This thesis is comprised of two related projects involving shifted tableaux. In the first project, we introduce a shifted analog of the plactic monoid of Lascoux and Schutzenberger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman's mixed insertion. Applications of the theory of the shifted plactic monoid include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux-Schutzenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more. In the second project, joint with T. K. Petersen, we show that the set of reduced expressions for the longest element in the hyperoctahedral group exhibits the cyclic sieving phenomenon. The proofs rely heavily on the theory of the shifted plactic monoid
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