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

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 $\tilde{\lambda }$. 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} $s_{\lambda |n}^{\ast }\in {\Lambda }^{\ast }(n)$. 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 $\mathbf{\zeta }(n)$ 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 $n\to \infty$ for the algebras $\mathbf{\zeta }(n)$ is obtained both as direct and inverse limit in the category of filtered algebras, via the \emph{Olshanski decomposition/projection}.

arXiv admin note: substantial text overlap with arXiv:1608.06780 TWO typos corrected at page 22