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

Abstract

In this paper, we introduced two classes of elements in the enveloping algebra [Formula: see text]: the double Young–Capelli bitableaux [Formula: see text] and the central Schur elements [Formula: see text], that act in a remarkable way on the highest weight vectors of irreducible Schur modules. Any element [Formula: see text] is the sum of all double Young–Capelli bitableaux [Formula: see text], [Formula: see text] row (strictly) increasing Young tableaux of shape [Formula: see text]. The Schur elements [Formula: see text] are proved to be the preimages — with respect to the Harish-Chandra isomorphism — of the shifted Schur polynomials [Formula: see text]. Hence, the Schur elements are the same as the Okounkov quantum immanants, recently described by the present authors as linear combinations of Capelli immanants. This new presentation of Schur elements/quantum immanants does not involve the irreducible characters of symmetric groups. The Capelli elements [Formula: see text] are column Schur elements and the Nazarov elements [Formula: see text] are row Schur elements. The duality in [Formula: see text] follows from a combinatorial description of the eigenvalues of the [Formula: see text] on irreducible modules that is dual (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the [Formula: see text]. The passage [Formula: see text] for the algebras [Formula: see text] is obtained both as direct and inverse limit in the category of filtered algebras, via the Olshanski decomposition/projection.