Split-Symmetry in Algebraic Combinatorics

Algebraic combinatorics has, at its core, the study of elements/bases of the ring of symmetric polynomials $\text{Sym}(n)$. Obversely, Lascoux--Schutzenberger introduced numerous asymmetric families in the polynomial ring $\text{Pol}(n)$. I will discuss an interpolation between $\text{Sym}(n)$ and $\text{Pol}(n)$ in the form of split-symmetric rings. We will see that many important families of polynomials, e.g. the key polynomials and Schubert polynomials, are split-symmetric.