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SuperPlancherelsticexpialidocious SuperPlancherelsticexpialidocious a.k.a. The SuperPlancherel measure on set partitions Dario De Stavola 29 November 2016 SuperPlancherelsticexpialidocious Structure of the talk 1 Quick summary on character

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SLIDE 1

SuperPlancherelsticexpialidocious

SuperPlancherelsticexpialidocious

a.k.a. The SuperPlancherel measure on set partitions Dario De Stavola 29 November 2016

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SLIDE 2

SuperPlancherelsticexpialidocious

Structure of the talk

1 Quick summary on character theory;

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SLIDE 3

SuperPlancherelsticexpialidocious

Structure of the talk

1 Quick summary on character theory; 2 problem: the upper unitriangular group;

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SLIDE 4

SuperPlancherelsticexpialidocious

Structure of the talk

1 Quick summary on character theory; 2 problem: the upper unitriangular group; 3 supercharacter theory;

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SLIDE 5

SuperPlancherelsticexpialidocious

Structure of the talk

1 Quick summary on character theory; 2 problem: the upper unitriangular group; 3 supercharacter theory; 4 the superplancherel measure;

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SLIDE 6

SuperPlancherelsticexpialidocious

Structure of the talk

1 Quick summary on character theory; 2 problem: the upper unitriangular group; 3 supercharacter theory; 4 the superplancherel measure; 5 limit shape for set partitions.

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SLIDE 7

SuperPlancherelsticexpialidocious Basic character theory

Frobenius scalar product

Let φ, ψ: G → C

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SLIDE 8

SuperPlancherelsticexpialidocious Basic character theory

Frobenius scalar product

Let φ, ψ: G → C φ, ψ := 1 |G|

  • g∈G

φ(g)ψ(g) ∈ C

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SLIDE 9

SuperPlancherelsticexpialidocious Basic character theory

Frobenius scalar product

Let φ, ψ: G → C φ, ψ := 1 |G|

  • g∈G

φ(g)ψ(g) ∈ C Consider the algebra of class functions of G, endowed with this product:

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SLIDE 10

SuperPlancherelsticexpialidocious Basic character theory

Frobenius scalar product

Let φ, ψ: G → C φ, ψ := 1 |G|

  • 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+.

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SLIDE 11

SuperPlancherelsticexpialidocious Basic character theory

Frobenius scalar product

Let φ, ψ: G → C φ, ψ := 1 |G|

  • 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).

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SLIDE 12

SuperPlancherelsticexpialidocious Basic character theory

The regular character

Define χreg : G → C: χreg(g) =

  • |G|

if g = idG

  • therwise
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SLIDE 13

SuperPlancherelsticexpialidocious Basic character theory

The regular character

Define χreg : G → C: χreg(g) =

  • |G|

if g = idG

  • therwise

χreg =

  • χ∈Irr(G)

χ(1)χ

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SLIDE 14

SuperPlancherelsticexpialidocious Basic character theory

The Plancherel measure

⇒

  • χ∈Irr(G)

χ(1)2 = |G|

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SLIDE 15

SuperPlancherelsticexpialidocious Basic character theory

The Plancherel measure

⇒

  • χ∈Irr(G)

χ(1)2 = |G| Measure on Irr(G) PlG(χ) = χ(1)2 |G|

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SLIDE 16

SuperPlancherelsticexpialidocious supercharacter theory

The upper unitriangular group

Fq the finite field with q elements, q a prime power

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SLIDE 17

SuperPlancherelsticexpialidocious supercharacter theory

The upper unitriangular group

Fq the finite field with q elements, q a prime power Un(Fq) :=

        

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

        

, ∗ ∈ Fq

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SLIDE 18

SuperPlancherelsticexpialidocious supercharacter theory

The upper unitriangular group

Fq the finite field with q elements, q a prime power Un(Fq) :=

        

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

        

, ∗ ∈ Fq Classifying the irreducible representations of Un(Fq) is a WILD problem

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SLIDE 19

SuperPlancherelsticexpialidocious supercharacter theory

OK, no irreducible characters. Now what?

      

G =

  • g∈G/∼

[g], G is an union of conjugacy classes Irr(G) = {χ: G/∼ → C orthonormal w.r.t. ·, ·}

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SLIDE 20

SuperPlancherelsticexpialidocious supercharacter theory

OK, no irreducible characters. Now what?

