Journal of Combinatorics

Volume 11 (2020)

Number 1

Fixed-point-free involutions and Schur $P$-positivity

Pages: 65 – 110



Zachary Hamaker (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)

Eric Marberg (Department of Mathematics. Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong)

Brendan Pawlowski (Department of Mathematics, University of Southern California, Los Angeles, Calif., U.S.A.)


The orbits of the symplectic group acting on the type A flag variety are indexed by the fixed-point-free involutions in a finite symmetric group. The cohomology classes of the closures of these orbits have polynomial representatives $\hat{\mathfrak{S}}^{\mathrm{FPF}}_{z}$ akin to Schubert polynomials. We show that the fixed-point-free involution Stanley symmetric functions $\hat{F}^{\mathrm{FPF}}_{z}$, which are stable limits of the polynomials $\hat{\mathfrak{S}}^{\mathrm{FPF}}_{z}$, are Schur $P$-positive. To do so, we construct an analogue of the Lascoux–Schützenberger tree, an algebraic recurrence that computes Schubert polynomials. As a byproduct of our proof, we obtain a Pfaffian formula of geometric interest for $\hat{\mathfrak{S}}^{\mathrm{FPF}}_{z}$ when $z$ is a fixed-point-free version of a Grassmannian permutation. We also classify the fixed-point-free involution Stanley symmetric functions that are single Schur $P$-functions, and show that the decomposition of $\hat{F}^{\mathrm{FPF}}_{z}$ into Schur $P$-functions is unitriangular with respect to dominance order on strict partitions. These results and proofs mirror previous work by the authors related to the orthogonal group action on the type A flag variety.

B. Pawlowski was partially supported by NSF grant 1148634.

Received 12 July 2017

Published 27 September 2019