RESEARCH ARTICLE

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

C. Choi^a, S. Kim^a, H. Seo<sup>b

a</sup> Department of Mathematics, Korea University, 145 Anam-ro Seongbuk-gu, Seoul 02841, South Korea
^b Department of Mathematics, University of Maryland, William E. Kirwan Hall, 4176 Campus Drive, College Park, MD 20742-4015, USA

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 $\operatorname{gr} L_n$ agrees with the direct sum decomposition of $\operatorname{gr} L_n$, 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 $\operatorname{gr} L_n$, we give some explicit constructions of weight multiplicity-free irreducible representations of $\mathfrak{gl}(n)$.

Keywords: Laurent polynomial, filtration, general linear Lie algebra, weight module.

MSC: 16S34, 16W70, 17B10, 17B45

Received: 13.12.2018
Revised: 24.02.2021

Language: English

DOI: 10.12958/adm1304