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Symmetric polynomials, for example, Schur, Jack, and Macdonald polynomials, are classical objects in the study of algebra, representation theory, and combinatorics. Interpolation polynomials are certain inhomogeneous versions of the classical Jack and Macdonald polynomials. In this talk, I will first review some basics of symmetric polynomials, introduce the interpolation polynomials, and discuss our recent work. As an application, I will talk about a characterization of the containment partial order in terms of Schur positivity or Jack positivity. This result parallels the works of Cuttler--Greene--Skandera, Sra, and Khare--Tao, which characterize two other partial orders in terms of Schur positivity.

This work is joint with my advisor, Prof. Siddhartha Sahi, and available on arXiv:2403.02490.