Polynomials in algebraic combinatorics

A long-standing theme in algebraic combinatorics is to study bases of the rings of symmetric functions, quasisymmetric functions, and polynomials. Classically, these bases are homogeneous functions, however, the introduction of K-theoretic combinatorics has led to increased interest in finding inhomogeneous deformations of classical bases. Joint with A. Yong and N. Tokcan, we introduce the notion of saturated Newton polytope (SNP), a property of polynomials, and study its prevalence in algebraic combinatorics. We find that many, but not all, of the families that arise in other contexts of algebraic combinatorics are SNP. We introduce a family of polytopes called the Schubitopes and connect it to the Newton polytopes of the Schubert polynomials and the key polynomials. Semistandard skyline fillings are a combinatorial model that arises from specializing the combinatorics of Macdonald polynomials. We define a set-valued extension which allows us to define inhomogeneous deformations of the Demazure atoms, key polynomials, and quasisymmetric Schur functions. We prove that these deformations act in many ways like their homogeneous counterparts. We then continue the work on set-valued skyline fillings. Joint with O. Pechenik and D. Searles, we provide deformations of the quasikey polynomials and the fundamental particles. This allows us to lift the quasisymmetric Grothendieck polynomials from the ring of quasisymmetric polynomials to the ring of polynomials and give expansions between the different bases under consideration that are analogous to the homogeneous case. We end with some conjectures on the structure constants of equivariant Schubert calculus in Type B and C, including a generalization of the Horn inequalities to this setting