@let@token representation theories of the symmetric group ... · representation theories of the...

Representation Theories of the Symmetric Group and the Rook Monoid

Representation Theories of the Symmetric

Group and the Rook Monoid

Tânia Soa Zaragoza Cotrim Silva

Young Women in Representation Theory

Mathematical Institute of the University of Bonn

June 24, 2016

The Symmetric Group

Introduction

The special case of the Symmetric Group

n.o of isoclasses of Irr. Rep.of G = n.o of conjugacy classes in G

Two permutations are conjugate if and only if they have the same

cycle type

n.o of isoclasses of Irr. Rep. of Sn = n.o of partitions of n

The Symmetric Group

Introduction

cycle type

The Symmetric Group

Introduction

cycle type

The Symmetric Group

Introduction

Isoclasses of Irreducible Representations of Sn

(1) n = 1

(1,1)(2) n = 2

(2,1)(3) (1,1,1) n = 3

(3,1) (2,2) (2,1,1)(4) (1,1,1,1) n = 4

(5) (4,1) (3,2) (3,1,1) (2,2,1) (2,1,1,1) (1,1,1,1,1) n = 5

The Symmetric Group

Introduction

(1) n = 1

(1,1)(2) n = 2

(2,1)(3) (1,1,1) n = 3

(3,1) (2,2) (2,1,1)(4) (1,1,1,1) n = 4

(5) (4,1) (3,2) (3,1,1) (2,2,1) (2,1,1,1) (1,1,1,1,1) n = 5

The Symmetric Group

Introduction

(1) n = 1

(1,1)(2) n = 2

(2,1)(3) (1,1,1) n = 3

(3,1) (2,2) (2,1,1)(4) (1,1,1,1) n = 4

(5) (4,1) (3,2) (3,1,1) (2,2,1) (2,1,1,1) (1,1,1,1,1) n = 5

The Symmetric Group

Introduction

(1) n = 1

(1,1)(2) n = 2

(2,1)(3) (1,1,1) n = 3

(3,1) (2,2) (2,1,1)(4) (1,1,1,1) n = 4

(5) (4,1) (3,2) (3,1,1) (2,2,1) (2,1,1,1) (1,1,1,1,1) n = 5

The Symmetric Group

Introduction

(1) n = 1

(1,1)(2) n = 2

(2,1)(3) (1,1,1) n = 3

(3,1) (2,2) (2,1,1)(4) (1,1,1,1) n = 4

(5) (4,1) (3,2) (3,1,1) (2,2,1) (2,1,1,1) (1,1,1,1,1) n = 5

The Symmetric Group

Introduction

IrrRep(Sn)

construction

cardinali

Young diagrams

The Symmetric Group

Introduction

IrrRep(Sn)

construction

cardinali

Young diagrams

The Symmetric Group

Classic Approach

partition of n

←→

Specht module

←→

classe of irreducible

Sn−representations

Young Symmetrizer V (2,1)

Young diagram of

the partition(2,1)Irreducible representation

The Symmetric Group

Classic Approach

partition of n

←→

Specht module

←→

classe of irreducible

Sn−representations

Young Symmetrizer V (2,1)

Young diagram of

the partition(2,1)Irreducible representation

The Symmetric Group

Classic Approach

Constructing the Specht Modules of S3λ1 = (3) λ2 = (2, 1) λ3 = (1, 1, 1)

1 2 31 23

Rλ1 = S3

Cλ1 = (1)Rλ2 = (1), (1 2)Cλ3 = (1), (1 3)

Rλ3 = (1)Cλ3 = S3

Young symmetrizers

sλi =

( ∑σ∈Rλi

)( ∑σ∈Cλi

sign(σ)σ

)∈ CS3

CS3 · sλ1 CS3 · sλ2 CS3 · sλ3C∑

σ∈S3σ Csλ2+Cx C

∑σ∈S3

sgn(σ)σ

The Symmetric Group

Classic Approach

1 2 31 23

Rλ1 = S3

Cλ1 = (1)Rλ2 = (1), (1 2)Cλ3 = (1), (1 3)