      

G =

  • g∈G/∼

[g], G is an union of conjugacy classes Irr(G) = {χ: G/∼ → C orthonormal w.r.t. ·, ·} idea:

      

G =

  • K∈K

K,

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SLIDE 21

SuperPlancherelsticexpialidocious supercharacter theory

OK, no irreducible characters. Now what?

      

G =

  • g∈G/∼

[g], G is an union of conjugacy classes Irr(G) = {χ: G/∼ → C orthonormal w.r.t. ·, ·} idea:

      

G =

  • K∈K

K, H = {ψ: K → C orthogonal w.r.t. ·, ·}

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SLIDE 22

SuperPlancherelsticexpialidocious supercharacter theory

OK, no irreducible characters. Now what?

      

G =

  • K∈K

K, H = {ψ: K → C orthogonal w.r.t. ·, ·}

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SLIDE 23

SuperPlancherelsticexpialidocious supercharacter theory

OK, no irreducible characters. Now what?

      

G =

  • K∈K

K, H = {ψ: K → C orthogonal w.r.t. ·, ·} Each K is a union of conjugacy classes;

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SLIDE 24

SuperPlancherelsticexpialidocious supercharacter theory

OK, no irreducible characters. Now what?

      

G =

  • K∈K

K, H = {ψ: K → C orthogonal w.r.t. ·, ·} Each K is a union of conjugacy classes; thus, ψ are class functions (constant on conjugacy classes);

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SLIDE 25

SuperPlancherelsticexpialidocious supercharacter theory

OK, no irreducible characters. Now what?

      

G =

  • K∈K

K, H = {ψ: K → C orthogonal w.r.t. ·, ·} 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;

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SLIDE 26

SuperPlancherelsticexpialidocious supercharacter theory

OK, no irreducible characters. Now what?

      

G =

  • K∈K

K, H = {ψ: K → C orthogonal w.r.t. ·, ·} 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; ψ =

  • χ∈I(ψ)

c(χ)χ, I(ψ) ⊆ Irr(G), c(χ) = 0

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SLIDE 27

SuperPlancherelsticexpialidocious supercharacter theory

OK, no irreducible characters. Now what?

      

G =

  • K∈K

K, H = {ψ: K → C orthogonal w.r.t. ·, ·} 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; ψ =

  • χ∈I(ψ)

c(χ)χ, I(ψ) ⊆ Irr(G), c(χ) = 0 if ψ1 = ψ2 then I(ψ1) ∩ I(ψ2) = ∅

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SLIDE 28

SuperPlancherelsticexpialidocious supercharacter theory

Supercharacter theory

A supercharacter theory is a pair (K, H) where K is a set partition

  • f G and H is an orthogonal set of characters such that
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SLIDE 29

SuperPlancherelsticexpialidocious supercharacter theory

Supercharacter theory

A supercharacter theory is a pair (K, H) where K is a set partition

  • f G and H is an orthogonal set of characters such that

1 |K| = |H|;

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SLIDE 30

SuperPlancherelsticexpialidocious supercharacter theory

Supercharacter theory

A supercharacter theory is a pair (K, H) where K is a set partition

  • f G and H is an orthogonal set of characters such that

1 |K| = |H|; 2 if ψ ∈ H then ψ constant on K, ∀K ∈ K;

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SLIDE 31

SuperPlancherelsticexpialidocious supercharacter theory

Supercharacter theory

A supercharacter theory is a pair (K, H) where K is a set partition

  • f G and H 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 that χ, ψ = 0.

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SLIDE 32

SuperPlancherelsticexpialidocious supercharacter theory

Supercharacter theory

A supercharacter theory is a pair (K, H) where K is a set partition

  • f G and H 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 that χ, ψ = 0.

Given a suitable K then H is fixed (and viceversa).

