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

TL;DR

This paper introduces new classes of elements in the universal enveloping algebra of gl(n), such as Capelli bitableaux and immanants, and demonstrates their properties, computational efficiency, and relation to quantum immanants.

Contribution

It develops a unified framework for determinants, permanents, and immanants in polynomial and Lie algebra contexts, introducing new basis elements and their properties.

Findings

Standard Capelli bitableaux form a basis of U(gl(n)).

Capelli immanants can be efficiently computed and generate U(gl(n)).

Okounkov quantum immanants are linear combinations of Capelli 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.