Rλ3 = (1)Cλ3 = S3

Young symmetrizers

sλi =

( ∑σ∈Rλi

)( ∑σ∈Cλi

sign(σ)σ

)∈ CS3

∑σ∈S3

sgn(σ)σ

The Symmetric Group

Classic Approach

1 2 31 23

Rλ1 = S3

Cλ1 = (1)Rλ2 = (1), (1 2)Cλ3 = (1), (1 3)

Rλ3 = (1)Cλ3 = S3

Young symmetrizers

sλi =

( ∑σ∈Rλi

)( ∑σ∈Cλi

sign(σ)σ

)∈ CS3

∑σ∈S3

sgn(σ)σ

The Symmetric Group

Classic Approach

1 2 31 23

Rλ1 = S3

Cλ1 = (1)Rλ2 = (1), (1 2)Cλ3 = (1), (1 3)

Rλ3 = (1)Cλ3 = S3

Young symmetrizers

sλi =

( ∑σ∈Rλi

)( ∑σ∈Cλi

sign(σ)σ

)∈ CS3

∑σ∈S3

sgn(σ)σ

The Symmetric Group

Classic Approach

1 2 31 23

Rλ1 = S3

Cλ1 = (1)Rλ2 = (1), (1 2)Cλ3 = (1), (1 3)

Rλ3 = (1)Cλ3 = S3

Young symmetrizers

sλi =

( ∑σ∈Rλi

)( ∑σ∈Cλi

sign(σ)σ

)∈ CS3

∑σ∈S3

sgn(σ)σ

The Symmetric Group

Classic Approach

Theorem

When λ ranges over all distinct partitions of n, CSn · sλ is a full

set of non-isomorphic simple CSn-modules.

The Symmetric Group

Classic Approach

The Symmetric Group

Dierent Approach

Branching Graph

Theorem

Let V be a simple CSn-module.

The restrictionV |Sn−1 is multiplicity-free.

The Symmetric Group

Dierent Approach

Branching Graph

Theorem

The Symmetric Group

Dierent Approach

Branching Graph

Theorem

The Symmetric Group

Dierent Approach

Branching Graph

Theorem

The Symmetric Group

Dierent Approach

Branching Graph

Theorem

The Symmetric Group

Dierent Approach

Branching Graph

Theorem

The Symmetric Group

Dierent Approach

GZ-basis

weights

contents

Inductive Family

Coxeter Groups

The Symmetric Group

Dierent Approach

GZ-basis

weights

contents

Inductive Family

Coxeter Groups

The Symmetric Group

Dierent Approach

Example: The standard representation of S4

S4Permutation matrices

ρ : S4 → GL(V )

where for all σ ∈ S4

ρ(σ)(x1, x2, x3, x4) = (xσ−1(1), xσ−1(2), xσ−1(3), xσ−1(4)).

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

ρ is an irreducible representation of S4

The Symmetric Group

Dierent Approach

ρ : S4 → GL(V )

ρ(σ)(x1, x2, x3, x4) = (xσ−1(1), xσ−1(2), xσ−1(3), xσ−1(4)).

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

The Symmetric Group

Dierent Approach

ρ : S4 → GL(V )

ρ(σ)(x1, x2, x3, x4) = (xσ−1(1), xσ−1(2), xσ−1(3), xσ−1(4)).

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

The Symmetric Group

Dierent Approach

ρ : S4 → GL(V )

ρ(σ)(x1, x2, x3, x4) = (xσ−1(1), xσ−1(2), xσ−1(3), xσ−1(4)).

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

The Symmetric Group

Dierent Approach

ρ : S4 → GL(V )

ρ(σ)(x1, x2, x3, x4) = (xσ−1(1), xσ−1(2), xσ−1(3), xσ−1(4)).

