Asymmetric function theory

TL;DR

This paper surveys recent advancements in extending the classical symmetric function theory to encompass asymmetric polynomials, highlighting new algebraic structures and potential applications in combinatorics and related fields.

Contribution

It introduces a comprehensive overview of how symmetric function theory has been generalized to asymmetric polynomials, expanding its scope and applicability.

Findings

Extension of symmetric functions to asymmetric polynomials

Connections to algebraic combinatorics and representation theory

Potential new applications in enumerative geometry

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.