Young-Capelli bitableaux, Capelli immanants in U(gl(n)) and the Okounkov quantum immanants

Abstract

We propose a new approach to a unified study of determinants, permanents, immanants, (determinantal) bitableaux and symmetrized bitableaux in the polynomial algebra $C[M_{n, n}]$ as well as of their Lie analogues in the enveloping algebra $U(gl(n))$. This leads to new relevant classes of elements in $U(gl(n))$: Capelli bitableaux, right Young-Capelli bitableaux and Capelli immanants. The set of standard Capelli bitableaux and the set of standard right Young-Capelli bitableaux are bases of $U(gl(n))$, whose action on the Gordan-Capelli basis of $C[M_{n, n}]$ have remarkable properties. Capelli immanants can be efficiently computed and provide a system of generators of $U(gl(n))$. The Okounkov quantum immanants are proved to be simple linear combinations of Capelli immanants. Several examples are provided throughout the paper.