Upper bounds of Schubert polynomials

Let w be a permutation of { 1, 2 ... ,}, and let D(w) be the Rothe diagram of w. The Schubert polynomial zeta(w)(x) can be realized as the dual character of the flagged Weyl module associated with D(w). This implies the following coefficient-wise inequality: Min(w)(x) 6 Sw(x) 6 Max(w) (x), where both Min(w)(x) and Max(w)(x) are polynomials determined by D(w). Fink et al. (2018) found that zeta(w)(x) equals the lower bound Min(w)(x) if and only if w avoids twelve permutation patterns. In this paper, we show that zeta(w)(x) reaches the upper bound Max(w) (x) if and only if w avoids two permutation patterns 1432 and 1423. Similarly, for any given composition alpha is an element of Z(n) >0, one can define a lower bound Min(alpha)(x) and an upper bound Max(alpha)(x) for the key polynomial kappa(alpha)(x). Hodges and Yong (2020) established that kappa(alpha)(x) equals Min(alpha)(x) if and only if alpha avoids five composition patterns. We show that kappa(alpha)(x) equals Max(alpha)(x) if and only if ff avoids a single composition pattern (0, 2). As an application, we obtain that when alpha avoids (0; 2), the key polynomial kappa(alpha)(x) is Lorentzian, partially verifying a conjecture of Huh et al. (2019).

Automated summary

This paper explores the relationship between permutations and composition patterns, determining lower and upper bounding polynomials for different permutations and composition patterns. The results show that under specific conditions of avoiding certain permutation and composition patterns, the minimum and maximum values of key polynomials can be determined.