Zero-one Schubert polynomials

Abstract

We prove that if $$\sigma \in S_m$$ σ ∈ S m is a pattern of $$w \in S_n$$ w ∈ S n , then we can express the Schubert polynomial $$\mathfrak {S}_w$$ S w as a monomial times $$\mathfrak {S}_\sigma $$ S σ (in reindexed variables) plus a polynomial with nonnegative coefficients. This implies that the set of permutations whose Schubert polynomials have all their coefficients equal to either 0 or 1 is closed under pattern containment. Using Magyar’s orthodontia, we characterize this class by a list of twelve avoided patterns. We also give other equivalent conditions on $$\mathfrak {S}_w$$ S w being zero-one. In this case, the Schubert polynomial $$\mathfrak {S}_w$$ S w is equal to the integer point transform of a generalized permutahedron.

Slide Multiplicity Free Key Polynomials

“…This result joins a myriad of other multiplicity free classifications in algebraic combinatorics including multiplicity free products of Schur functions in terms of Schur functions [18], and analogously for multiplicity free Schur P -function products [3], multiplicity free skew Schur functions in terms of Schur functions [9,19], multiplicity free Schur P -functions in terms of Schur functions [17], multiplicity free Stanley symmetric functions in terms of monomials [5], multiplicity free Schubert polynomials in terms of monomials [7], and multiplicity free Schur functions in terms of fundamental quasisymmetric functions [4]. This latter result is as follows.…”

Multiplicity-free skew Schur polynomials

“…The skew Schur functions whose coefficients are all equal to 1 over an entire dominance order interval, and equal to 0 otherwise, are characterized in [2]. Other examples of multiplicity-free classifications include [3,6,15]. As the skew Schur polynomials are GL n characters of the skew Schur modules [5], this paper represents our contribution to this body of work.…”

“…4 #6 (2021) Let CS k (λ/µ) = |{r : (r, k) ∈ λ/µ}| be the number of boxes in the k-th column of λ/µ. Define (3) ρ(λ/µ) := max{CS k (λ/µ) : 1 k λ 1 } to be the maximal number of boxes in any column of λ/µ. In Example 1.1, ρ(λ/µ) = 2.…”

Determinantal Formulas for SEM Expansions of Schubert Polynomials

“…One can then ask similar questions about the class of Schubert polynomials satisfying this weaker property. (See Winkel [16] for some discussion, as well as [2,7] for some similar studies.) Question 5.2.…”