Slide multiplicity free key polynomials

TL;DR

This paper characterizes which fundamental slide polynomial refinements of key polynomials are multiplicity free, providing recursive algorithms and classifications for certain composition cases.

Contribution

It determines multiplicity free conditions for fundamental slide polynomial refinements of key polynomials and offers a recursive method to compute their terms.

Findings

Identifies multiplicity free fundamental slide polynomial refinements for strong compositions.

Provides a recursive algorithm to compute terms in these refinements.

Classifies cases with at most two nonzero parts or terms.

Abstract

Schubert polynomials are refined by the key polynomials of Lascoux-Sch\"{u}tzenberger, which in turn are refined by the fundamental slide polynomials of Assaf-Searles. In this paper we determine which fundamental slide polynomial refinements of key polynomials, indexed by strong compositions, are multiplicity free. We also give a recursive algorithm to determine all terms in the fundamental slide polynomial refinement of a key polynomial indexed by a strong composition. From here, we apply our results to begin to classify which fundamental slide polynomial refinements, indexed by weak compositions, are multiplicity free. We completely resolve the cases when the weak composition has at most two nonzero parts or the sum has at most two nonzero terms.