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Quantum immanants, double Young–Capelli bitableaux and Schur shifted symmetric functions.
- Source :
- Journal of Algebra & Its Applications; Nov2023, Vol. 22 Issue 11, p1-48, 48p
- Publication Year :
- 2023
-
Abstract
- 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]
Details
- Language :
- English
- ISSN :
- 02194988
- Volume :
- 22
- Issue :
- 11
- Database :
- Complementary Index
- Journal :
- Journal of Algebra & Its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 173273199
- Full Text :
- https://doi.org/10.1142/S0219498823502286