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

    https://doi.org/10.1142/S0219498823502286Cited by:0 (Source: Crossref)

    In this paper, we introduced two classes of elements in the enveloping algebra U(gl(n)): the double Young–Capelli bitableaux [S|T] and the central Schur elements Sλ(n), that act in a remarkable way on the highest weight vectors of irreducible Schur modules.

    Any element Sλ(n) is the sum of all double Young–Capelli bitableaux [S|S], S row (strictly) increasing Young tableaux of shape λ̃. The Schur elements Sλ(n) are proved to be the preimages — with respect to the Harish-Chandra isomorphism — of the shifted Schur polynomials sλ|nΛ(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 Hk(n) are column Schur elements and the Nazarov elements Ik(n) are row Schur elements. The duality in ζ(n) follows from a combinatorial description of the eigenvalues of the Hk(n) on irreducible modules that is dual (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the Ik(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.

    Communicated by M. Gorelik

    AMSC: 17B10, 05E10, 17B35

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