# Skew key polynomials and a generalized Littlewood-Richardson rule

## TL;DR

This paper introduces a new partial order called the key poset on weak compositions, generalizes skew Schur functions to skew key polynomials, and provides a nonnegative Littlewood-Richardson rule for their expansion.

## Contribution

It defines the key poset and skew key polynomials, extending classical combinatorial objects and rules to a nonsymmetric setting with a new Littlewood-Richardson rule.

## Key findings

- Defined the key poset on weak compositions.
- Introduced skew key polynomials as a generalization of skew Schur functions.
- Provided a nonnegative Littlewood-Richardson rule for skew key polynomial expansion.

## Abstract

Young's lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing Young's lattice, we introduce a new partial order on weak compositions that we call the key poset. Saturated chains in this poset correspond to standard key tableaux, the combinatorial objects that generate the key polynomials, a nonsymmetric polynomial generalization of the Schur basis. Generalizing skew Schur functions, we define skew key polynomials in terms of this new poset. Using weak dual equivalence, we give a nonnegative weak composition Littlewood-Richardson rule for the key expansion of skew key polynomials, generalizing the flagged Littlewood-Richardson rule of Reiner and Shimozono.