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

The Symmetric Group

Dierent Approach

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

S4S3S2S1

The Symmetric Group

Dierent Approach

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

S3S2S1

The Symmetric Group

Dierent Approach

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

The Symmetric Group

Dierent Approach

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

S4S3S2

The Symmetric Group

Dierent Approach

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

S4S3S2S1

The Symmetric Group

Dierent Approach

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

S4S3S2S1

The Symmetric Group

Dierent Approach

Branching Graph

u, v ,w

u v ,w

The Symmetric Group

Dierent Approach

Branching Graph

n = 4u, v ,w

u v ,w

The Symmetric Group

Dierent Approach

Branching Graph

n = 4u, v ,w

u v ,w

The Symmetric Group

Dierent Approach

Branching Graph

n = 4u, v ,w

u v ,w

The Symmetric Group

Dierent Approach

Branching Graph

n = 4u, v ,w

u v ,w

The Symmetric Group

Dierent Approach

Proposition

All simple CSn-module V has a canonical decomposition

V =⊕T

indexed by all possible chains

T = λ1 . . . λn

where λi ∈ S∧n , V ∈ λn and all V T are simple S1-modules.

Gelfand-Zetlin Basis

For each T we can choose a vector vT form each V T obtaining a

basis vT of V ,which we call Gelfand-Zetlin basis.

The Symmetric Group

Dierent Approach

Proposition

All simple CSn-module V has a canonical decomposition

V =⊕T

indexed by all possible chains

T = λ1 . . . λn

where λi ∈ S∧n , V ∈ λn and all V T are simple S1-modules.

Gelfand-Zetlin Basis

For each T we can choose a vector vT form each V T obtaining a

basis vT of V ,which we call Gelfand-Zetlin basis.

The Symmetric Group

Dierent Approach

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

V = <(1, 1, 1,−3)>⊕<(1, 1,−2, 0)>⊕<(1,−1, 0, 0)>.

Let u = (1, 1, 1,−3), v = (1, 1,−2, 0), w = (1,−1, 0, 0) and x

the basis GZ = u, v ,w for V .

(1) (1 2) (2 3) (3 4)1 0 0

0 0 −1

0 −12

43 0 0

0 13 1

The Symmetric Group

Dierent Approach

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

V = <(1, 1, 1,−3)>⊕<(1, 1,−2, 0)>⊕<(1,−1, 0, 0)>.

Let u = (1, 1, 1,−3), v = (1, 1,−2, 0), w = (1,−1, 0, 0) and x

the basis GZ = u, v ,w for V .

(1) (1 2) (2 3) (3 4)1 0 0

0 0 −1

0 −12

43 0 0

0 13 1

The Symmetric Group

Dierent Approach

V = (x1, x2, x3, x4) ∈ C4 : x1 + x2 + x3 + x4 = 0

J4 u v w

J1 = 0

J2 = (1 2)J3 = (1 3) + (2 3)J4 = (1 4) + (2 4) + (3 4)Jucys-Murphy elements

The Symmetric Group

Dierent Approach

0 (1 2) (1 3) + (2 3) (1 4) + (2 4) + (3 4)0 0 0

0 0 −1

0 −1 0

−1 0 0

.γ(u) = (0, 1, 2,−1)

γ(v) = (0, 1,−1, 2)

γ(w) = (0,−1, 1, 2)

weights

The Symmetric Group

Dierent Approach

0 (1 2) (1 3) + (2 3) (1 4) + (2 4) + (3 4)0 0 0

0 0 −1

0 −1 0

−1 0 0

.γ(u) = (0, 1, 2,−1)

γ(v) = (0, 1,−1, 2)

