Quasisymmetric divided differences

We develop a quasisymmetric analogue of the combinatorial theory of Schubert polynomials and the associated divided difference operators. Our counterparts are “forest polynomials”, and a new family of linear operators, whose theory of compositions is governed by forests and the “Thompson monoid”. Our approach extends naturally to $m$-colored quasisymmetric functions. We then give several applications of our theory to fundamental quasisymmetric functions, the study of quasisymmetric coinvariant rings and their associated harmonics, and positivity results for various expansions. In particular we resolve a conjecture of Aval-Bergeron-Li regarding quasisymmetric harmonics.

💡 Research Summary

The paper develops a comprehensive quasisymmetric analogue of the classical theory of Schubert polynomials and divided‑difference operators. In the symmetric setting, Schubert polynomials (S_w) form a (\mathbb Z)-basis of the polynomial ring (\mathrm{Pol}n) and interact with the nil‑Coxeter divided‑difference operators (\partial_i) via the simple rule (\partial_i S_w = S{ws_i}) when (i) is a descent of (w). These operators generate the nil‑Hecke algebra and descend to endomorphisms of the coinvariant algebra (\mathrm{Coinv}_n = \mathrm{Pol}_n / \mathrm{Sym}^+_n).

The authors ask the natural quasisymmetric counterpart: replace the symmetric ring (\mathrm{Sym}_n) by the quasisymmetric ring (\mathrm{QSym}_n) and study the quotient (\mathrm{QSCoinv}_n = \mathrm{Pol}_n / \mathrm{QSym}^+_n). Directly transplanting (\partial_i) fails because the Hivert quasisymmetrizing action (\sigma_i) does not respect multiplication, and the resulting operators do not compose nicely.

To overcome this, the authors introduce two new families:

1. Forest polynomials (P_F) – originally defined in