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SLIDE 33

SuperPlancherelsticexpialidocious supercharacter theory

Examples

1 (K, H) = (G/∼, Irr(G)), trivial;

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SLIDE 34

SuperPlancherelsticexpialidocious supercharacter theory

Examples

1 (K, H) = (G/∼, Irr(G)), trivial; 2 (K, H) = ({1G, G \ {1G}}, {id, χreg − id}), trivial;

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SLIDE 35

SuperPlancherelsticexpialidocious supercharacter theory

Examples

1 (K, H) = (G/∼, Irr(G)), trivial; 2 (K, H) = ({1G, G \ {1G}}, {id, χreg − id}), trivial; 3 suppose A acts on G, φ: A → Aut(G)

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SLIDE 36

SuperPlancherelsticexpialidocious supercharacter theory

Examples

1 (K, H) = (G/∼, Irr(G)), trivial; 2 (K, H) = ({1G, G \ {1G}}, {id, χreg − id}), trivial; 3 suppose A acts on G, φ: A → Aut(G)

the superclasses are {φ(A)([g1]), . . . , φ(A)([gr])}

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SLIDE 37

SuperPlancherelsticexpialidocious supercharacter theory

Examples

1 (K, H) = (G/∼, Irr(G)), trivial; 2 (K, H) = ({1G, G \ {1G}}, {id, χreg − id}), trivial; 3 suppose A acts on G, φ: A → Aut(G)

the superclasses are {φ(A)([g1]), . . . , φ(A)([gr])} but A acts also on Irr(G); call Ω1, . . . , Ωr the orbits, then the supercharacters are

  • χ∈Ωi

χ(1)χ. Brauer This is a supercharacter theory

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SLIDE 38

SuperPlancherelsticexpialidocious supercharacter theory

(Bergeron and Thiem) A supercharacter theory for Un(Fq)

A = Un(Fq) × Un(Fq) × Dn(Fq) acts on Un(Fq):

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SLIDE 39

SuperPlancherelsticexpialidocious supercharacter theory

(Bergeron and Thiem) A supercharacter theory for Un(Fq)

A = Un(Fq) × Un(Fq) × Dn(Fq) acts on Un(Fq): φ(g1, g2, t)(h) = 1 + g1t(h − 1)t−1g−1

2

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SLIDE 40

SuperPlancherelsticexpialidocious supercharacter theory

(Bergeron and Thiem) A supercharacter theory for Un(Fq)

A = Un(Fq) × Un(Fq) × Dn(Fq) acts on Un(Fq): φ(g1, g2, t)(h) = 1 + g1t(h − 1)t−1g−1

2

h =

             

1 1 1 4 3 1 5 2 3 6 1 4 3 1 3 1 4 1 1 1

             

,

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SLIDE 41

SuperPlancherelsticexpialidocious supercharacter theory

(Bergeron and Thiem) A supercharacter theory for Un(Fq)

A = Un(Fq) × Un(Fq) × Dn(Fq) acts on Un(Fq): φ(g1, g2, t)(h) = 1 + g1t(h − 1)t−1g−1

2

h =

             

1 1 1 4 3 1 5 2 3 6 1 4 3 1 3 1 4 1 1 1

             

,

             

1 · ⋆ · · · 1 ⋆ · · · · · 1 · ⋆ 1 · 1 ⋆ · 1 1 1

             

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SLIDE 42

SuperPlancherelsticexpialidocious supercharacter theory

(Bergeron and Thiem) A supercharacter theory for Un(Fq)

A = Un(Fq) × Un(Fq) × Dn(Fq) acts on Un(Fq): φ(g1, g2, t)(h) = 1 + g1t(h − 1)t−1g−1

2

h =

             

1 1 1 4 3 1 5 2 3 6 1 4 3 1 3 1 4 1 1 1

             

,

             

1 · ⋆ · · · 1 ⋆ · · · · · 1 · ⋆ 1 · 1 ⋆ · 1 1 1

             

π = 1 2 3 4 5 6 7 8

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SLIDE 43

SuperPlancherelsticexpialidocious supercharacter theory

Why this supercharacter theory?

1 Superclasses (and supercharacters) are indexed by nice

combinatorial objects;

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SLIDE 44

SuperPlancherelsticexpialidocious supercharacter theory

Why this supercharacter theory?

