Upper bounds of Schubert polynomials

Abstract

Let w be a permutation of {1, 2, …, n}, and let D(w) be the Rothe diagram of w. The Schubert polynomial \({\mathfrak{S}_w}\left(x \right)\) can be realized as the dual character of the flagged Weyl module associated with D(w). This implies the following coefficient-wise inequality: $${\rm{Mi}}{{\rm{n}}_w}\left(x \right) \le {\mathfrak{S}_w}\left(x \right) \le {\rm{Ma}}{{\rm{x}}_w}\left(x \right),$$where both Minw(x) and Maxw(x) are polynomials determined by D(w). Fink et al. (2018) found that \({\mathfrak{S}_w}\left(x \right)\) equals the lower bound Minw(x) if and only if w avoids twelve permutation patterns. In this paper, we show that \({\mathfrak{S}_w}\left(x \right)\) reaches the upper bound Maxw(x) if and only if w avoids two permutation patterns 1432 and 1423. Similarly, for any given composition α ∈ ℤ n≽0 , one can define a lower bound Minα(x) and an upper bound Maxα(x) for the key polynomial κα(x). Hodges and Yong (2020) established that κα(x) equals Minα(x) if and only if α avoids five composition patterns. We show that κα(x) equals Maxα(x) if and only if α avoids a single composition pattern (0, 2). As an application, we obtain that when α avoids (0, 2), the key polynomial κα(x) is Lorentzian, partially verifying a conjecture of Huh et al. (2019).