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

Abstract

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