1 Superclasses (and supercharacters) are indexed by nice

combinatorial objects;

2 the supercharacters have an explicit formula;

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SLIDE 45

SuperPlancherelsticexpialidocious 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;

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SLIDE 46

SuperPlancherelsticexpialidocious 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;

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SLIDE 47

SuperPlancherelsticexpialidocious 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; m13/24(x1, x2, . . .) = x1x2x1x2 + x2x1x2x1 + x1x3x1x3+ x3x1x3x1 + x2x3x2x3 + x3x2x3x2 + . . . =

  • i=j

xixjxixj

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SLIDE 48

SuperPlancherelsticexpialidocious 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; m13/24(x1, x2, . . .) = x1x2x1x2 + x2x1x2x1 + x1x3x1x3+ x3x1x3x1 + x2x3x2x3 + x3x2x3x2 + . . . =

  • i=j

xixjxixj

5 Nice decomposition of the supercharacter table.

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SLIDE 49

SuperPlancherelsticexpialidocious supercharacter theory

Set partitions notation

π = 1 2 3 4 5 6 7 8 Arcs(π) = {(1, 5), (2, 3), (3, 8), (5, 7)};

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SLIDE 50

SuperPlancherelsticexpialidocious 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;

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SLIDE 51

SuperPlancherelsticexpialidocious 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) =

  • (i,j)∈Arcs(π)

j − i = 12;

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SLIDE 52

SuperPlancherelsticexpialidocious 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) =

  • (i,j)∈Arcs(π)

j − i = 12; crs(π) = ♯crossings of π = 1;

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SLIDE 53

SuperPlancherelsticexpialidocious supercharacter theory

The dimension of a supercharacter is χπ(1) = (q − 1)a(π) · qdim(π)−a(π);

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SLIDE 54

SuperPlancherelsticexpialidocious supercharacter theory

The dimension of a supercharacter is χπ(1) = (q − 1)a(π) · qdim(π)−a(π); χπ, χπ = (q − 1)a(π) · qcrs(π);

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SLIDE 55

SuperPlancherelsticexpialidocious supercharacter theory

The dimension of a supercharacter is χπ(1) = (q − 1)a(π) · qdim(π)−a(π); χπ, χπ = (q − 1)a(π) · qcrs(π); The superplancherel measure

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SLIDE 56

SuperPlancherelsticexpialidocious supercharacter theory

The dimension of a supercharacter is χπ(1) = (q − 1)a(π) · qdim(π)−a(π); χπ, χπ = (q − 1)a(π) · qcrs(π); The superplancherel measure SPlG(χ) = 1 |G| χ(1)2 χ, χ

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SLIDE 57

SuperPlancherelsticexpialidocious supercharacter theory

The dimension of a supercharacter is χπ(1) = (q − 1)a(π) · qdim(π)−a(π); χπ, χπ = (q − 1)a(π) · qcrs(π); The superplancherel measure SPlG(χ) = 1 |G| χ(1)2 χ, χ = 1 q

n(n−1) 2

(q − 1)a(π) · q2 dim(π)−2a(π) qcrs(π)

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SLIDE 58

SuperPlancherelsticexpialidocious supercharacter theory

Plan

1 See set partitions as objects of the same space

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SLIDE 59

SuperPlancherelsticexpialidocious supercharacter theory

Plan

1 See set partitions as objects of the same space (some

renormalization happens);

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SLIDE 60

SuperPlancherelsticexpialidocious supercharacter theory

Plan

1 See set partitions as objects of the same space (some

renormalization happens);

2 interpret your statistics w.r.t. this new setting;

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SLIDE 61

SuperPlancherelsticexpialidocious supercharacter theory

Plan

1 See set partitions as objects of the same space (some

renormalization happens);

2 interpret your statistics w.r.t. this new setting; 3 let n → ∞;

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SLIDE 62

SuperPlancherelsticexpialidocious supercharacter theory

First step to look for a limit

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SLIDE 63

SuperPlancherelsticexpialidocious supercharacter theory

First step to look for a limit

→

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SLIDE 64

SuperPlancherelsticexpialidocious supercharacter theory

First step to look for a limit

→ →

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SLIDE 65

SuperPlancherelsticexpialidocious supercharacter theory

First step to look for a limit

→ → →

slide-66
SLIDE 66

SuperPlancherelsticexpialidocious supercharacter theory

First step to look for a limit

→ → →

             

