# Two Bases of the Polynomial Ring: Schubert Polynomials and Key Polynomials

The polynomial ring has two distinct bases: Schubert polynomials and key polynomials. In this talk, we will first study the vector subspace spanned by Schubert polynomials in \(S_n\), and describe another basis consisting of key polynomials. Then we will explore the process of expanding a Schubert polynomial into a sum of key polynomials, demonstrating how these two bases are interrelated.

**Date published:**Wednesday, March 22, 2023