γ(w) = (0,−1, 1, 2)

weights

The Symmetric Group

Dierent Approach

Theorem

The Symmetric Group

Dierent Approach

The Symmetric Group

Dierent Approach

Example

→ → →

!1 2 34

→ → → !1 2 43

→ → → !1 3 42

The Symmetric Group

Dierent Approach

Example

→ → → !1 2 34

→ → → !1 2 43

→ → → !1 3 42

The Symmetric Group

Dierent Approach

Example

→ → → !1 2 34

→ → →

!1 2 43

→ → → !1 3 42

The Symmetric Group

Dierent Approach

Example

→ → → !1 2 34

→ → → !1 2 43

→ → → !1 3 42

The Symmetric Group

Dierent Approach

Example

→ → → !1 2 34

→ → → !1 2 43

→ → →

!1 3 42

The Symmetric Group

Dierent Approach

Example

→ → → !1 2 34

→ → → !1 2 43

→ → → !1 3 42

The Symmetric Group

Dierent Approach

Example

Consider

Q1 = 1 2 34

, Q2 = 1 2 43

and Q3 = 1 3 42

We have:

δ(Q1) = (0, 1, 2,−1)

δ(Q2) = (0, 1,−1, 2)

δ(Q3) = (0,−1, 1, 2)

contents = weights

γ(u) = (0, 1, 2,−1)

γ(v) = (0, 1,−1, 2)

γ(w) = (0,−1, 1, 2)

The Symmetric Group

Dierent Approach

Example

Consider

Q1 = 1 2 34

, Q2 = 1 2 43

and Q3 = 1 3 42

We have:

δ(Q1) = (0, 1, 2,−1)

δ(Q2) = (0, 1,−1, 2)

δ(Q3) = (0,−1, 1, 2)

contents = weights

γ(u) = (0, 1, 2,−1)

γ(v) = (0, 1,−1, 2)

γ(w) = (0,−1, 1, 2)

The Symmetric Group

Dierent Approach

Example

Consider

Q1 = 1 2 34

, Q2 = 1 2 43

and Q3 = 1 3 42

We have:

δ(Q1) = (0, 1, 2,−1)

δ(Q2) = (0, 1,−1, 2)

δ(Q3) = (0,−1, 1, 2)

contents

= weights

γ(u) = (0, 1, 2,−1)

γ(v) = (0, 1,−1, 2)

γ(w) = (0,−1, 1, 2)

The Symmetric Group

Dierent Approach

Example

Consider

Q1 = 1 2 34

, Q2 = 1 2 43

and Q3 = 1 3 42

We have:

δ(Q1) = (0, 1, 2,−1)

δ(Q2) = (0, 1,−1, 2)

δ(Q3) = (0,−1, 1, 2)

contents = weights

γ(u) = (0, 1, 2,−1)

γ(v) = (0, 1,−1, 2)

γ(w) = (0,−1, 1, 2)

The Symmetric Group

Dierent Approach

The Spectrum and the Resume

Theorem

Let Ω be the set of all elements

x = (x1, . . . , xn) ∈ Cn

which verify the following properties:

1. x1 = 0;

2. xi − 1, xi + 1 ∩ x1, . . . , xi−1 6= ∅, ∀i ∈ 2, . . . , n;3. If xi = xj = a, for some i < j , then

a−1, a + 1 ⊆ xi+1, . . . , xj−1.

We have

Ω = En = Rn.

The Symmetric Group

Dierent Approach

The Spectrum and the Resume

Theorem

Let Ω be the set of all elements

x = (x1, . . . , xn) ∈ Cn

which verify the following properties:

1. x1 = 0;

2. xi − 1, xi + 1 ∩ x1, . . . , xi−1 6= ∅, ∀i ∈ 2, . . . , n;3. If xi = xj = a, for some i < j , then

a−1, a + 1 ⊆ xi+1, . . . , xj−1.

We have

Ω = En = Rn.

