The Sperner property for $132$-avoiding intervals in the weak order
| Autor: | Katherine Tung, Christian Gaetz |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Mathematics::Combinatorics
General Mathematics 010102 general mathematics Schubert polynomial 01 natural sciences Bruhat order Action (physics) Combinatorics Symmetric group FOS: Mathematics Interval (graph theory) Order (group theory) Mathematics - Combinatorics Combinatorics (math.CO) 0101 mathematics Quotient Word (group theory) Mathematics |
| DOI: | 10.48550/arxiv.2006.16359 |
| Popis: | A well-known result of Stanley from 1980 implies that the weak order on a maximal parabolic quotient of the symmetric group $S_n$ has the Sperner property; this same property was recently established for the weak order on all of $S_n$ by Gaetz and Gao, resolving a long-open problem. In this paper we interpolate between these results by showing that the weak order on any parabolic quotient of $S_n$ (and more generally on any $132$-avoiding interval) has the Sperner property. This result is proven by exhibiting an action of $\mathfrak{sl}_2$ respecting the weak order on these intervals. As a corollary we obtain a new formula for principal specializations of Schubert polynomials. Our formula can be seen as a strong Bruhat order analogue of Macdonald's reduced word formula. This proof technique and formula generalize work of Hamaker, Pechenik, Speyer, and Weigandt and Gaetz and Gao. Comment: 18 pages. v2: fixed typos |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|