What is... a supercharacter? Dario De Stavola 11 October 2016 What - - PowerPoint PPT Presentation

What is... a supercharacter?

What is... a supercharacter?

Dario De Stavola 11 October 2016

What is... a supercharacter? Basic representation theory

G finite group, V finite dimensional C- vector space of dimension d

What is... a supercharacter? Basic representation theory

G finite group, V finite dimensional C- vector space of dimension d X : G → GL(V ) is a C-linear representation.

What is... a supercharacter? Basic representation theory

G finite group, V finite dimensional C- vector space of dimension d X : G → GL(V ) is a C-linear representation. Fix a basis for V ∼ = Cd, then X : G → GLd(C).

What is... a supercharacter? Basic representation theory

Examples

The trivial representation: ∀g ∈ G, X(g) = 1d;

What is... a supercharacter? Basic representation theory

Examples

The trivial representation: ∀g ∈ G, X(g) = 1d; G = Z/4Z = {0, 1, 2, 3}, V = C, GL(V ) = C×, then

What is... a supercharacter? Basic representation theory

Examples

The trivial representation: ∀g ∈ G, X(g) = 1d; G = Z/4Z = {0, 1, 2, 3}, V = C, GL(V ) = C×, then 1 = X(0) = X(1 + 1 + 1 + 1) = X(1)4

What is... a supercharacter? Basic representation theory

Examples

The trivial representation: ∀g ∈ G, X(g) = 1d; G = Z/4Z = {0, 1, 2, 3}, V = C, GL(V ) = C×, then 1 = X(0) = X(1 + 1 + 1 + 1) = X(1)4 X(1) ∈ {1, i, −1, −i}

What is... a supercharacter? Basic representation theory

Examples

The trivial representation: ∀g ∈ G, X(g) = 1d; G = Z/4Z = {0, 1, 2, 3}, V = C, GL(V ) = C×, then 1 = X(0) = X(1 + 1 + 1 + 1) = X(1)4 X(1) ∈ {1, i, −1, −i} X(1) = i, X(2) = −1, X(3) = −i, X(0) = 1.

What is... a supercharacter? Basic representation theory

Examples

G = Z/2Z × Z/2Z, X : G → GL2(C)

What is... a supercharacter? Basic representation theory

Examples

G = Z/2Z × Z/2Z, X : G → GL2(C) X(0, 0) =

• 1

1

• ,

X(1, 0) =

• −i

i

• ,

X(0, 1) =

• −1

−1

• ,

X(1, 1) =

• i

−i

• .

What is... a supercharacter? Basic representation theory

Irreducibles

X : G → GLd(C), X(g): V → V

What is... a supercharacter? Basic representation theory

Irreducibles

X : G → GLd(C), X(g): V → V W ≤ V ⇒ X(g)|W : W → V

What is... a supercharacter? Basic representation theory

Irreducibles

X : G → GLd(C), X(g): V → V W ≤ V ⇒ X(g)|W : W → V If X(g)|W : W → W ⇒ XW : G → GL(W ) is a subrepresentation

What is... a supercharacter? Basic representation theory

Irreducibles

X : G → GLd(C), X(g): V → V W ≤ V ⇒ X(g)|W : W → V If X(g)|W : W → W ⇒ XW : G → GL(W ) is a subrepresentation if W ≤ V ⇒ V = W ⊕ U

What is... a supercharacter? Basic representation theory

Irreducibles

X : G → GLd(C), X(g): V → V W ≤ V ⇒ X(g)|W : W → V If X(g)|W : W → W ⇒ XW : G → GL(W ) is a subrepresentation if W ≤ V ⇒ V = W ⊕ U Manschke’s theorem U is also a subrepresentation.

What is... a supercharacter? Basic representation theory

Irreducibles

V = W ⊕ U, W , U subrepresentations

What is... a supercharacter? Basic representation theory

Irreducibles

V = W ⊕ U, W , U subrepresentations X(g)(v) = X(g)(w + u) = XW (g)(w) ⊕ XU(g)(u)

What is... a supercharacter? Basic representation theory

Irreducibles

V = W ⊕ U, W , U subrepresentations X(g)(v) = X(g)(w + u) = XW (g)(w) ⊕ XU(g)(u) X(g) ∼ =

• XW (g)

XU(g)

What is... a supercharacter? Basic representation theory

Irreducibles

V = W ⊕ U, W , U subrepresentations X(g)(v) = X(g)(w + u) = XW (g)(w) ⊕ XU(g)(u) X(g) ∼ =

• XW (g)

XU(g)

• We need to study only representations without nontrivial

subrepresentations! These representations are called Irreducible.

