Quantum immanants, double Young-Capelli bitableaux and Schur shifted symmetric functions

TL;DR

This paper introduces new algebraic elements called double Young-Capelli bitableaux and Schur elements in the enveloping algebra of gl(n), linking them to shifted Schur polynomials and quantum immanants, with implications for representation theory.

Contribution

It presents a novel presentation of Schur elements and quantum immanants without using symmetric group characters, and explores their duality and limits in algebraic structures.

Findings

Schur elements are sums of double Young-Capelli bitableaux.

Schur elements correspond to shifted Schur polynomials via Harish-Chandra isomorphism.

Duality of eigenvalues for Capelli and Nazarov-Umeda elements is established.

Abstract

In this paper are introduced two classes of elements in the enveloping algebra $\mathbf{U}(gl(n))$: the \emph{double Young-Capelli bitableaux} $[\ \fbox{$S \ | \ T$}\ ]$ and the \emph{central} \emph{Schur elements} $\mathbf{S}_{\lambda}(n)$, that act in a remarkable way on the highest weight vectors of irreducible Schur modules. Any element $\mathbf{S}_{\lambda}(n)$ is the sum of all double Young-Capelli bitableaux $[\ \fbox{$S \ | \ S$}\ ]$, $S$ row (strictly) increasing Young tableaux of shape $\widetilde{\lambda}$. The Schur elements $\mathbf{S}_\lambda(n)$ are proved to be the preimages - with respect to the Harish-Chandra isomorphism - of the \emph{shifted Schur polynomials} $s_{\lambda|n}^* \in \Lambda^*(n)$. Hence, the Schur elements are the same as the Okounkov \textit{quantum immanants}, recently described by the present authors as linear combinations of \emph{Capelli immanants}. This new presentation of Schur elements/quantum immanants doesn't involve the irreducible characters of symmetric groups. The Capelli elements $\mathbf{H}_k(n)$ are column Schur elements and the Nazarov-Umeda elements $\mathbf{I}_k(n)$ are row Schur elements. The duality in $\boldsymbol{\zeta}(n)$ follows from a combinatorial description of the eigenvalues of the $\mathbf{H}_k(n)$ on irreducible modules that is {\it{dual}} (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the $\mathbf{I}_k(n)$. The passage $n \rightarrow \infty$ for the algebras $\boldsymbol{\zeta}(n)$ is obtained both as direct and inverse limit in the category of filtered algebras, via the \emph{Olshanski decomposition/projection}.