Gelfand-Kirillov dimensions of the $Z^2$-graded oscillator representations of sl(n)

TL;DR

This paper derives an exact formula for the Gelfand-Kirillov dimensions of certain infinite-dimensional irreducible modules of sl(n), extending classical harmonic polynomial theorems with new explicit calculations.

Contribution

It provides a precise formula for Gelfand-Kirillov dimensions of Z^2-graded oscillator modules of sl(n), including identification of subfamilies with minimal dimensions.

Findings

Exact Gelfand-Kirillov dimension formula derived

Identification of three subfamilies with minimal dimensions

Modules include unbounded weight multiplicities and pointed modules

Abstract

We find an exact formula of Gelfand-Kirillov dimensions for the infinite-dimensional explicit irreducible sl(n, F)-modules that appeared in the Z^2-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. Three infinite subfamilies of these modules have the minimal Gelfand-Kirillov dimension. They contain weight modules with unbounded weight multiplicities and completely pointed modules.