Pattern-avoidance characterization of elementary-symmetric factorizations of Schubert polynomials

Establish that a Schubert polynomial associated with a permutation factors as a product of elementary symmetric polynomials if and only if the permutation avoids the patterns 1432, 1423, 4132, and 3142.

Background

The paper studies when a Schubert polynomial $\mathfrak{S}_w$ can be expressed in the form $\prod_{t=1}^r e_{b_t}(x_1,\ldots,x_{a_t})$, where the factors are elementary symmetric polynomials, potentially in different sets of initial variables. The proposed characterization is in terms of avoidance of the four permutation patterns 1432, 1423, 4132, and 3142.

The paper proves one direction: avoiding these four patterns implies the desired factorization, using conditions on the Lehmer code and a diagonal separation property of pipe dreams. The converse remains the unresolved part addressed by the conjecture; the paper provides progress toward it by proving that permutations whose Lehmer codes contain a rectangular block cannot yield such a factorization.