Generalized exponents of small representations. I

TL;DR

This paper introduces non-negative formulas for generalized exponents of small representations across all types, focusing on type A and connecting with the charge formula.

Contribution

It provides the first explicit, manifestly non-negative formulas for generalized exponents of small representations in all Lie types.

Findings

Structure of the argument for root systems of type A

Relationship with Lascoux-Schutzenberger charge formula

Foundation for future formulas in other types

Abstract

This is the first paper in a sequence devoted to giving manifestly non-negative formulas for generalized exponents of small representations in all types. The main part of this paper illustrates the overall structure of the argument on root systems of type A and discusses the relationship with the Lascoux-Schutzenberger charge formula.