Asymmetric function theory

Abstract

The classical theory of symmetric functions has a central position in algebraic combinatorics, bridging aspects of representation theory, combinatorics, and enumerative geometry. More recently, this theory has been fruitfully extended to the larger ring of quasisymmetric functions, with corresponding applications. Here, we survey recent work extending this theory further to general asymmetric polynomials.

Coxeter combinatorics and spherical Schubert geometry

“…1 borrowing the terminology of [PS19] Although (V) does not give, per se, an algorithm to decide sphericality, we suggest (VIII) Conjecture 4.19, which asserts that checking the "staircase" key polynomial suffices. This conjecture reduces to a combinatorial question about the split symmetry of key polynomials; see Conjecture 4.20, Conjecture 4.21 and Proposition 4.22.…”

“…Algebraic combinatorics has, at its core, the study of elements/bases of the ring of symmetric polynomials Sym(n) (see, e.g., [S99, Chapter 7]). Obversely, A. Lascoux-M.-P. Sch ützenberger introduced numerous asymmetric families in the polynomial ring Pol(n); see, e.g., [L13,PS19] and the references therein. We now discuss an interpolation between Sym(n) and Pol(n):…”