Queer dual equivalence graphs

We introduce a new paradigm for proving the Schur $P$‑positivity of a given quasi-symmetric function. Generalizing dual equivalence, we give an axiomatic definition for a family of involutions on a set of objects to be a queer dual equivalence, and we prove whenever such a family exists, the fundamental quasisymmetric generating function is Schur $P$‑positive. In contrast with shifted dual equivalence, the queer dual equivalence involutions restrict to a dual equivalence when the queer involution is omitted. We highlight the difference between these two generalizations with a new application to the product of Schur $P$‑functions.

Keywords

shifted tableaux, Schur $P$-functions, Schur $Q$-functions, dual equivalence graph

2010 Mathematics Subject Classification

Primary 05E05. Secondary 05A05, 05E10.

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The author’s work was supported in part by NSF grant DMS-1763336.

Received 6 May 2021

Accepted 28 September 2021

Published 19 August 2022