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

In this paper, we introduced two classes of elements in the enveloping algebra U (g l (n)) : the double Young–Capelli bitableaux  [ S | T ] and the central Schur elements  S λ (n) , that act in a remarkable way on the highest weight vectors of irreducible Schur modules. Any element S λ (n) is the sum of all double Young–Capelli bitableaux [ S | S ] , S row (strictly) increasing Young tableaux of shape λ ̃. The Schur elements S λ (n) are proved to be the preimages — with respect to the Harish-Chandra isomorphism — of the shifted Schur polynomials  s λ | n ∗ ∈ Λ ∗ (n). 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 H k (n) are column Schur elements and the Nazarov elements I k (n) are row Schur elements. The duality in ζ (n) follows from a combinatorial description of the eigenvalues of the H k (n) on irreducible modules that is dual (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the I k (n). The passage n → ∞ for the algebras ζ (n) is obtained both as direct and inverse limit in the category of filtered algebras, via the Olshanski decomposition/projection. [ABSTRACT FROM AUTHOR]