By John J. Graham

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Observe that a reflection s ∈ R swapping b < c in c′ , is a descent of Q(θ′′ ) iff Q(θ′′ )(b) ∈ Y . On the other hand, if col ◦Q(θ′′ )(w(b)) = 1, then s is a descent of Q(θ′′ ). It follows that there are at most k = |Y \ {yk }| + 1 elements in the first column of {S2 (w(c)) | c ∈ c \ {a}} = {Q(θ′′ )(w(c)) | c ∈ c′ }. Therefore w(a) is inserted into the first column of S2 at a row not exceeding k + 1. 6, each c displaces b in a row whose index is 30 Chapter 3 no larger. As row(x˙ j ) ≥ k + 2, no element of X is displaced.

If Φ is empty, the result is trivial. Assume Φ is nonempty, let λ be the right most index in the product and let Φ′ = Φ \ {λ}. We have an exact sequence, 0 → H(Φ′ )/ rad(Φ′ ) → H(Φ)/ rad(Φ′ ) → H({λ}) → 0 43 Cellular Algebras of (H, H)-modules. Assume (by induction) that eΦ′ is the identity of the algebra H(Φ′ )/ rad(Φ′ ). If a ∈ H, then aeΦ′ = eΦ′ aeΦ′ = eΦ′ a mod rad(Φ′ ). Therefore x + rad(Φ′ ) → eΦ′ x + rad(Φ′ ) is an (H, H)-module homomorphism which splits the sequence above. The corresponding injection H({λ}) → H(Φ)/ rad(Φ′ ) takes x to (1− eΦ′ )x+ rad(Φ′ ).

Applications of our next lemma provide examples. 6. Suppose x, y ∈ Dom(λ) and col(x) < col(y). Assume x˙ = y˙ and let µ = λ \ {x, x, ˙ y, y}. ˙ If there exists z ∈ Dom(µ) such that z ≥ (row(y), col(x)) (the meet of x and y), then x ≈ y. z z˙ x y y˙ x˙ Proof. Let m′′ < m′ < m be the largest three elements of x. Construct an x′′′′ -tableau S of shape µ such that S(m′′ ) = z˙ by Res′′ (S) = (µ, S′′ ) where ˙ Observe that λ \ {x, x} ˙ S′′ is an arbitrary x′′′ ′′′ -tableau of shape µ \ {z, z}. and λ \ {y, y} ˙ are the only Young diagrams ν such that µ ⊂ ν ⊂ λ and the complements λ \ ν and ν \ µ are dominos.