Multiplicative structures of the immaculate basis of non-commutative symmetric functions

TL;DR

This paper develops a new basis for non-commutative symmetric functions, analogous to Schur functions, with novel combinatorial rules, surprising coefficient relations, and geometric interpretations.

Contribution

It introduces a non-commutative basis with properties similar to Schur functions, including new rules and geometric insights, advancing the understanding of non-commutative symmetric functions.

Findings

Non-commutative Littlewood-Richardson rule formulated

New relations among non-commutative coefficients discovered

Coefficients interpreted as lattice points in polytopes

Abstract

We continue our development of a new basis for the algebra of non-commutative symmetric functions. This basis is analogous to the Schur basis for the algebra of symmetric functions, and it shares many of its wonderful properties. For instance, in this article we describe non-commutative versions of the Littlewood-Richardson rule and the Murnaghan-Nakayama rule. A surprising relation develops among non-commutative Littlewood-Richardson coefficients, which has implications to the commutative case. Finally, we interpret these new coefficients geometrically as the number of integer points inside a certain polytope.