The Rook Monoid

Introduction

Finite Semigroups Representation Theory

S Finite Semigroup

J1 J2 . . . JsU(S)

Regular J -classes

GJ1 GJ2. . . GJs Maximal Subgroups

Irreducible Representations

The Rook Monoid

Introduction

S Finite Semigroup

J1 J2 . . . JsU(S)

Regular J -classes

The Rook Monoid

Introduction

S Finite Semigroup

J1 J2 . . . JsU(S)

Regular J -classes

The Rook Monoid

Introduction

S Finite Semigroup

J1 J2 . . . JsU(S)

Regular J -classes

The Rook Monoid

Introduction

Theorem[Cliord,Munn,Ponizovskii]

The number of irreducible representations of S (up to isomorphism)

is equal to the number of irreducible representations of its maximal

subgroups GJ , with J ∈ U(S).

The Rook Monoid

Introduction

The Rook Monoid

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 0 1 0 0

1 0 0 0 0

0 0 0 0 0

0 0 0 1 0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

1 2 3 4 5

[2 1] [2 1 3][5 4] (1)(2)(3)(4)(5)

The Rook Monoid

Introduction

The Rook Monoid

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 0 1 0 0

1 0 0 0 0

0 0 0 0 0

0 0 0 1 0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

1 2 3 4 5

[2 1] [2 1 3][5 4] (1)(2)(3)(4)(5)

The Rook Monoid

Introduction

The Rook Monoid

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 0 1 0 0

1 0 0 0 0

0 0 0 0 0

0 0 0 1 0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

1 2 3 4 5

[2 1] [2 1 3][5 4] (1)(2)(3)(4)(5)

The Rook Monoid

Classical Approach

01 2 3

(1) [1 2] [1 3] [2 1] (2) [2 3] [3 1] [3 2] (3)1 2 3

(1)(2) (1)[2 3] (1)[3 2] (1)(3) (1 2) [1 2 3] [3 1 2] [1 2](3) [2 1 3]1 2 3

(2)[1 3] (1 3) [1 3 2] [3 2 1] (3)[2 1] (2)[3 1] (2)(3) (1)[2 3] (2 3)1 2 3

(1)(2)(3) (1)(2 3) (1 2)(3) (1 2 3) (1 3 2) (1 3)(2)

The Rook Monoid

Classical Approach

The special case of the Rook Monoid

n.o of isoclasses of Irr. Rep. = sum of the n.o of isoclasses of Irr.

Rep of its maximal subgroup GJ

The list of the maximal subgroups GJ of In will be isomorphic to

S0,S1, . . . ,Sn.

|IrrRep(In)| =n∑

|IrrRep(Sk)|

The Rook Monoid

Classical Approach

S0,S1, . . . ,Sn.

|IrrRep(In)| =n∑

|IrrRep(Sk)|

The Rook Monoid

Classical Approach

S0,S1, . . . ,Sn.

|IrrRep(In)| =n∑

|IrrRep(Sk)|

The Rook Monoid

Classical Approach

Isoclasses of Irr. Rep. of In

∅ n = 0

∅ n = 1

∅ n = 2

∅ n = 3

The Rook Monoid

Classical Approach

∅ n = 0

∅ n = 1

∅ n = 2

∅ n = 3

The Rook Monoid

Classical Approach

∅ n = 0

∅ n = 1

∅ n = 2

∅ n = 3

The Rook Monoid

Classical Approach

∅ n = 0

∅ n = 1

∅ n = 2

∅ n = 3

The Rook Monoid

Dierent Approach

01 2 3

(1) [1 2] [1 3] [2 1] (2) [2 3] [3 1] [3 2] (3)1 2 3

(1)(2) (1)[2 3] (1)[3 2] (1)(3) (1 2) [1 2 3] [3 1 2] [1 2](3) [2 1 3]1 2 3

(2)[1 3] (1 3) [1 3 2] [3 2 1] (3)[2 1] (2)[3 1] (2)(3) (1)[2 3] (2 3)1 2 3

(1)(2)(3) (1)(2 3) (1 2)(3) (1 2 3) (1 3 2) (1 3)(2)

