  In previous work with Mikhail Khovanov and Aaron Lauda we introduced two odd
analogues of the Schur functions: one via the combinatorics of Young tableaux
(odd Kostka numbers) and one via the odd symmetrization operator. In this paper
we introduce a third analogue, the plactic Schur functions. We show they
coincide with both previously defined types of Schur function, confirming a
conjecture. Using the plactic definition, we establish an odd
Littlewood-Richardson rule. We also re-cast this rule in the language of
polytopes, via the Knutson-Tao hive model.
