Newton’s formula for 𝔤𝔩_{𝔫}

Abstract

This paper presents an explicit relation between the two sets which are well-known generators of the center of the universal enveloping algebra U ( g l n ) U(\mathfrak {gl}_{n}) of the Lie algebra g l n \mathfrak {gl}_{n} : one by Capelli (1890) and the other by Gelfand (1950). Our formula is motivated to give an exact analogy for the classical Newton’s formula connecting the elementary symmetric functions and the power sum symmetric functions. The formula itself can be deduced from a more general result on Yangians obtained by Nazarov. Our proof is elementary and has an advantage in its direct accessibility.