Word reading is a crystal morphism

The symplectic plactic monoid, crystals, and MV cycles

“…This question was answered positively in [6] and [20] for G ∨ = SL(n, C). In the rest of this paper we do so as well for G ∨ = SP(2n, C) and γ a readable gallery.…”

“…Word Reading is a Crystal Morphism. This subsection is the 'symplectic' version of Proposition 2.5 in [20]. Since the root operators are type preserving (see 3.1), the set of words W Cn is naturally endowed with a crystal structure.…”

“…In [5] they consider one skeleton galleries, which are piecewise linear paths in X ∨ ⊗ Z R. Such galleries can be interpreted in terms of Young tableaux for types A, B and C. For G ∨ = SL(n, C), Gaussent, Littelmann and Nguyen show in [6] that for any fixed point δ ∈ Σ T γ λ , the closure π(C δ ) is in fact an MV cycle. They achieve this using combinatorics of Young tableaux such as word reading and the well known Knuth relations, and by relating them to the Chevalley relations for root subgroups which hold in the affine Grassmannian G. In [20] it is observed that word reading is a crystal morphism, and this allows one to prove that in this case, the map from all galleries to MV cycles is in fact a morphism of crystals.…”