By Ben Brubaker

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9 4 2 . 10 We have indicated the circling and boxing of entries. >=< 4 9 10 4 12 . 3 We observe the following points. • The first row of Γt is decreasing and the bottom row is increasing; these are reversed for ∆t , just as the boxing and circling conventions are reversed. • The involution does not preserve strictness. If t is strict, no element can be both boxed and circled, but if t is not strict, an entry in the bottom row is both boxed and circled, and the same is true for ∆t : if t is not strict, then an entry in the bottom row of ∆t is both boxed and circled.

This means that we may generate Γ(Tv ) and ∆(Tv ) from the same element v and the same word ΩΓ , but applying fi successively to generate the entries bi of Γ(Tv ) and applying the ei (in the same order) to generate the entries li of ∆(Tv ). The boxing and circling rules are defined analogously for the ∆(Tv ) as for Γ(Tv ). Now box-circle duality means that there a bijection between the bi and the li in which bi is circled if 33 and only if the corresponding li is boxed. Note that this bijection changes the order: b1 b2 b3 b4 b5 b6 · · · l1 l3 l2 l6 l5 l4 · · · This is a rather striking property of the crystal graph.

But in this summation, because τ and σ are both 48 CHAPTER 5. OUTLINE OF THE PROOF required to be compatible with the same accordion a, only between σ and τ in the τ that appear in this definition. Thus in the above example ΛΓ (f, ∗) = GΓ (f ∗) − GΓ (f ∗∗ ) ’s are changed to ∗ − GΓ (f∗ ∗ ) + GΓ (f∗∗∗ ) . Now let η be a nodal signature, and we may take ci = ∞ if ηi = ❡. Let a ∈ CPη (c0 , · · · , cd ). 21) is strict. Statement E. Assume that n|s. We have ΛΓ (a, σ) = a∈CPη (c0 ,··· ,cd ) Λ∆ (a , σ).

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