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Séminaire Lotharingien de Combinatoire, 80B.53 (2018), 12 pp.

# Dominic Searles

# Combinatorial Bases of Polynomials

**Abstract.**
We establish a poset structure on combinatorial bases of polynomials, defined by positive expansions. These bases include the well-studied Schubert polynomials, Demazure characters and Demazure atoms, as well as the recently-introduced slide and quasi-key bases. The product of a Schur polynomial and an element of a basis in the poset expands positively in that basis; in particular we give the first Littlewood-Richardson rule for the product of a Schur polynomial and a quasi-key polynomial, extending the rule of Haglund, Luoto, Mason and van Willigenburg for quasi-Schur polynomials. We also establish a bijection connecting the combinatorial models of semi-skyline fillings and quasi-key tableaux for these polynomials.

Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.

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