Stanley Symmetric Functions Research Articles

B n Stanley symmetric functions

We use the Kraśkiewicz insertion algorithm to show that the B n Stanley symmetric function can be expressed as a nonnegative integer combination of Schur P-functions. It turns out that the decreasing parts of the insertion tableau form a shifted tableau, and Edelman-Greene insertion can be regarded as a special case of Kraśkiewicz insertion. Using these results, we obtain simple forms for some of the B n Stanley symmetric functions. We show that the recording tableau of a reduced word a under Kraśkiewicz insertion is the evacuation of the recording tableau of the reverse of a. We also establish a connection between Kraśkiewicz insertion and promotion sequence.

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

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 $\hat{\mathfrak{S}}^{\tt{FPF}}_z$ akin to Schubert polynomials. We show that the fixed-point-free involution Stanley symmetric functions $\hat{F}^{\tt{FPF}}_z$, which are stable limits of the polynomials $\hat{\mathfrak{S}}^{\tt{FPF}}_z$, are Schur $P$-positive. To do so, we construct an analogue of the Lascoux-Sch\"utzenberger tree, an algebraic recurrence that computes Schubert polynomials. As a byproduct of our proof, we obtain a Pfaffian formula of geometric interest for $\hat{\mathfrak{S}}^{\tt{FPF}}_z$ 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 $\hat{F}^{\tt{FPF}}_z$ 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.