Central elements in U(gl(n)), shifted symmetric functions and the superalgebraic Capelli's method of virtual variables

TL;DR

This paper introduces a unified, representation-theoretic approach to the center of U(gl(n)) and shifted symmetric polynomials using superalgebraic methods, extending Capelli's auxiliary variable technique.

Contribution

It develops a new superalgebraic method of virtual variables that unifies the study of the center of U(gl(n)) and shifted symmetric functions.

Findings

Provides a transparent, concise framework for the theory of the center of U(gl(n))

Extends Capelli's method using superalgebraic virtual variables

Enhances understanding of shifted symmetric polynomials in representation theory

Abstract

In this work, we propose a new method for a unified study of some of the main features of the theory of the center of the enveloping algebra U(gl(n)) and of the algebra of shifted symmetric polynomials, that allows the whole theory to be developed, in a transparent and concise way, from the representation-theoretic point of view, that is entirely in the center of U(gl(n)). Our methodological innovation is the systematic use of the superalgebraic method of virtual variables for gl(n), which is, in turn, an extension of Capelli's method of "variabili ausiliarie".