representations of s_n
TRANSCRIPT
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Goal: Partitions Irred. Reps.
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Integer
Partitions
Natural:
cycle types
Order:
decreasing
Young diagrams
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Domination:
Anti - Symmetric
(λ ≥ µ) and (µ ≥ λ)
→ µ = λ
Reflexive
Transitive
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Best Tetris Block?
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S3, Represent!
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Sprecht mit Ihrer Seele.
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Tableaux, Tabloids
Equivalence ~ :
row elements
Tabloids:
[t] Є T λ
Well-defined:
σ[t] = [σt]
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Column Stabilizers
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Synthetic Vector Spaces
¢T λ
Permutation representations
φ λ : Sn → GL ( ¢T λ )
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Polytabloids
Linear Operator
At : ¢T λ → ¢T λ
Polytabloid
et = At[t]
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Sprecht Representations
Represent polytabloids
φ λ (σ) et = e σ t
Subspace
S λ = span {et}
Represent (Sprecht!)
ψ λ : Sn → GL ( S λ )
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Properties of At
At [s] = ± et
Im(At) = ¢et
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The Irreducible Mr. Sprecht
S λ : no “good” subspaces
(proper, nonzero, Sn-invariant)
ψ λ : irreducible
(by definition)
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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)
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Sprecht : a Complete Set
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Tetris …
Treats PTSD
Represents all symmetries