1 · ⋆ · · · 1 ⋆ · · · · · 1 · ⋆ 1 · 1 ⋆ · 1 1 1

             

slide-67
SLIDE 67

SuperPlancherelsticexpialidocious supercharacter theory

First step to look for a limit

→ → →

             

1 · ⋆ · · · 1 ⋆ · · · · · 1 · ⋆ 1 · 1 ⋆ · 1 1 1

             

→

             

1 0 · 0 ⋆ · · · 1 ⋆ · · · · · 1 0 0 0 · ⋆ 1 0 0 · 1 0 ⋆ · 1 0 0 1 0 1

             

slide-68
SLIDE 68

SuperPlancherelsticexpialidocious supercharacter theory

First step to look for a limit

→ → →

             

1 · ⋆ · · · 1 ⋆ · · · · · 1 · ⋆ 1 · 1 ⋆ · 1 1 1

             

→

             

1 0 · 0 ⋆ · · · 1 ⋆ · · · · · 1 0 0 0 · ⋆ 1 0 0 · 1 0 ⋆ · 1 0 0 1 0 1

             

→

1 1

slide-69
SLIDE 69

SuperPlancherelsticexpialidocious supercharacter theory

First step to look for a limit

→ → →

             

1 · ⋆ · · · 1 ⋆ · · · · · 1 · ⋆ 1 · 1 ⋆ · 1 1 1

             

→

             

1 0 · 0 ⋆ · · · 1 ⋆ · · · · · 1 0 0 0 · ⋆ 1 0 0 · 1 0 ⋆ · 1 0 0 1 0 1

             

→

1 1

→ ?

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SLIDE 70

SuperPlancherelsticexpialidocious supercharacter theory

Our setting

∆ =

1 1

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SLIDE 71

SuperPlancherelsticexpialidocious supercharacter theory

Our setting

∆ =

1 1

Γ =

    

measures µ on ∆ s.t.

  • ∆ µ ≤ 1 (subprobability)

µ has sub-uniform marginals

    

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SLIDE 72

SuperPlancherelsticexpialidocious supercharacter theory

Our setting

∆ =

1 1

Γ =

    

measures µ on ∆ s.t.

  • ∆ µ ≤ 1 (subprobability)

µ has sub-uniform marginals

     1 1 a b c d

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SLIDE 73

SuperPlancherelsticexpialidocious supercharacter theory

Our setting

∆ =

1 1

Γ =

    

measures µ on ∆ s.t.

  • ∆ µ ≤ 1 (subprobability)

µ has sub-uniform marginals

     1 1 a b c d y=1

y=0

x=b

x=a

dµ = b − a UNIFORM

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SLIDE 74

SuperPlancherelsticexpialidocious supercharacter theory

Our setting

∆ =

1 1

Γ =

    

measures µ on ∆ s.t.

  • ∆ µ ≤ 1 (subprobability)

µ has sub-uniform marginals

     1 1 a b c d y=1

y=0

x=b

x=a

dµ = b − a UNIFORM

x=1

x=0

y=d

y=c

dµ ≤ d − c SUB-UNIFORM

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SLIDE 75

SuperPlancherelsticexpialidocious supercharacter theory

Theorem (DDS) There exists a measure Ω ∈ Γ such that µπ → Ω almost surely.

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SLIDE 76

SuperPlancherelsticexpialidocious supercharacter theory

Theorem (DDS) There exists a measure Ω ∈ Γ such that µπ → Ω almost surely.

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SLIDE 77

SuperPlancherelsticexpialidocious supercharacter theory

Theorem (DDS) There exists a measure Ω ∈ Γ such that µπ → Ω almost surely. 1 2 3 n

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SLIDE 78

SuperPlancherelsticexpialidocious supercharacter theory

Interpretation of statistics: a(π)

π = 1 2 3 4 5 6 7 8

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SLIDE 79

SuperPlancherelsticexpialidocious supercharacter theory

Interpretation of statistics: a(π)

π = 1 2 3 4 5 6 7 8 µπ =

1 1

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SLIDE 80

SuperPlancherelsticexpialidocious supercharacter theory

Interpretation of statistics: a(π)

π = 1 2 3 4 5 6 7 8 µπ =

1 1

Each box has mass 1

n, there are a(π) boxes, ⇒

a(π) =

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SLIDE 81

SuperPlancherelsticexpialidocious supercharacter theory

Interpretation of statistics: a(π)

