• 検索結果がありません。

# A superplancherel measure associated to set partitions and its limit

N/A
N/A
Protected

シェア "A superplancherel measure associated to set partitions and its limit"

Copied!
73
0
0

(1)

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

Affiliation: University of Zürich

(2)

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

## Frobenius scalar product

Letφ, ψ:GC

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).

(3)

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

## Frobenius scalar product

Letφ, ψ:GC

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).

(4)

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

## Frobenius scalar product

Letφ, ψ:GC

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).

(5)

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

## Frobenius scalar product

Letφ, ψ:GC

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).

(6)

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

## Frobenius scalar product

Letφ, ψ:GC

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).

(7)

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|

(8)

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|

(9)

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

(10)

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

(11)

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

(12)

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

## OK, no irreducible characters. Now what?

G = F

gG/

[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}

(13)

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

## OK, no irreducible characters. Now what?

G = F

gG/

[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}

(14)

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

## OK, no irreducible characters. Now what?

G = F

gG/

[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}

(15)

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;

(16)

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;

(17)

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;

(18)

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;

(19)

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 χIrr(G) then ∃!ψ such thathχ, ψi 6= 0. Given a suitableK then His fixed

(and viceversa).

(20)

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 χIrr(G) then ∃!ψ such thathχ, ψi 6= 0. Given a suitableK then His fixed

(and viceversa).

(21)

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 χIrr(G) then ∃!ψ such thathχ, ψi 6= 0. Given a suitableK then His fixed

(and viceversa).

(22)

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 χIrr(G) then∃!ψ such thathχ, ψi 6= 0.

Given a suitableK then His fixed (and viceversa).

(23)

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 χIrr(G) then∃!ψ such thathχ, ψi 6= 0.

Given a suitableK then His fixed (and viceversa).

(24)

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

n

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

(25)

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

n

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

(26)

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

n

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

(27)

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).

(28)

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).

(29)

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).

(30)

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).

(31)

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).

(32)

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)

ji = 12; crs(π) =]crossings ofπ = 1;

(33)

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)

ji = 12; crs(π) =]crossings ofπ = 1;

(34)

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)

ji = 12;

crs(π) =]crossings ofπ = 1;

(35)

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)

ji = 12;

crs(π) =]crossings ofπ = 1;

(36)

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);

π, 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)

(37)

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);

π, 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)

(38)

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);

π, 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)

(39)

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);

π, 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)

(40)

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);

π, 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)

(41)

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→ ∞;

(42)

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→ ∞;

(43)

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→ ∞;

(44)

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→ ∞;

(45)

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

→ ?

(46)

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

→ ?

(47)

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

→ ?

(48)

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

→ ?

(49)

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

→ ?

(50)

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

→ ?

(51)

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

→ ?

(52)

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

→ ?

(53)

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

dc SUB-UNIFORM

(54)

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

dc SUB-UNIFORM

(55)

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

dc SUB-UNIFORM

(56)

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

dc SUB-UNIFORM

(57)

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

(58)

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

(59)

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

(60)

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

SPlnπ) = 1 qn(n−1)2

q2 dim(π)−2a(π)

(q−1)a(π)qcrs(π) =

exp

−n2logq 1

2 −2Idimπ) +Icrsπ)

+O(n)

dim(π)→Idimπ) := Z

(y−x)dµ crs(π)→

Z

2

1[x1 <x2<y1 <y2]π(x1,y1)π(x2,y2)

H(µ) := 1

2−2Idim(µ) +Icrs(µ)

(61)

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

SPlnπ) = 1 qn(n−1)2

q2 dim(π)−2a(π)

(q−1)a(π)qcrs(π) =

exp

−n2logq 1

2−2Idimπ) +Icrsπ)

+O(n)

dim(π)→Idimπ) := Z

(y−x)dµ crs(π)→

Z

2

1[x1 <x2<y1 <y2]π(x1,y1)π(x2,y2)

H(µ) := 1

2−2Idim(µ) +Icrs(µ)

(62)

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

SPlnπ) = 1 qn(n−1)2

q2 dim(π)−2a(π)

(q−1)a(π)qcrs(π) =

exp

−n2logq 1

2−2Idimπ) +Icrsπ)

+O(n)

dim(π)→Idimπ) :=

Z

(y−x)dµ

crs(π)→ Z

2

1[x1 <x2<y1 <y2]π(x1,y1)π(x2,y2)

H(µ) := 1

2−2Idim(µ) +Icrs(µ)

(63)

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

SPlnπ) = 1 qn(n−1)2

q2 dim(π)−2a(π)

(q−1)a(π)qcrs(π) =

exp

−n2logq 1

2−2Idimπ) +Icrsπ)

+O(n)

dim(π)→Idimπ) :=

Z

(y−x)dµ crs(π)→

Z

2

1[x1 <x2<y1 <y2]π(x1,y1)π(x2,y2)

H(µ) := 1

2−2Idim(µ) +Icrs(µ)

(64)

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

SPlnπ) = 1 qn(n−1)2

q2 dim(π)−2a(π)

(q−1)a(π)qcrs(π) =

exp

−n2logq 1

2−2Idimπ) +Icrsπ)

+O(n)

dim(π)→Idimπ) :=

Z

(y−x)dµ crs(π)→

Z

2

1[x1 <x2<y1 <y2]π(x1,y1)π(x2,y2)

H(µ) := 1

2−2Idim(µ) +Icrs(µ)

(65)

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

dim

R

## (y − x ) dµ

7→

µ φ(µ)

(66)

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

## Playing with crossings

Icrsπ) = Z

2

1[x1<x2 <y1<y2]π(x1,y1)π(x2,y2)

Proposition

Ifµhas mass in andµ has uniform marginals then Icrs(µ) = 0⇔µ= = Ω

(67)

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

## Playing with crossings

Icrsπ) = Z

2

1[x1<x2 <y1<y2]π(x1,y1)π(x2,y2)

Proposition

Ifµhas mass in andµ has uniform marginals then Icrs(µ) = 0⇔µ= = Ω

(68)

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.

(69)

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.

(70)

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.

(71)

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.

(72)

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.

, 6, then L(7) 6= 0; the origin is a fine focus of maximum order seven, at most seven small amplitude limit cycles can be bifurcated from the origin.. Sufficient

Thus, starting with a bivariate function which is a tensor- product of finitely supported totally positive refinable functions, the new functions are obtained by using the

CHANDRA, On the degree of approximation of a class of functions by means of Fourier series, Acta Math. CHANDRA, A note on the degree of approximation of continuous functions,

Then the strongly mixed variational-hemivariational inequality SMVHVI is strongly (resp., weakly) well posed in the generalized sense if and only if the corresponding inclusion

In the proofs of these assertions, we write down rather explicit expressions for the bounds in order to have some qualitative idea how to achieve a good numerical control of the

We note that in the case m = 1, the class K 1,n (D) properly contains the classical Kato class K n (D) introduced in [1] as the natural class of singular functions which replaces the

A class F of real or complex valued functions is said to be inverse closed if 1/f remains in the class whenever f is in the class and it does not vanish, and it is said to

Isaacs generalized Andr´e’s theory to the notion of a su- percharacter theory for arbitrary finite groups, where irreducible characters are replaced by supercharacters and