Fixed-point-free involutions and Schur P-positivity
| Autor: | Zachary Hamaker, Eric Marberg, Brendan Pawlowski |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Symplectic group
Mathematics::Combinatorics 010102 general mathematics Schubert polynomial Stanley symmetric function 0102 computer and information sciences Type (model theory) 16. Peace & justice 01 natural sciences Combinatorics Tree (descriptive set theory) 010201 computation theory & mathematics Symmetric group FOS: Mathematics Dominance order Mathematics - Combinatorics Orthogonal group Combinatorics (math.CO) Representation Theory (math.RT) 0101 mathematics Mathematics::Representation Theory Mathematics - Representation Theory Mathematics |
| Popis: | 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. Comment: 34 pages, 1 figure. This article was formerly the second half of arXiv:1701.02824; v2: revised introduction, expanded proofs and examples, added index of notation, minor corrections |
| Databáze: | OpenAIRE |
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