Representations of S_n

Goal: Partitions Irred. Reps.

Integer

Partitions

Natural:

cycle types

Order:

decreasing

Young diagrams

Domination:

Anti - Symmetric

(λ ≥ µ) and (µ ≥ λ)

→ µ = λ

Reflexive

Transitive

Best Tetris Block?

S3, Represent!

Sprecht mit Ihrer Seele.

Tableaux, Tabloids

Equivalence ~ :

row elements

Tabloids:

[t] Є T λ

Well-defined:

σ[t] = [σt]

Column Stabilizers

Synthetic Vector Spaces

¢T λ

Permutation representations

φ λ : Sn → GL ( ¢T λ )

Polytabloids

Linear Operator

At : ¢T λ → ¢T λ

Polytabloid

et = At[t]

Sprecht Representations

Represent polytabloids

φ λ (σ) et = e σ t

Subspace

S λ = span {et}

Represent (Sprecht!)

ψ λ : Sn → GL ( S λ )

Properties of At

At [s] = ± et

Im(At) = ¢et

The Irreducible Mr. Sprecht

S λ : no “good” subspaces

(proper, nonzero, Sn-invariant)

ψ λ : irreducible

(by definition)

A Schur Win

T Є Hom(φ λ, φ λ)

→ T|S λ = kI

→ dim(Hom(ψ λ, φ λ)) = 1

Hom(ψ λ, φ λ) ≠ 0 → λ ≥ µ

(needed for final theorem)

Sprecht : a Complete Set

Tetris …

Treats PTSD

Represents all symmetries