Skew Littlewood-Richardson rules from Hopf algebras

TL;DR

This paper develops new skew Littlewood-Richardson rules for various symmetric functions using Hopf algebra techniques, confirming conjectures and extending combinatorial rules to broader algebraic contexts.

Contribution

It introduces a Hopf algebra-based framework to derive skew Littlewood-Richardson rules for multiple classes of symmetric functions, including Schur P- and Q-functions.

Findings

Proves a version of the Littlewood-Richardson rule for skew Schur functions.

Establishes skew Littlewood-Richardson rules for Schur P- and Q-functions.

Derives skew Pieri rules for k-Schur functions and related structures.

Abstract

We use Hopf algebras to prove a version of the Littlewood-Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood-Richardson rules for Schur P- and Q-functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group.