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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