{a(π)}*q*dim(π)−a(π);

hχ^{π}*, χ** ^{π}*i= (q−1)

*·*

^{a(π)}*q*

*;*

^{crs(π)}The superplancherel measure

SPl*G*(χ) = 1

|G|
*χ(1)*^{2}
hχ, χi

= 1

*q*^{n(n−1)}^{2}

(q−1)* ^{a(π)}*·

*q*2 dim(π)−2a(π)

*q*^{crs(π)}

A superplancherel measure associated to set partitions and its limit supercharacter theory

The dimension of a supercharacter is
*χ** ^{π}*(1) = (q−1)

*·*

^{a(π)}*q*dim(π)−a(π);

hχ^{π}*, χ** ^{π}*i= (q−1)

*·*

^{a(π)}*q*

*;*

^{crs(π)}The superplancherel measure

SPl*G*(χ) = 1

|G|
*χ(1)*^{2}

hχ, χi = 1
*q*^{n(n−1)}^{2}

(q−1)* ^{a(π)}*·

*q*2 dim(π)−2a(π)

*q*^{crs(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 *µ*on ∆ s.t.
R

∆*µ*≤1 (subprobability)
*µ*has*sub-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 *µ*on ∆ s.t.

R

∆*µ*≤1 (subprobability)
*µ*has*sub-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 *µ*on ∆ s.t.

R

∆*µ*≤1 (subprobability)
*µ*has*sub-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 *µ*on ∆ s.t.

R

∆*µ*≤1 (subprobability)
*µ*has*sub-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 that*µ**π* →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 that*µ**π* →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 that*µ**π* →Ω almost surely.

1 2 3 n

A superplancherel measure associated to set partitions and its limit supercharacter theory

SPl* _{n}*(χ

*) = 1*

^{π}*q*

^{n(n−1)}^{2}

*q*2 dim(π)−2a(π)

(q−1)^{a(π)}*q** ^{crs(π)}* =

exp

−n^{2}log*q*
1

2 −2I_{dim}(µ* _{π}*) +

*I*

*(µ*

_{crs}*)*

_{π}+*O(n)*

dim(π)→*I*_{dim}(µ* _{π}*) :=
Z

∆

(y−*x)dµ*
*crs*(π)→

Z

∆^{2}

**1[x**1 *<x*2*<y*1 *<y*2]*dµ**π*(x1*,y*1)*dµ**π*(x2*,y*2)

H(µ) := 1

2−2I_{dim}(µ) +*I** _{crs}*(µ)

A superplancherel measure associated to set partitions and its limit supercharacter theory

SPl* _{n}*(χ

*) = 1*

^{π}*q*

^{n(n−1)}^{2}

*q*2 dim(π)−2a(π)

(q−1)^{a(π)}*q** ^{crs(π)}* =

exp

−n^{2}log*q*
1

2−2I_{dim}(µ* _{π}*) +

*I*

*(µ*

_{crs}*)*

_{π}+*O(n)*

dim(π)→*I*_{dim}(µ* _{π}*) :=
Z

∆

(y−*x)dµ*
*crs*(π)→

Z

∆^{2}

**1[x**1 *<x*2*<y*1 *<y*2]*dµ**π*(x1*,y*1)*dµ**π*(x2*,y*2)

H(µ) := 1

2−2I_{dim}(µ) +*I** _{crs}*(µ)

A superplancherel measure associated to set partitions and its limit supercharacter theory

SPl* _{n}*(χ

*) = 1*

^{π}*q*

^{n(n−1)}^{2}

*q*2 dim(π)−2a(π)

(q−1)^{a(π)}*q** ^{crs(π)}* =

exp

−n^{2}log*q*
1

2−2I_{dim}(µ* _{π}*) +

*I*

*(µ*

_{crs}*)*

_{π}+*O(n)*

dim(π)→*I*_{dim}(µ* _{π}*) :=

Z

∆

(y−*x)dµ*

*crs*(π)→
Z

∆^{2}

**1[x**1 *<x*2*<y*1 *<y*2]*dµ**π*(x1*,y*1)*dµ**π*(x2*,y*2)

H(µ) := 1

2−2I_{dim}(µ) +*I** _{crs}*(µ)

A superplancherel measure associated to set partitions and its limit supercharacter theory