What is... a supercharacter? Basic representation theory

Irreducibles

V = W ⊕ U, W , U subrepresentations X(g)(v) = X(g)(w + u) = XW (g)(w) ⊕ XU(g)(u) X(g) ∼ =

• XW (g)

XU(g)

• We need to study only representations without nontrivial

subrepresentations! These representations are called Irreducible. The number of irreducible representations is equal to the number

• f conjugacy classes of G

What is... a supercharacter? Basic representation theory

Characters

X : G → GLd(C) The character χ: G → C is defined as χ(g) := tr(X(g)).

What is... a supercharacter? Basic representation theory

Characters

X : G → GLd(C) The character χ: G → C is defined as χ(g) := tr(X(g)). X(0, 0) =

• 1

1

• ,

X(1, 0) =

• −i

i

• ,

X(0, 1) =

• −1

−1

• ,

X(1, 1) =

• i

−i

• .

What is... a supercharacter? Basic representation theory

Characters

X : G → GLd(C) The character χ: G → C is defined as χ(g) := tr(X(g)). X(0, 0) =

• 1

1

• ,

X(1, 0) =

• −i

i

• ,

X(0, 1) =

• −1

−1

• ,

X(1, 1) =

• i

−i

• .

χ(0, 0) = 2, χ(1, 0) = 0, χ(0, 1) = −2, χ(1, 1) = 0.

What is... a supercharacter? Basic representation theory

Properties

χ(1G) = d,

What is... a supercharacter? Basic representation theory

Properties

χ(1G) = d, called the degree of the representation;

What is... a supercharacter? Basic representation theory

Properties

χ(1G) = d, called the degree of the representation; χ(g−1) = χ(g);

What is... a supercharacter? Basic representation theory

Properties

χ(1G) = d, called the degree of the representation; χ(g−1) = χ(g); χ(h−1gh) = χ(g),

What is... a supercharacter? Basic representation theory

Properties

χ(1G) = d, called the degree of the representation; χ(g−1) = χ(g); χ(h−1gh) = χ(g), the character is a class function.

What is... a supercharacter? Basic representation theory

Properties

χ(1G) = d, called the degree of the representation; χ(g−1) = χ(g); χ(h−1gh) = χ(g), the character is a class function. A character is irreducible if it is the trace of an irreducible representation.

What is... a supercharacter? Basic representation theory

Frobenius scalar product

Let φ, ψ: G → C

What is... a supercharacter? Basic representation theory

Frobenius scalar product

Let φ, ψ: G → C φ, ψ := 1 |G|

• g∈G

φ(g)ψ(g) ∈ C

What is... a supercharacter? Basic representation theory

Frobenius scalar product

Let φ, ψ: G → C φ, ψ := 1 |G|

• g∈G

φ(g)ψ(g) ∈ C Irreducible characters are orthonormal w.r.t. this product: if χ1, χ2 are irreducible characters then χ1, χ2 = δ{χ1=χ2}

What is... a supercharacter? Basic representation theory

Class functions

∼: conjugation (we say g ∼ g′ if g′ = h−1gh)

What is... a supercharacter? Basic representation theory

Class functions

∼: conjugation (we say g ∼ g′ if g′ = h−1gh) ClG := {f : G → C class function } ∼ = {f : G/∼ → C} = (G/∼)∗

What is... a supercharacter? Basic representation theory

Class functions

∼: conjugation (we say g ∼ g′ if g′ = h−1gh) ClG := {f : G → C class function } ∼ = {f : G/∼ → C} = (G/∼)∗ Since |G/∼| = |Irr(G)|, and irreducible characters are othonormal w.r.t ·, ·

What is... a supercharacter? Basic representation theory

Class functions

∼: conjugation (we say g ∼ g′ if g′ = h−1gh) ClG := {f : G → C class function } ∼ = {f : G/∼ → C} = (G/∼)∗ Since |G/∼| = |Irr(G)|, and irreducible characters are othonormal w.r.t ·, · Irr(G) is an orthonormal basis for ClG.

What is... a supercharacter? Basic representation theory

Character table

K1 K2 K3 · · · χ1 χ1(K1) χ1(K2) χ1(K3) χ2 χ2(K1) χ2(K2) χ2(K3) χ3 χ3(K1) χ3(K2) χ3(K3) . . . . . .

What is... a supercharacter? Basic representation theory

Character table

K1 K2 K3 · · · χ1 χ1(K1) χ1(K2) χ1(K3) χ2 χ2(K1) χ2(K2) χ2(K3) χ3 χ3(K1) χ3(K2) χ3(K3) . . . . . . This table is orthonormal!