The Rook Monoid

Dierent Approach

rank 0 rank 1 rank 2 rank 31 2 3

ε∅ ε1 ε2 ε3 ε1,2 ε1,3 ε2,3 ε1,2,3

η0 = ε∅

η1 = −3 ε∅ + ε1 + ε2 + ε3

η2 = 3 ε∅ − 2 ε1 − 2 ε2 − 2 ε3 + ε1,2 + ε1,3 + ε2,3

η3 = −ε∅ + ε1 + ε2 + ε3 − ε1,2 − ε1,3 − ε2,3 + ε1,2,3

In this case:

CI3 ' M1(CS0)⊕M3(CS1)⊕M3(CS2)⊕M1(CS3)

CIn ' CInη0 ⊕ . . .⊕ CInηn ' M(n0)(CS0)⊕ . . .⊕M(nn)

∴ CIn is semisimple.

The Rook Monoid

Dierent Approach

ε∅ ε1 ε2 ε3 ε1,2 ε1,3 ε2,3 ε1,2,3

η0 = ε∅

η1 = −3 ε∅ + ε1 + ε2 + ε3

η2 = 3 ε∅ − 2 ε1 − 2 ε2 − 2 ε3 + ε1,2 + ε1,3 + ε2,3

η3 = −ε∅ + ε1 + ε2 + ε3 − ε1,2 − ε1,3 − ε2,3 + ε1,2,3

In this case:

CI3 ' M1(CS0)⊕M3(CS1)⊕M3(CS2)⊕M1(CS3)

The Rook Monoid

Dierent Approach

ε∅ ε1 ε2 ε3 ε1,2 ε1,3 ε2,3 ε1,2,3

η0 = ε∅

η1 = −3 ε∅ + ε1 + ε2 + ε3

η2 = 3 ε∅ − 2 ε1 − 2 ε2 − 2 ε3 + ε1,2 + ε1,3 + ε2,3

η3 = −ε∅ + ε1 + ε2 + ε3 − ε1,2 − ε1,3 − ε2,3 + ε1,2,3

In this case:

CI3 ' M1(CS0)⊕M3(CS1)⊕M3(CS2)⊕M1(CS3)

The Rook Monoid

Dierent Approach

ε∅ ε1 ε2 ε3 ε1,2 ε1,3 ε2,3 ε1,2,3

η0 = ε∅

η1 = −3 ε∅ + ε1 + ε2 + ε3

η2 = 3 ε∅ − 2 ε1 − 2 ε2 − 2 ε3 + ε1,2 + ε1,3 + ε2,3

η3 = −ε∅ + ε1 + ε2 + ε3 − ε1,2 − ε1,3 − ε2,3 + ε1,2,3

In this case:

CI3 ' M1(CS0)⊕M3(CS1)⊕M3(CS2)⊕M1(CS3)

References

Main References

I Alexander Kleshchev. Linear and Projective Representations of Symmetric

Groups. Cambridge University Press. 2005.

I Andrei Okounkov, Anatoly Vershik. A New Approach to Repesentation

Theory of Symmetric Groups. Selecta Mathematica, New Series, vol. 2,

No. 4,581-605. 1996.

I Louis Solomon. Representations of the rook monoid. Journal of Algebra

256, 309-342. 2002.

I W.D.Munn. The characters of the symmetric inverse semigroup. Proc.

Cambridge Philos. Soc. 1957.

References

Other References

I James Alexander "Sandy" Green. Polynomial representations of GLn.

Springer. Second edition. 2006.

I Benjamin Steinberg. The Representation Theory of Finite Monoids,

preliminary version. Springer, 2016.

I C. Curtis, I. Reiner. Representation Theory of Finite Groups and

Associative Algebras. American Mathematical Society. 1962.

I William Fulton and Joe Harris. Representation theory, volume 129 of

Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. A rst

course, Readings in Mathematics.

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