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607ee45 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 | import Mathlib
import OAI.RepresentationTheory.Saxl.Main
import OAI.RepresentationTheory.UniversalSquare.Specht.RepeatedColumns
namespace OAI
/-! General Detection. -/
section
noncomputable section
open scoped TensorProduct
namespace Saxl
lemma wordMap_pair_transpose {n d e : ℕ} (L : Fin d → Fin e → ℂ)
(x : WordSpace n d) (y : WordSpace n e) :
dotProduct (wordMap L x) y = dotProduct x (wordMap (fun b a => L a b) y) := by
classical
change (∑ b, (∑ a, x a * ∏ i, L (a i) (b i)) * y b) =
∑ a, x a * ∑ b, y b * ∏ i, L (a i) (b i)
simp_rw [Finset.sum_mul, Finset.mul_sum]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro a ha
apply Finset.sum_congr rfl
intro b hb
ring
theorem specht_to_cyclic_of_pair {n d : ℕ} {μ : YoungDiagram}
(t : Tableau n μ) (L : Fin (μ.colLen 0) → Fin d → ℂ)
(u : WordSpace n d)
(h : dotProduct (wordMap L (polytabloid t)) u ≠ 0) :
∃ F : Representation.IntertwiningMap (spechtRep t)
(cyclic (wordRep n d) u).toRepresentation, F ≠ 0 := by
let pr := subrepProject (spechtSub t)
(fun _ hx => cyclic_star _ (polytabloid_real t) hx)
let F : Representation.IntertwiningMap
(cyclic (wordRep n d) u).toRepresentation (spechtRep t) :=
pr.comp ((wordMap (fun b a => L a b)).comp (subrepInclusion _))
apply specht_map_back_of_nonzero _ t F
intro hz
have hp := subrepProject_pair (spechtSub t)
(fun _ hx => cyclic_star _ (polytabloid_real t) hx)
(wordMap (fun b a => L a b) u) ⟨polytabloid t, mem_cyclic _ _⟩
have he : pr (wordMap (fun b a => L a b) u) = 0 :=
congrArg (fun H : Representation.IntertwiningMap
(cyclic (wordRep n d) u).toRepresentation (spechtRep t) =>
H ⟨u, mem_cyclic _ _⟩) hz
rw [he] at hp
change dotProduct (0 : WordSpace n (μ.colLen 0)) (polytabloid t) =
dotProduct (wordMap (fun b a => L a b) u) (polytabloid t) at hp
rw [zero_dotProduct] at hp
rw [wordMap_pair_transpose, dotProduct_comm]
at h
exact h hp.symm
end Saxl
end
end
end OAI
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