Superalgebraic methods in the classical theory of representations. Capelli's identity, the Koszul map and the center of the enveloping algebra U(gl(n))

In this paper we essentially deal with Capelli’s identity and with the center of the enveloping algebra of the general linear Lie algebra gl(n). By way of elementary motivation, we show that the proof of Capelli’s identity can be reduced to a straightforward computation just by using a touch of superalgebraic notions. This point of view is extended to the study of the enveloping algebra U(gl(n)). The notions of determinantal and permanental Capelli bitableaux provide two relevant classes of bases that arise from Straightening Laws. In the final section, we submit new results on the center of U(gl(n)). These results - which cannot even be expressed without appealing to the superalgebraic notation - allows a variety of classical and recent results to be almost trivially proved and be put under one roof.