A filtration on the ring of Laurent polynomials and representations of the general linear Lie algebra

TL;DR

This paper introduces a filtration on Laurent polynomial rings that aligns with the decomposition of their associated graded rings into simple modules over gl(n), and constructs explicit weight multiplicity-free irreducible representations.

Contribution

It presents a new filtration on Laurent polynomial rings compatible with gl(n) module decomposition and constructs explicit weight multiplicity-free irreducible representations.

Findings

Filtration on Laurent polynomial ring compatible with gl(n) decomposition

Explicit constructions of weight multiplicity-free irreducible representations

Alignment of graded ring decomposition with module structure

Abstract

We first present a filtration on the ring L of Laurent polynomials such that the direct sum decomposition of its associated graded ring gr L agrees with the direct sum decomposition of gr L, as a module over the complex general linear Lie algebra gl(n), into its simple submodules. Next, generalizing the simple modules occurring in gr L, we give some explicit constructions of weight multiplicity-free irreducible representations of gl(n).