SPl* _{n}*(χ

*) = 1*

^{π}*q*

^{n(n−1)}^{2}

*q*2 dim(π)−2a(π)

(q−1)^{a(π)}*q** ^{crs(π)}* =

exp

−n^{2}log*q*
1

2−2I_{dim}(µ* _{π}*) +

*I*

*(µ*

_{crs}*)*

_{π}+*O(n)*

dim(π)→*I*_{dim}(µ* _{π}*) :=

Z

∆

(y−*x)dµ*
*crs*(π)→

Z

∆^{2}

**1[x**1 *<x*2*<y*1 *<y*2]*dµ**π*(x1*,y*1)*dµ**π*(x2*,y*2)

H(µ) := 1

2−2I_{dim}(µ) +*I** _{crs}*(µ)

A superplancherel measure associated to set partitions and its limit supercharacter theory

SPl* _{n}*(χ

*) = 1*

^{π}*q*

^{n(n−1)}^{2}

*q*2 dim(π)−2a(π)

(q−1)^{a(π)}*q** ^{crs(π)}* =

exp

−n^{2}log*q*
1

2−2I_{dim}(µ* _{π}*) +

*I*

*(µ*

_{crs}*)*

_{π}+*O(n)*

dim(π)→*I*_{dim}(µ* _{π}*) :=

Z

∆

(y−*x)dµ*
*crs*(π)→

Z

∆^{2}

**1[x**1 *<x*2*<y*1 *<y*2]*dµ**π*(x1*,y*1)*dµ**π*(x2*,y*2)

H(µ) := 1

2−2Idim(µ) +*I**crs*(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

*I**crs*(µ*π*) =
Z

∆^{2}

**1[x**1*<x*2 *<y*1*<y*2]*dµ**π*(x1*,y*1)*dµ**π*(x2*,y*2)

Proposition

If*µ*has mass in and*µ* has uniform marginals then
*I**crs*(µ) = 0⇔*µ*= = Ω

A superplancherel measure associated to set partitions and its limit supercharacter theory

## Playing with crossings

*I**crs*(µ*π*) =
Z

∆^{2}

**1[x**1*<x*2 *<y*1*<y*2]*dµ**π*(x1*,y*1)*dµ**π*(x2*,y*2)

Proposition

If*µ*has mass in and*µ* has uniform marginals then
*I**crs*(µ) = 0⇔*µ*= = Ω

A superplancherel measure associated to set partitions and its limit supercharacter theory

## Summary

H(µ) = 0⇔

*µ*has uniform marginals

*µ*inside the top left square
*I** _{crs}*(µ) = 0

⇔*µ*= Ω

The last step (technical) is to prove that*µ** _{π}*(n) →Ω iff
H(µ

*(n))→ H(Ω) = 0.*

_{π}A superplancherel measure associated to set partitions and its limit supercharacter theory

## Summary

H(µ) = 0⇔

*µ*has uniform marginals
*µ*inside the top left square

*I** _{crs}*(µ) = 0

⇔*µ*= Ω

The last step (technical) is to prove that*µ** _{π}*(n) →Ω iff
H(µ

*(n))→ H(Ω) = 0.*

_{π}A superplancherel measure associated to set partitions and its limit supercharacter theory

## Summary

H(µ) = 0⇔

*µ*has uniform marginals
*µ*inside the top left square
*I** _{crs}*(µ) = 0

⇔*µ*= Ω

The last step (technical) is to prove that*µ** _{π}*(n) →Ω iff
H(µ

*(n))→ H(Ω) = 0.*

_{π}A superplancherel measure associated to set partitions and its limit supercharacter theory

## Summary

H(µ) = 0⇔

*µ*has uniform marginals
*µ*inside the top left square
*I** _{crs}*(µ) = 0

⇔*µ*= Ω

The last step (technical) is to prove that*µ** _{π}*(n) →Ω iff
H(µ

*(n))→ H(Ω) = 0.*

_{π}A superplancherel measure associated to set partitions and its limit supercharacter theory

## Summary

H(µ) = 0⇔

*µ*has uniform marginals
*µ*inside the top left square
*I** _{crs}*(µ) = 0

⇔*µ*= Ω

The last step (technical) is to prove that*µ** _{π}*(n) →Ω iff
H(µ

*(n))→ H(Ω) = 0.*

_{π}