Shifted dual equivalence and Schur $P$-positivity

By considering type $\mathrm{B}$ analogs of permutations and tableaux, we extend abstract dual equivalence to type $\mathrm{B}$ in two directions. In one direction, we define involutions on shifted tableaux that give a dual equivalence, thereby giving a new combinatorial 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. As a first application, we give a new combinatorial proof that the product of Schur $P$-functions is Schur $P$-positive.

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The author’s work was supported in part by NSF grant DMS-1265728.

Received 12 August 2015

Published 22 January 2018