What is... a supercharacter? Basic representation theory

Example

G = Z/3Z

What is... a supercharacter? Basic representation theory

Example

G = Z/3Z 1 2 χ1 1 1 1 χ2 1 e

πi 3

e

2πi 3

χ3 1 e

2πi 3

e

πi 3

What is... a supercharacter? Basic representation theory

X : G → GL(V ), Y : G → GL(V ), χ = tr(X), γ = tr(Y )

What is... a supercharacter? Basic representation theory

X : G → GL(V ), Y : G → GL(V ), χ = tr(X), γ = tr(Y ) X ∼ = Y ⇔ χ = γ

What is... a supercharacter? Basic representation theory

X ∼ = Y ⇔ χ = γ proof [⇒] X ∼ = Y ⇒

What is... a supercharacter? Basic representation theory

X ∼ = Y ⇔ χ = γ proof [⇒] X ∼ = Y ⇒ X(g) ∼ = Y (g) ∀g ⇒

What is... a supercharacter? Basic representation theory

X ∼ = Y ⇔ χ = γ proof [⇒] X ∼ = Y ⇒ X(g) ∼ = Y (g) ∀g ⇒ X(g) = A−1Y (g)A ⇒

What is... a supercharacter? Basic representation theory

X ∼ = Y ⇔ χ = γ proof [⇒] X ∼ = Y ⇒ X(g) ∼ = Y (g) ∀g ⇒ X(g) = A−1Y (g)A ⇒ tr(X(g)) = tr(Y (g)) ∀g

What is... a supercharacter? Basic representation theory

X ∼ = Y ⇔ χ = γ proof [⇐] {(Xi, Vi), i ∈ I} set of irreducible representations (finite!),

What is... a supercharacter? Basic representation theory

X ∼ = Y ⇔ χ = γ proof [⇐] {(Xi, Vi), i ∈ I} set of irreducible representations (finite!), χi the irreducible representations,

What is... a supercharacter? Basic representation theory

X ∼ = Y ⇔ χ = γ proof [⇐] {(Xi, Vi), i ∈ I} set of irreducible representations (finite!), χi the irreducible representations, X ∼ =

• i∈I

niXi Y ∼ =

• i∈I

miXi

What is... a supercharacter? Basic representation theory

X ∼ = Y ⇔ χ = γ proof [⇐] {(Xi, Vi), i ∈ I} set of irreducible representations (finite!), χi the irreducible representations, X ∼ =

• i∈I

niXi Y ∼ =

• i∈I

miXi χ =

• i

niχi γ =

• i

miχi; ni = χ, χi = γ, χi = mi

What is... a supercharacter? Basic representation theory

X ∼ = Y ⇔ χ = γ proof [⇐] {(Xi, Vi), i ∈ I} set of irreducible representations (finite!), χi the irreducible representations, X ∼ =

• i∈I

niXi Y ∼ =

• i∈I

miXi χ =

• i

niχi γ =

• i

miχi; ni = χ, χi = γ, χi = mi X ∼ =

• i∈I

niXi ∼ = Y

What is... a supercharacter? Basic representation theory

Recap

Groups are hard, we map them into group of matrices

What is... a supercharacter? Basic representation theory

Recap

Groups are hard, we map them into group of matrices (X : G → GL(V )) ;

What is... a supercharacter? Basic representation theory

Recap

Groups are hard, we map them into group of matrices (X : G → GL(V )) ; we don’t have to study all representations, only the irreducible

• nes

What is... a supercharacter? Basic representation theory

Recap

Groups are hard, we map them into group of matrices (X : G → GL(V )) ; we don’t have to study all representations, only the irreducible

• nes

 X ∼

=

• i∈I

niXi

  ;

What is... a supercharacter? Basic representation theory

Recap

Groups are hard, we map them into group of matrices (X : G → GL(V )) ; we don’t have to study all representations, only the irreducible

• nes

 X ∼

=

• i∈I

niXi

  ;

knowing the representation is equivalent to knowing the character

What is... a supercharacter? Basic representation theory

Recap

Groups are hard, we map them into group of matrices (X : G → GL(V )) ; we don’t have to study all representations, only the irreducible

• nes

 X ∼

=

• i∈I

niXi

  ;

knowing the representation is equivalent to knowing the character (X ∼ = Y ⇔ χ = γ) .

What is... a supercharacter? supercharacter theory

The upper unitriangular group

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

What is... a supercharacter? supercharacter theory

The upper unitriangular group

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

        

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

        

, ∗ ∈ Fq

What is... a supercharacter? supercharacter theory

The upper unitriangular group

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

        

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

        

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

What is... a supercharacter? supercharacter theory

OK, no irreducible characters. Now what?

      

G =

• g∈G/∼

[g], G is an union of conjugacy classes

What is... a supercharacter? 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. ·, ·}

What is... a supercharacter? 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,

What is... a supercharacter? 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. ·, ·}

What is... a supercharacter? supercharacter theory

OK, no irreducible characters. Now what?

