The "Young" and "reverse" dichotomy of polynomials

TL;DR

This paper introduces Young analogues of polynomial bases using a flip-and-reversal involution, providing combinatorial formulas, representation-theoretic interpretations, and exploring their relationships with reverse bases.

Contribution

It develops Young analogues of key and Schubert polynomials, offering new combinatorial, algebraic, and representation-theoretic insights into their structure.

Findings

Young key polynomials as generating functions for left keys

Combinatorial formulas for Young analogues of Schubert polynomials

Representation-theoretic interpretation of Young key polynomials

Abstract

A "flip-and-reversal" involution arising in the study of quasisymmetric Schur functions provides a passage between what we term "Young" and "reverse" variants of bases of polynomials or quasisymmetric functions. Building on this perspective, which has found recent application in the study of $q$-analogues of combinatorial Hopf algebras and generalizations of dual immaculate functions, we develop and explore Young analogues of well-known bases for polynomials. We prove several combinatorial formulas for the Young analogue of the key polynomials, show that they form the generating functions for left keys, and provide a representation-theoretic interpretation of Young key polynomials as traces on certain modules. We also give combinatorial formulas for the Young analogues of Schubert polynomials, including their crystal graph structure. We moreover determine the intersections of (reverse) bases and their Young counterparts, further clarifying their relationships to one another.