2021/07/21 by Andrea Brini, Brini, Andrea, Antonio Teolis +1
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2107.10205
In this paper are introduced two classes of elements in the enveloping algebra U(gl(n)): the double Young-Capelli bitableaux [ \fboxS | 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 [ \fboxS | S ], S row (strictly) increasing Young tableaux of shape \widetildeλ. The Schur elements Sλ(n) are proved to be the preimages - with respect to the Harish-Chandra isomorphism - of the shifted Schur polynomials sλ|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 doesn't involve the irreducible characters of symmetric groups. The Capelli elements Hk(n) are column Schur elements and the Nazarov-Umeda elements Ik(n) are row Schur elements. The duality in \boldsymbolζ(n) follows from a combinatorial description of the eigenvalues of the Hk(n) on irreducible modules that is \itdual (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the Ik(n). The passage n → ∞ for the algebras \boldsymbolζ(n) is obtained both as direct and inverse limit in the category of filtered algebras, via the Olshanski decomposition/projection.