      

G =

• K∈K

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

What is... a supercharacter? 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;

What is... a supercharacter? 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);

What is... a supercharacter? 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;

What is... a supercharacter? 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

What is... a supercharacter? 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) = ∅

What is... a supercharacter? 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

What is... a supercharacter? 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|;

What is... a supercharacter? 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;

What is... a supercharacter? 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.

What is... a supercharacter? supercharacter theory

Examples

1 (K, H) = (G/∼, Irr(G)), trivial;

What is... a supercharacter? supercharacter theory

Examples

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

• χ∈Irr(G)

χ(1)χ − id}), trivial;

What is... a supercharacter? supercharacter theory

Examples

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

• χ∈Irr(G)

χ(1)χ − id}), trivial;

3 suppose A acts on G, φ: A → Aut(G)

What is... a supercharacter? supercharacter theory

Examples

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

• χ∈Irr(G)

χ(1)χ − id}), trivial;

3 suppose A acts on G, φ: A → Aut(G)

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

What is... a supercharacter? supercharacter theory

Examples

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

• χ∈Irr(G)

χ(1)χ − 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

What is... a supercharacter? supercharacter theory

A nice supercharacter theory for Un(Fq)

A = Un(Fq) × Un(Fq) × F×

q acts on Un(Fq):

What is... a supercharacter? supercharacter theory

A nice supercharacter theory for Un(Fq)

A = Un(Fq) × Un(Fq) × F×

q acts on Un(Fq):

φ(g1, g2, t)(h) = 1 + tg1(h − 1)g−1

2

What is... a supercharacter? supercharacter theory

A nice supercharacter theory for Un(Fq)

A = Un(Fq) × Un(Fq) × F×

q acts on Un(Fq):

φ(g1, g2, t)(h) = 1 + tg1(h − 1)g−1

2

             

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

             

,

What is... a supercharacter? supercharacter theory

A nice supercharacter theory for Un(Fq)

A = Un(Fq) × Un(Fq) × F×

q acts on Un(Fq):

φ(g1, g2, t)(h) = 1 + tg1(h − 1)g−1

2

             

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

             

;

What is... a supercharacter? supercharacter theory

A nice supercharacter theory for Un(Fq)

A = Un(Fq) × Un(Fq) × F×

q acts on Un(Fq):

φ(g1, g2, t)(h) = 1 + tg1(h − 1)g−1

2

             

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

What is... a supercharacter? supercharacter theory

For each set partitions π and σ the value of χπ(σ) is explicit!

What is... a supercharacter? supercharacter theory

For each set partitions π and σ the value of χπ(σ) is explicit! χπ(σ) =

  

qdim(π) qnstπ(σ)

• q−1

q

d(π) (

1 1−q)d(π∩σ)

if D(σ) ⊆ Regπ;

• therwise.

What is... a supercharacter? supercharacter theory

Symmetric functions in noncommuting variables

NCSymn(x1, x2, . . .) = symmetric functions in non commuting variables

What is... a supercharacter? supercharacter theory

Symmetric functions in noncommuting variables

NCSymn(x1, x2, . . .) = symmetric functions in non commuting variables Equivalently, the linear span of {mπ}π∈Pn, where Pn= set partitions of {1, . . . , n}, and

What is... a supercharacter? supercharacter theory

Symmetric functions in noncommuting variables

NCSymn(x1, x2, . . .) = symmetric functions in non commuting variables Equivalently, the linear span of {mπ}π∈Pn, where Pn= set partitions of {1, . . . , n}, and m13/24(x1, x2, . . .) = x1x2x1x2 + x2x1x2x1 + x1x3x1x3+ x3x1x3x1 + x2x3x2x3 + x3x2x3x2 + . . . =

• i=j

xixjxixj

What is... a supercharacter? supercharacter theory

Superclass functions

Pn = { set partitions of n} SCln = {f : Pn → C} is an algebra, and {χπ}π∈Pn is an orthogonal basis

What is... a supercharacter? supercharacter theory

Superclass functions

Pn = { set partitions of n} SCln = {f : Pn → C} is an algebra, and {χπ}π∈Pn is an orthogonal basis SCln and NCSymn are isomorphic

What is... a supercharacter? supercharacter theory

Possible directions

Properties of Un(Fq) exposed by this supercharacter theory (random walks on Un(Fq); random statistics for set partitions)

What is... a supercharacter? supercharacter theory

Possible directions

Properties of Un(Fq) exposed by this supercharacter theory (random walks on Un(Fq); random statistics for set partitions) Supercharacter theories in general (how many? Classification? Minimal integral?)

What is... a supercharacter? supercharacter theory

Possible directions

Properties of Un(Fq) exposed by this supercharacter theory (random walks on Un(Fq); random statistics for set partitions) Supercharacter theories in general (how many? Classification? Minimal integral?) Cool identities (involving Ramanujan sums) easily proved (???)

What is... a supercharacter? supercharacter theory

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