π = 1 2 3 4 5 6 7 8 µπ =

1 1

Each box has mass 1

n, there are a(π) boxes, ⇒

a(π) = n

  • ∆

dµπ(x, y)

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SLIDE 82

SuperPlancherelsticexpialidocious supercharacter theory

Interpretation of statistics: dim(π)

π = 1 2 3 4 5 6 7 8

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SLIDE 83

SuperPlancherelsticexpialidocious supercharacter theory

Interpretation of statistics: dim(π)

π = 1 2 3 4 5 6 7 8 µπ =

1 1

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SLIDE 84

SuperPlancherelsticexpialidocious supercharacter theory

Interpretation of statistics: dim(π)

π = 1 2 3 4 5 6 7 8 µπ =

1 1

dim(π) =

  • (i,j)∈Arcs(π)

j − i

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SLIDE 85

SuperPlancherelsticexpialidocious supercharacter theory

Interpretation of statistics: dim(π)

π = 1 2 3 4 5 6 7 8 µπ =

1 1

dim(π) =

  • (i,j)∈Arcs(π)

j − i = n2

  • ∆

(y − x) dµπ(x, y)

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SLIDE 86

SuperPlancherelsticexpialidocious supercharacter theory

Interpretation of statistics: dim(π)

π = 1 2 3 4 5 6 7 8 µπ =

1 1

dim(π) =

  • (i,j)∈Arcs(π)

j − i = n2

  • ∆

(y − x) dµπ(x, y) y ∼ 1 nj x ∼ 1 ni each box has mass 1

n

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SLIDE 87

SuperPlancherelsticexpialidocious supercharacter theory

Interpretation of statistics: crs(π)

π = 1 2 3 4 5 6 7 8

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SLIDE 88

SuperPlancherelsticexpialidocious supercharacter theory

Interpretation of statistics: crs(π)

π = 1 2 3 4 5 6 7 8 µπ =

1 1

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SLIDE 89

SuperPlancherelsticexpialidocious supercharacter theory

Interpretation of statistics: crs(π)

π = 1 2 3 4 5 6 7 8 µπ =

1 1

crs(π) = ♯{(i1, j1), (i2, j2) ∈ Arcs(π) s.t. i1 < i2 < j1 < j2}

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SLIDE 90

SuperPlancherelsticexpialidocious supercharacter theory

Interpretation of statistics: crs(π)

π = 1 2 3 4 5 6 7 8 µπ =

1 1

crs(π) = ♯{(i1, j1), (i2, j2) ∈ Arcs(π) s.t. i1 < i2 < j1 < j2} = n2

  • ∆2 1[x1 < x2 < y1 < y2] dµπ(x1, y1) dµπ(x2, y2) + O(n)
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SLIDE 91

SuperPlancherelsticexpialidocious supercharacter theory

SPln(χπ) = 1 q

n(n−1) 2

q2 dim(π)−2a(π) (q − 1)a(π)qcrs(π) =

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SLIDE 92

SuperPlancherelsticexpialidocious supercharacter theory

SPln(χπ) = 1 q

n(n−1) 2

q2 dim(π)−2a(π) (q − 1)a(π)qcrs(π) = = exp

  • log q
  • −n2

2 + n 2 + log(q − 1) log q a(π) − 2a(π) + 2 dim(π) − crs(π)

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SLIDE 93

SuperPlancherelsticexpialidocious supercharacter theory

SPln(χπ) = 1 q

n(n−1) 2

q2 dim(π)−2a(π) (q − 1)a(π)qcrs(π) = = exp

  • log q
  • −n2

2 + n 2 + log(q − 1) log q a(π) − 2a(π) + 2 dim(π) − crs(π)

  • exp
  • −n2 log q

1

2 − 2Idim(µπ) + Icrs(µπ)

  • + O(n)
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SLIDE 94

SuperPlancherelsticexpialidocious supercharacter theory

exp

  • −n2 log q

1

2 − 2Idim(µπ) + Icrs(µπ)

  • + O(n)
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SLIDE 95

SuperPlancherelsticexpialidocious supercharacter theory

exp

  • −n2 log q

1

2 − 2Idim(µπ) + Icrs(µπ)

