Shifted dual equivalence and Schur P-positivity

By considering type B analogs of permutations and tableaux, we extend abstract dual equivalence to type B in two directions. In one direction, we define involutions on signed permutations and shifted tableaux that give a weak dual equivalence, thereby giving another proof of the Schur positivity of Schur Q- and P-functions. In another direction, we define an abstract shifted dual equivalence parallel to dual equivalence and prove that it can be used to establish Schur P-positivity of a function expressed as a sum of shifted fundamental quasisymmetric functions.