Crystal graphs for general linear Lie superalgebras and quasi-symmetric functions

TL;DR

This paper introduces a novel representation-theoretic interpretation of quasi-symmetric functions via crystal bases for the Lie superalgebra gl_{n|n}, linking algebraic structures to combinatorial functions.

Contribution

It provides the first realization of super quasi-symmetric functions as characters of irreducible crystals for Lie superalgebras, expanding the understanding of their algebraic and combinatorial properties.

Findings

Super quasi-symmetric functions are realized as characters of irreducible crystals.

A new algebraic characterization of super quasi-symmetric functions is provided.

The approach connects Lie superalgebra representations with combinatorial functions.

Abstract

We give a new representation theoretic interpretation of the ring of quasi-symmetric functions. This is obtained by showing that the super analogue of the Gessel's fundamental quasi-symmetric function can be realized as the character of an irreducible crystal for the Lie superalgebra $\frak{gl}_{n|n}$ associated to its non-standard Borel subalgebra with a maximal number of odd isotropic simple roots. We also present an algebraic characterization of these super quasi-symmetric functions.