  • + O(n)
  • H(µ) := 1

2 − 2Idim(µ) + Icrs(µ)

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SLIDE 96

SuperPlancherelsticexpialidocious supercharacter theory

exp

  • −n2 log q

1

2 − 2Idim(µπ) + Icrs(µπ)

  • + O(n)
  • H(µ) := 1

2 − 2Idim(µ) + Icrs(µ) IDEA Find Ω s.t. H(Ω) = 0 (⇒ Ω candidate limit shape)

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SLIDE 97

SuperPlancherelsticexpialidocious supercharacter theory

Playing with Idim(µ) =

  • ∆(y − x) dµ

Proposition φ: Γ → Γ s.t. φ(µ) has uniform marginals

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SLIDE 98

SuperPlancherelsticexpialidocious supercharacter theory

Playing with Idim(µ) =

  • ∆(y − x) dµ

Proposition φ: Γ → Γ s.t. φ(µ) has uniform marginals Idim(φ(µ)) ≥ Idim(µ)

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SLIDE 99

SuperPlancherelsticexpialidocious supercharacter theory

Playing with Idim(µ) =

  • ∆(y − x) dµ

Proposition φ: Γ → Γ s.t. φ(µ) has uniform marginals Idim(φ(µ)) ≥ Idim(µ) Idim(φ(µ)) = Idim(µ) ⇔ µ has uniform marginals

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SLIDE 100

SuperPlancherelsticexpialidocious supercharacter theory

Playing with Idim(µ) =

  • ∆(y − x) dµ

Proposition φ: Γ → Γ s.t. φ(µ) has uniform marginals Idim(φ(µ)) ≥ Idim(µ) Idim(φ(µ)) = Idim(µ) ⇔ µ has uniform marginals → µ φ(µ)

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SLIDE 101

SuperPlancherelsticexpialidocious supercharacter theory

Playing with Idim(µ) =

  • ∆(y − x) dµ

Proposition φ: Γ → Γ s.t. φ(µ) has uniform marginals Idim(φ(µ)) ≥ Idim(µ) Idim(φ(µ)) = Idim(µ) ⇔ µ has uniform marginals

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SLIDE 102

SuperPlancherelsticexpialidocious supercharacter theory

Playing with Idim(µ) =

  • ∆(y − x) dµ

Proposition φ: Γ → Γ s.t. φ(µ) has uniform marginals Idim(φ(µ)) ≥ Idim(µ) Idim(φ(µ)) = Idim(µ) ⇔ µ has uniform marginals Corollary Idim(µ) is maximal ⇔ µ has uniform marginals µ has all the mass in the top left square

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SLIDE 103

SuperPlancherelsticexpialidocious supercharacter theory

Playing with Idim(µ) =

  • ∆(y − x) dµ

Proposition φ: Γ → Γ s.t. φ(µ) has uniform marginals Idim(φ(µ)) ≥ Idim(µ) Idim(φ(µ)) = Idim(µ) ⇔ µ has uniform marginals Corollary Idim(µ) is maximal ⇔ µ has uniform marginals µ has all the mass in the top left square In this case Idim(µ) = 1

4.

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SuperPlancherelsticexpialidocious supercharacter theory

Playing with crossings

Proposition If µ has mass in and µ has uniform marginals then Icrs(µ) = 0 ⇔ µ = = Ω

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SLIDE 105

SuperPlancherelsticexpialidocious supercharacter theory

Summary

H(µ) = 0 ⇔ µ has uniform marginals

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SLIDE 106

SuperPlancherelsticexpialidocious supercharacter theory

Summary

H(µ) = 0 ⇔ µ has uniform marginals µ inside the top left square

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SLIDE 107

SuperPlancherelsticexpialidocious supercharacter theory

Summary

H(µ) = 0 ⇔ µ has uniform marginals µ inside the top left square Icrs(µ) = 0

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SLIDE 108

SuperPlancherelsticexpialidocious supercharacter theory

Summary

H(µ) = 0 ⇔ µ has uniform marginals µ inside the top left square Icrs(µ) = 0 ⇔ µ = Ω

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SuperPlancherelsticexpialidocious supercharacter theory

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